Hong Kong Advanced Level Examination
Pure Mathematics Past Paper ( 1990 – 2006 ) Sorted by Topic
Mathematical Induction
……………………………………………………
P.2
Polynomials
……………………………………………………
P.10
Binomial Theorem
……………………………………………………
P.28
Inequalities
……………………………………………………
P.33
Complex Number
……………………………………………………
P.55
Determinants and Matrices
……………………………………………………
P.75
Functions
…………………………………………………… P.115
Sequences
…………………………………………………… P.124
Limit
…………………………………………………… P.145
Differentiation
…………………………………………………… P.151
Integration
…………………………………………………… P.191
Coordinate Geometry
…………………………………………………… P.237
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
Contents: Classification
Chronological Order
Type 1: Verify a Formula 1. 2. 3. 4. 5. 6.
92I05 90I07 93I02 02I01 06I01 97I02 Type 2: Divisibility / Rationality 7. 98I05 8. 04I06 9. 05I01 Type 3: Derive a Formula 10. 96I06 11. 01I02 Type 4: Strong Induction 12. 94I05 13. 95I06 Type 5: Binomial Expansion 14. 91I11 15. 97I12 Type 6: Others 16. 03I04 17. 94I13
P. 2
90I07
Q. 2
91I11 92I05 93I02 94I05 94I13 95I06 96I06 97I02 97I12 98I05 01I02 02I01 03I04 04I06 05I01 06I01
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
14 1 3 12 17 13 10 6 15 7 11 4 16 8 9 5
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
1.
Type 1: Verify a Formula (92I05) Consider the sequence
un
in which
u1 0 , un1 2n un for n = 1, 2, … . Using mathematical induction or otherwise, show that 2un 2n 1 (1)n for n = 1, 2, … .
Hence find lim
n
un . n (4 marks)
2.
(90I07) A sequence
a0 , a1, a2 ,
of real numbers is defined by
a0 0 , a1 1 and an an1 an2 for all n 2, 3, . Show that for all non-negative integers n , an
1 ( n n ) , 5
where , are roots of x2 x 1 0 with 0 , 0 . a Also prove that lim n 1 . n a n (7 marks) 3.
(93I02) Let u1 1 , u2 3 and un un2 un1 for n 3 . Using mathematical induction, or otherwise, prove that un n n for n 1 ,
where and are the roots of x2 x 1 = 0 . (5 marks) 4.
(02I01) A sequence {an} is defined by a1 = 1 , a2 = 3 and
an2 2an1 an for n 1, 2, 3,... . Prove by mathematical induction that an
(1 2)n (1 2) n for n 1, 2, 3,... . 2
(5 marks) P. 3
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
5.
(06I01) A sequence {an} is defined by a1 = 1 , a2 = 3 and an + 2 = 3an + 1 + 2an for all n = 1 , 2 , 3 , … . Using mathematical induction,
1 prove that an 17
3 17 n 3 17 n for any positive 2 2
integer n . (6 marks) 6.
(97I02) Let {an} be a sequence of real numbers, where
a0 1 , a1 6 , a2 45 and 1 1 an an1 an 2 an3 0 for n = 0, 1, 2, … . 3 27 Using mathematical induction, or otherwise, show that an 3n (n2 1) for n = 0, 1, 2, … .
(4 marks) Type 2: Divisibility / Rationality 7.
(98I05) Let , be the roots of x2 14x + 36 = 0 . Show that n + n
is divisible by 2n for n = 1, 2, 3, … . (5 marks)
8.
(04I06) Let R . For each n N , define xn sin n cosn . (a) Find a function f ( ) , which is independent of n , such that
xn1 x1 xn2 f ( ) xn . Also express f ( ) in terms of x1 . (b) Suppose that x1 is a rational number. Using mathematical induction, prove that xn is a rational number for every n . (7 marks)
P. 4
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
9.
(05I01) For each positive integer n , define Sn (1 5)n (1 5)n . Prove that (a) Sn2 2Sn1 4Sn , (b) S n is divisible by 2n . (6 marks)
Type 3: Derive a Formula 10. (96I06) A sequence
xn
is defined by x0 1 , x1 2 and xn
xn 1 xn 2 2
for n 2 . (a) Write down the values of x2 x1 , x3 x2 and x4 x3 . (b) For n = 1, 2, 3, … , guess an expression for xn xn1 in terms of n and prove it. Hence find lim xn . n
(7 marks) 11. (01I02) (a) Show that
n
r
2
r 1
(b) A sequence
an
1 n(n 1)(2n 1) . 6
is defined as follows:
a1 6 , ak 1 ak 3k 2 9k 6 for k = 1, 2, 3, … .
Find an in terms of n . (5 marks)
P. 5
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
Type 4: Strong Induction 12. (94I05) Let {an} be a sequence of positive numbers such that 1 an a1 a2 an 2 for n = 1, 2, 3, … .
2
Prove by induction that an = 2n 1 for n = 1, 2, 3, … . (5 marks) 13. (95I06) Let
an
be a sequence of non-negative integers such that n
n ak 2 n 1 (1)n for n = 1, 2, 3, … . k 1
Prove that an 1 for n 1 . (4 marks)
P. 6
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
Type 5: Binomial Expansion 14. (91I11) (a) For n = 1, 2, … , prove that there exist unique positive integers pn and qn such that
( 3 2)2n pn qn 6 ……… (*) and
( 3 2)2n pn qn 6 .
(b) Hence deduce that 2 pn 1 ( 3 2)2n 2 pn . [Hint: Use the fact that 0 3 2 1 .] (6 marks) (c) For n = 1, 2, … , show that the following integers are positive multiples of 10 : (i)
25n 2n ,
(ii)
34 n 1 ,
(iii)
2 p2 n (23n1 )(3n ) where p2n is given by (*) .
(5 marks) (d) By using (a) and (b) or otherwise, find the unit of digit when ( 3 2)100 is expressed in the decimal form.
(4 marks)
P. 7
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
15. (97I12) (a) Show that for any positive integer n , there exist unique positive integers an and bn such that
(1 3)2n an bn 3 and that (i) an 2 3bn 2 22 n , (ii) an and bn are both divisible by 2n . (8 marks) (b) For an and bn as determined in (a), show that
(1 3)2n an bn 3 . (2 marks) an 3 . n b n
(c) Using (b), or otherwise, prove that lim
(3 marks) (d) Using (a) and (b), or otherwise, prove that for any positive integer n , the smallest integer greater than (1 3)2 n is divisible by 2n + 1 . (2 marks) Type 6: Others 16. (03I04) Let {xn} be a sequence of positive real numbers, where x1 = 2 and xn1 xn 2 xn 1 for all n = 1, 2, 3, … . n
Define Sn i 1
1 for all n = 1, 2, 3, … xi
(a) Using mathematical induction, prove that for any positive integer n , (i)
xi > n ,
(ii)
Sn 1
1 xn 1 1
.
(b) Using (a), or otherwise, prove that lim Sn exists. n
(7 marks)
P. 8
HKAL Pure Mathematics Past Paper Topic: Mathematical Induction
17. (94I13) Let Z + be the set of all positive integers and m, n Z + .
A(m, n) (1 xm )(1 xm1 )(1 xmn1 ) ,
Let
B(n) (1 x)(1 x 2 )(1 x n ) . (a) Show that A(m 1, n 1) A(m, n 1) is divisible by
(1 xn1 )A(m 1, n) . (2 marks) (b) Suppose P(m, n) denote the statement. “ A(m, n) is divisible by B(n) .” (i)
Show that P(1, n) and P(m, 1) are true.
(ii)
Using (a), or otherwise, show that if P(m, n 1) and P(m 1, n) are true, then P(m 1, n 1) is also true.
(iii)
Let k be a fixed positive integer such that P(m, k ) is true for all m Z . Show by induction that P(m, k + 1) is true for all m Z (10 marks)
(c) Using (b), or otherwise, show that P(m, n) is true for all
m, n Z . (3 marks)
P. 9
HKAL Pure Mathematics Past Paper Topic: Polynomials
Contents: 1.
Classification Type 1: Coefficients and Roots 90I03
2. 3. 4. 5. 6. 7. 8. 9.
94I03 95I03 96I07 00I07 03I06 98I12 06I09 94I12 Type 2: Degree 10. 06I03 11. 91I04 12. 01I06 Type 3: H.C.F. 13. 98I07 14. 00I05 15. 02I05 16. 93I12 Type 4: Remainder Theorem / Division Algorithm 17. 99I04 18. 04I04 19. 05I04 20. 95I10 Type 5: Partial Fractions 21. 00I12 Type 6: Repeated Roots 22. 04I09 23. 98I11 Type 7: Solving Polynomial Equations 24. 97I04 25. 26. 27. 28. 29. 30.
96I11 03I11 05I10 02I11 99I11 01I13
P. 10
Chronological Order 90I03 Q. 1 91I04 Q. 11 93I12 Q. 16 94I03 Q. 2 94I12 Q. 9 95I03 Q. 3 95I10 Q. 20 96I07 Q. 4 96I11 Q. 25 97I04 Q. 24 98I07 Q. 13 98I11 Q. 23 98I12 Q. 7 99I04 Q. 17 99I11 Q. 29 00I05 Q. 14 00I07 Q. 5 00I12 Q. 21 01I06 Q. 12 01I13 Q. 30 02I05 02I11 03I06 03I11 04I04 04I09 05I10 06I03 06I09
Q. Q. Q. Q. Q. Q. Q. Q. Q.
15 28 6 26 18 22 19 27 10
HKAL Pure Mathematics Past Paper Topic: Polynomials
1.
Type 1: Coefficients and Roots (90I03) (a) If , and are the roots of x3 Ax2 Bx C 0 , express
2 2 2 and 2 2 2 2 2 2 in terms of A , B and C . (b) Find a cubic equation whose roots are the squares of the roots of x 3 3x 1 0 .
(5 marks) 2.
(94I03) (a) If , and are the roots of x3 + px + q = 0 , find a cubic equation whose roots are 2 , 2 and 2 .
x 2 3 (b) Solve the equation 2 x 3 0 . 2 3 x
3.
Hence, or otherwise, solve the equation x3 38x2 361x 900 0 . (6 marks) (95I03) (a) If a1 , a2 , a3 , a4 , p, q, , are real numbers such that
x4 a1 x3 a2 x2 a3 x a4 ( x 2 px q)2 ( x )2 for all x ,
show that
2 a12 2q a2 4 1 (a1q a3 ) . 2 2 q 2 a4
(b) Find the possible real values of p, q, , such that
x4 4 x3 12 x2 24 x 9 ( x px q)2 ( x )2 for all x . (c) Solve x4 4 x3 12 x2 24 x 9 0 . (7 marks)
P. 11
HKAL Pure Mathematics Past Paper Topic: Polynomials
4.
(96I07) Let the roots of x3 px2 qx r 0 be , and . (a) Show that p q 2 . (b) If the roots of x3 Px 2 Qx R 0 are 2 , 2 and
2 , express P , Q and R in terms of p , q and r . (7 marks) 5.
(00I07) Suppose the equation x3 + px2 + qx + 1 = 0 has three real roots. (a) If the roots of the equation can be written as
a , a and ar , r
show that p q . (b) If p = q , show that 1 is a root of the equation and the three roots of the equation can form a geometric sequence. (7 marks) 6.
(03I06) (a) Suppose the cubic equation x3 px2 qx r 0 , where p , q and r are real numbers, has three real roots. Using relations between coefficients and roots, or otherwise, prove that the three roots form an arithmetic sequence if and only if
p is a root of the equation. 3
(b) Find the two values of p such that the equation
x3 px2 21x p 0 has three real roots that form an arithmetic sequence. (8 marks)
P. 12
HKAL Pure Mathematics Past Paper Topic: Polynomials
7.
(98I12) (a) Let , and be positive numbers. Suppose
( x )( x ) x2 2 px q and ( x )( x )( x ) x3 3bx 2 3cx d for all x . (i)
Show that p2 q .
(ii)
By expressing b , c and d in terms of , p and q , or otherwise, show that b2 c > 0 and c2 bd . Hence, or otherwise, show that b c 3 d . (10 marks)
(b) Let A , B and C be the angles of a triangle. Show that
tan
A B B C C A tan tan tan tan tan 1 . 2 2 2 2 2 2
Using (a), or otherwise, show that
tan
A B C 3 A B C tan tan 3 and tan tan tan . 2 2 2 9 2 2 2 (5 marks)
8.
(06I09) (a) Let b , c , d R and , and be the roots of the equation x3 + bx2 + cx + d = 0 . For every positive integer k , define Sk k k k . (i)
Using relations between coefficients and roots, express S1 , S2 and S3 in terms of b , c and d .
(ii)
Prove that Sk + 3 + bSk + 2 + cSk + 1 + dSk = 0 for any positive integer k .
(iii)
Suppose d = bc . Using the results of (a)(i) and (a)(ii) , prove that S2n + 1 + bS2n = (1)n 2bcn and S2n + bS2n 1 = (1)n 2cn for any positive integer n . (11 marks)
(b) Find three numbers such that their sum is 3 , the sum of their squares is 3203 and the sum of their cubes is 9603 . (4 marks)
P. 13
HKAL Pure Mathematics Past Paper Topic: Polynomials
9.
(94I12) Let p( x) x4 a1 x3 a2 x2 a3 x a4 , where a1 , a2 , a3 , a4 R . Suppose z1 cos 1 i sin 1 and z2 cos 2 i sin 2 are two roots of p( x) 0 , where 0 1 2 .
(a) Show that (i)
p( x) ( x2 2 x cos 1 1)( x 2 2 x cos 2 1) ,
(ii)
x cos 1 x cos 2 p' ( x) 2p( x) 2 2 . x 2 x cos 1 1 x 2 x cos 2 1
(5 marks) (b) Suppose p(w) = 0 , by considering p(x) w(x) , show that
p( x) x3 ( w a1 ) x 2 (w2 a1w a2 ) x (w3 a1w2 a2 w a3 ) . xw (3 marks) (c) Let sn z1n z1n z2n z2n , using (a)(ii) and (b) , show that
p' ( x) 4 x3 (s1 4a1 ) x2 (s2 a1s1 4a2 ) x (s3 s2a1 s1a2 4a3 ) . [Hint:
2( x cos r ) 1 1 , r = 1, 2 . ] x 2 x cos r 1 x zr x zr 2
Hence show that sn a1sn1 an1s1 nan 0 for n 1, 2, 3, 4 . (7 marks)
P. 14
HKAL Pure Mathematics Past Paper Topic: Polynomials
Type 2: Degree 10. (06I03) Let p(x) be a polynomial of degree 4 with real coefficients satisfying p(0) 0 , p(1) =
1 2 3 4 , p(2) = , p(3) = and p(4) = . 2 3 4 5
(a) Let q(x) = (x + 1)p(x) x . (i)
Evaluate q(0) , q(1) , q(2) , q(3) and q(4) .
(ii)
Express q(x) as a product of linear polynomials.
(b) Evaluate p(5) . (6 marks) 11. (91I04) Let a1 , a2 ,, an be n distinct non-zero real numbers, where n 2 . (a) Define P( x) a1
( x a2 ) ( x an ) (a1 a2 ) (a1 an )
ai
( x a1 ) ( x ai 1 )( x ai 1 ) ( x an ) (ai a1 ) (ai ai 1 )(ai ai 1 ) (ai an )
an
( x a1 ) ( x an 1 ) . (an a1 ) (an an 1 )
(i)
Evaluate P(ai ) for i = 1, 2, … , n .
(ii)
Show that the equation P(x) x = 0 has n distinct roots.
(iii)
Deduce that P( x) x 0 for all x R .
(b) Prove that 1 1 (a1 a2 ) (a1 an ) (ai a1 ) (ai ai 1 )(ai ai 1 ) (ai an )
1 0 . (an a1 ) (an an 1 ) (5 marks)
12. (01I06) Let f (x) = ax2 + bx + c where a, b, c are real numbers and a 0 . Show that if f[f ( x)] [f ( x)]2 for all x , then f (x) = x2 . (4 marks)
P. 15
HKAL Pure Mathematics Past Paper Topic: Polynomials
Type 3: H.C.F 13. (98I07) It is given that f (x) = 2x4 + x3 + 10x2 + 2x + 15 and g(x) = x3 + 2x 3 . Let d(x) be the H.C.F. of f (x) and g(x) . (a) Using Euclidean Algorithm, or otherwise, find d(x) . (b) Find polynomials u(x) and v(x) of degree 1 such that u( x)f ( x) v( x)g( x) d( x) for all x . (6 marks) 14. (00I05) Let f (x) = 2x4 x3 + 3x2 2x + 1 and g(x) = x2 x + 1 . (a) Show that f (x) and g(x) have no non-constant common factors. (b) Find a polynomial p(x) of the lowest degree such that f (x) + p(x) is divisible by g(x) . (5 marks) 15. (02I05) (a) Let f (x) and g(x) be polynomials. Prove that a non-zero polynomial u(x) is a common factor of f (x) and g(x) if and only if u(x) is a common factor of f (x) g(x) and g(x) . (b) Let f (x) = x4 3x3 6 x2 5x 3 and g( x) x 4 4 x3 8x 2 7 x 4 . Using (a) or otherwise, find the H.C.F. of f (x) and g(x) . (7 marks)
P. 16
HKAL Pure Mathematics Past Paper Topic: Polynomials
16. (93I12) Let be the set of all polynomials with real coefficients. Let f , g \{0} and A = {mf + ng : m , n } . Suppose r A\{0} has the property that deg r deg p for all p A\{0} . (a) Show that r divides every polynomial in A . Deduce that r is a G.C.D. of f and g ( i.e. r divides both f and g , and if h divides both f and g then h divides r ). (6 marks) (b) Let B = {hr : h } . Show that A = B . (4 marks) (c) If deg r = 0 , i.e. r is a non-zero constant, show that there exist m0 , n0 such that m0f + n0g = 1 , and also A = . (5 marks) Type 4: Remainder Theorem / Division Algorithm 17. (99I04) Let P(x) be a polynomial. When P(x) is divided by x2 4x 21 , the remainder is 11x 10 . When P(x) is divided by x2 6x 7 . the remainder is 9x + c , where c is a constant. (a) Find a common factor of x2 4x 21 and x2 6x 7 . Hence find c . (b) Find the remainder when P(x) is divided by x2 + 4x + 3 . (6 marks)
P. 17
HKAL Pure Mathematics Past Paper Topic: Polynomials
18. (04I04) Let f ( x) x3 px 2 qx r , where p , q and r are non-zero real numbers. (a) If f (x) is divisible by x2 + q , find r in terms of p and q . (b) Suppose that f (x) is divisible by both x a and x + a , where a is a non-zero real number. (i)
Factorize f (x) as a product of three linear polynomials with real coefficients.
(ii)
If f (x) and f (x + a) have a non-constant common factor, find p in terms of a . (7 marks)
19. (05I04) Let f (x) be a polynomial of degree 4 with real coefficients. When f (x) is divided by x 2 , the remainder is 4 . When f (x) is divided by x + 3 , the remainder is 6 . Let r(x) be the remainder when f (x) is divided by (x 2)(x + 3) . (a) Find r(x) . (b) Let g(x) = f (x) r(x) . It is known that g(x) is divisible by x2 + 1 and g(1) = 16 . Find g(x) . (7 marks)
P. 18
HKAL Pure Mathematics Past Paper Topic: Polynomials
20. (95I10) Let , and be real and distinct and
( x )( x )( x ) x3 px 2 qx r . (a) Show that (i)
1 1 1 3x 2 2 px q 3 ; x x x x px 2 qx r
(ii)
3 2 2 p q ( )( ) . (4 marks)
(b) Let f (x) be a real polynomial. Suppose Ax2 + Bx + C is the remainder when (3x2 2 px q)f ( x) is divided by x3 px2 qx r . (i)
f ( ) f ( ) f ( ) Ax 2 Bx C Prove that . x x x x3 px 2 qx r
(ii)
Express A , B and C in terms of , , , f ( ) , f ( ) and f ( ) . (11 marks)
P. 19
HKAL Pure Mathematics Past Paper Topic: Polynomials
Type 5: Partial Fractions 21. (00I12) (a) Resolve
x3 x 2 3x 2 into partial fractions. x 2 ( x 1)2 (3 marks)
(b) Let P(x) = m(x 1)(x 2)(x 3)(x 4) where
m, 1 , 2 , 3 , R and m 0 . Prove that (i)
n
i 1
(ii)
1
x
i
P' ( x ) , and P( x)
1 [P' ( x)]2 P( x)P'' ( x) . 2 [P( x)]2 i 1 ( x i ) 4
(3 marks) (c) Let f (x) = ax4 bx2 + a where ab > 0 and b2 > 4a2 . (i)
Show that the four roots of f (x) = 0 are real and none of them is equal to 0 or 1 .
(ii)
Denote the roots of f (x) = 0 by β1 , β2 , β3 and β4 . Find
i 3 i 2 3 i 2 in terms of a and b . i 2 ( i 1)2 i 1 4
(9 marks) Type 6: Repeated Roots 22. (04I09) (a) Let f (x) be a polynomial with real coefficients. Prove that a real number r is a repeated root of f (x) = 0 if and only if f (r ) f' (r ) 0 . (5 marks) (b) Let g( x) x3 ax 2 bx c , where a , b and c are real numbers. If a 2 3b , prove that all the roots of g(x) = 0 are distinct. (6 marks) (c) Let k be a real constant. If the equation 12 x3 8x2 x k 0 has a positive repeated root, find all the roots of the equation. (4 marks)
P. 20
HKAL Pure Mathematics Past Paper Topic: Polynomials
23. (98I11) Consider the equation x3 3px + 2q = 0
……… (*) ,
where p , q are real numbers. (a) (i)
If (*) has a repeated root, show that p3 = q2 .
(ii)
If q
p3 , show that
p is a repeated root of (*) .
(iii)
If q p3 , show that (*) has a repeated root. (8 marks)
(b) Consider the equation 2x3 + 3x2 + x + c = 0 where c is a real number.
……… (**) ,
(i)
Transform (**) into the form y3 3py + 2q = 0 by using the substitution x = y h for some constant h .
(ii)
Find c > 0 such that (**) has a repeated root. Solve (**) for this value of c . (7 marks)
Type 7: Solving Polynomial Equations 24. (97I04) Suppose r r is a root of the cubic equation x3 ax b 0 , where a , b , r are rational numbers and r is not a rational number. (a) Show that r 3 3r 2 ar b 0 and 3r 2 r a 0 . (b) Using (a), or otherwise, show that (i)
r r is also a root of the equation, and
(ii)
r
4 8a 9b if a . 3 2(3a 4) (7 marks)
P. 21
HKAL Pure Mathematics Past Paper Topic: Polynomials
25. (96I11) Suppose the equation x4 3x2 k 0 ……… (*) has two roots , such that + = 2 .
(a) Show that . (5 marks) (b) Show that 2 , 2 are two distinct roots of the equation
y2 3y k 0 . Hence find the value of k . (5 marks) (c) Solve (*) and express the roots in the form
a b where a ,
b are rationals. Hence find the values of and . (5 marks)
P. 22
HKAL Pure Mathematics Past Paper Topic: Polynomials
26. (03I11) (a) Consider the equation …………. (*) , x4 ax2 bx c where a , b and c are real numbers. (i)
Suppose b = 0 . Solve (*) .
(ii)
Suppose b 0 . (1)
Prove that (*) can be written as
( x2 t )2 (a 2t ) x2 bx (c t 2 ) , where t is any real number. (2)
Prove that there exists a real number t0 such the equation
(a 2t0 ) x2 bx (c t02 ) 0 has a repeated root. Hence, deduce that (*) can be written as
( x2 t0 )2 (a 2t0 )( x )2 for some real number . (9 marks) (b) Consider the equation …………. (**) . x4 6 x2 12 x 8 Find a real value of t such that the equation
(6 2t ) x2 12 x (8 t 2 ) 0 has a repeated root. Hence solve (**) . (6 marks)
P. 23
HKAL Pure Mathematics Past Paper Topic: Polynomials
27. (05I10) (a) Let f (x) = x4 + 2ax2 + 4bx + c , where a , b , c R with b 0 . It is known that f ( x) ( x2 2tx r )( x 2 2tx s) , where r , s , t R . (i)
Prove that t 0 .
(ii)
Express r and s in terms of a , b and t .
(iii)
Prove that 4t 6 4at 4 (a 2 c)t 2 b2 0 . (6 marks)
(b) Consider the equation
y 4 4 y3 2 y 2 52 y 9 0 (i)
Find a constant h such that when y = x + h , (*) can be written as x4 8x2 64 x 48 0
(ii)
………(*) .
………(**) .
Using the results of (a), solve (**) in (b)(i). Hence write down all the roots of (*) . (9 marks)
P. 24
HKAL Pure Mathematics Past Paper Topic: Polynomials
28. (02I11) (a) Let f (x) = x3 3px + 1 , where p R . (i)
Show that the equation f (x) = 0 has at least one real root.
(ii)
Using differentiation or otherwise, show that if p 0 , then the equation f (x) = 0 has one and only one real root.
(iii)
If p > 0 , find the range of values of p for each of the following cases: (1) (2)
the equation f (x) = 0 has exactly one real root, the equation f (x) = 0 has exactly two distinct real roots, (3) the equation f (x) = 0 has three distinct real roots. (9 marks) (b) Let g(x) = x4 + 4x + a , where a R . (i)
Prove that the equation g(x) = 0 has at most two real roots.
(ii)
Prove that the equation g(x) = 0 has two distinct real roots if and only if a < 3 . (6 marks)
P. 25
HKAL Pure Mathematics Past Paper Topic: Polynomials
29. (99I11) Let be a complex cube root of 1 . (a) Prove the identities (i) a2 + b2 + c2 ab bc ca = (a + b + c 2)(a + b 2 + c ) , (ii) a3 + b3 + c3 3abc = (a + b + c)(a + b + c 2)(a + b 2 + c) . (4 marks) (b) Consider the equation z3 9z + 12 = 0
……… (*) .
(i)
Find real numbers p and q such that p3 + q3 = 12 and pq = 3 .
(ii)
Using (a) or otherwise, find the roots of (*) in terms of . (5 marks)
(c) Consider the equation
y3 3 y 2 12 y 10 5 14 0
……… (**) .
Using the substitution y = z h with a suitable constant h , rewrite (**) as z 3 sz t 0 where s , t are constants. Hence solve (**) . (6 marks)
P. 26
HKAL Pure Mathematics Past Paper Topic: Polynomials
30. (01I13) (a) Let P(x) = x4 + ax3 + bx2 + cx + d where a, b, c, d R . (i)
Show that if is a complex root of P(x) = 0 , then is also a root of P(x) = 0 .
(ii)
For any C , show that ( x )( x ) is a quadratic polynomial in x with real coefficients. Hence show that P(x) can be factorized as a product of two quadratic polynomials with real coefficients. (6 marks)
(b) Let f (x) = x4 + 8x3 +23x2 + 26x + 7 and g(x) = f (x + k) where k R and the coefficient of x3 in g(x) is zero. (i)
Find k and the coefficients of g(x) .
(ii)
Suppose g(x) = (x2 + px + q)(x2 + rx + s) where p, q, r , s R . By comparing coefficients or otherwise, show that p6 2 p 4 5 p 2 4 0 . Hence find p , q , r and s .
(iii)
Find all roots of f (x) = 0 . (9 marks)
P. 27
HKAL Pure Mathematics Past Paper Topic: Binomial Theorem
Contents: Classification
Chronological Order
Type 1: Properties of nCr
90I04
Q. 3
1.
03I02
92I04
2.
95I02
94I07
3.
90I04
95I02
4.
94I07
96I02
5.
00I03
99I02
6.
99I02
00I03
7.
06I02
01I05
Type 2: Inequalities
03I02
8.
01I05
05I02
9.
05I02
06I02
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
Type 3: Others 10.
92I04
11.
96I02
P. 28
10 4 2 11 6 5 8 1 9 7
HKAL Pure Mathematics Past Paper Topic: Binomial Theorem
Type 1: Properties of nCr (03I02) n For any positive integer n , let Ck be the coefficient of xk in the
expansion of (1 x)
n
. Evaluate
n n n n (a) C1 C2 C3 Cn ,
n n n n (b) C1 2C2 3C3 nCn ,
n 2 n 2 n 2 n (c) C1 2 C2 3 C3 n Cn .
(6 marks) 1.
(95I02) Let n be an integer and n > 1 . By considering the binomial expansion of (1 x)
n
, or otherwise,
n n n n n 1 (a) show that C1 2C2 3C3 nCn 2 n ;
(b) evaluate
1 2 3 (1) n1 n . (n 1)! 2!(n 2)! 3!(n 3)! n! (5 marks)
2.
(90I04) Let k and n be non-negative integers. Prove that
(a) Ckn (b)
n 1
(1) C k
k 0
(c)
k 1 n 1 Ck 1 , where 0 k n ; n 1 n 1 k
0 ;
(1) k n 1 Ck . n 1 k 0 k 1 n
P. 29
HKAL Pure Mathematics Past Paper Topic: Binomial Theorem
3.
(94I07) (a) Let m and n be positive integers. Using the identity
(1 x)n (1 x)n1 (1 x)n m
(1 x)n m1 (1 x)n , x
n n 1 nm n m1 where x 0 , show that Cn Cn Cn Cn1 .
(b) Using (a), or otherwise, show that m 4
r (r 1)(r 2)(r 3) 24 C
m 5 5
r 5
Hence evaluate
1 .
k
r (r 1)(r 2)(r 3)
for k 4 .
r 0
(7 marks) 4.
(00I03) Let n be a positive integer. (a) Expand
(1 x) n 1 in ascending powers of x . x
(b) Using (a) or otherwise, show that
C2n 2C3n 3C4n (n 1)Cnn (n 2)2n1 1 . (5 marks) 5.
(99I02) n For any positive integer n , let Ck be the coefficient of xk in the
expansion of (1 x)
n
.
n n n 1 (a) Show that Ck Ck 1 Ck 1 .
(b) By induction on m or otherwise, show that
Cnn Cnn1 Cnn2 Cnnm Cnn1m1 for any m 0 . (5 marks)
P. 30
HKAL Pure Mathematics Past Paper Topic: Binomial Theorem
6.
(06I02) n Let Ck be the coefficient of xk in the expansion of (1 + x)n .
(a) Using the identity (1 x2)n = (1 + x)n(1 x)n , prove that the coefficient of xn in the expansion of (1 x2)n is
n
(1) (C k
k 0
) .
n 2 k
(b) Evaluate (i)
2005
(1) (C k
k 0
(ii)
2005 2 k
)
2006
(1) (C k
k 0
2006 2 k
)
, . (7 marks)
Type 2: Inequalities 7.
(01I05) Let the kth term in the binomial expansion of (1 + x)2n in ascending 2 n k 1 powers of x be denoted by Tk , i.e. Tk Ck 1 x .
(a) If x
1 , find the range of values of k such that Tk 1 Tk . 3
(b) Find the greatest term in the expansion if x 8.
1 and n = 15 . 3
(05I02) For any two positive integers k and n , let Tr be the rth term in the kn kn r 1 expansion of (1 x) in ascending powers of x , i.e. Tr Cr 1 x .
(a) Suppose x
2 . Find, in terms of k and n , the range of values k
of r such that Tr 1 Tr .
(b) Suppose x
2 . Using the result of (a), find the greatest term in 3
the expansion of (1 x)
51
. (6 marks)
P. 31
HKAL Pure Mathematics Past Paper Topic: Binomial Theorem
Type 3: Others 9.
(92I04) By considering (1 i)
2n
, or otherwise, evaluate
n
(1) C r
r 0
n 1
(1) C r
r 0
2n 2 r 1
2n 2r
and
,where n is a positive integer. (5 marks)
10.
(96I02) (a) Let k and n be positive integers. If k > 1 , show that when (1+ k)n is divided by k , the remainder is 1 . (b) If today is Tuesday, what day of the week is 896 days after? (4 marks)
P. 32
HKAL Pure Mathematics Past Paper Topic: Inequalities
Contents: Classification
Chronological Order
Type 1: Elementary Methods
90I06
Q. 13
1.
91I06
90I12
2.
00I02
90II01
3.
04I05
91I06
4.
02I07
91I08
Type 2: Differentiation
92I07
5.
90II01
93I01
6.
97II02
93I07
7.
01I03
94I14
8.
98I06
95I13
9.
03II02
96I04
10.
00II02
96I13
11.
96I04
97I05
12.
90I12
97I09
Type 3: Absolute Values
97I13
13.
90I06
97II02
14.
93I07
98I06
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
12 5 1 26 17 22 14 33 28 11 29 16 18 32 6 8
15.
03I01
99I12
Type 4: A.M. G.M.
00I02
16.
97I05
00I11
17.
92I07
00II02
18.
97I09
01I03
19.
01I14
01I14
20.
03I10
02I07
21.
05I11
02I10
Type 5: Cauchy-Schwarz’s
03I01
Inequality
03I10
93I01
03II02
23.
06I06
04I05
24.
99I12
04I10
25.
02I10
05I11
26.
91I08
06I06
Type 6: Method of Difference
06I10
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
24 2 27 10 7 19 4 25 15 20 9 3 31 21 23 30
22.
27.
00I11 Type 7: Others
28.
95I13
P. 33
HKAL Pure Mathematics Past Paper Topic: Inequalities
29.
96I13
30.
06I10
31.
04I10
32.
97I13
33.
94II14
P. 34
HKAL Pure Mathematics Past Paper Topic: Inequalities
Type 1: Elementary Methods 1.
(91I06) (a) Let a , b and c be real numbers. (i)
Show that a2 b2 c2 ab bc ca .
(ii)
Hence deduce that if a b c 0 then
a3 b3 c3 3abc . (b) Let | x | ln 2 . (i)
Show that 1
1
1
(e x ) 3 (2 e x ) 3 (e x e x 1) 3 0 . (ii)
Using (a) or otherwise, show that
e x (2 e x )(e x e x 1) 1 . (7 marks) 2.
(00I02) (a) Le p and q be positive numbers. Using the fact that ln x is increasing on (0, ) , show that (p q)(ln p ln q) 0 . (b) Let a , b and c be positive numbers. Using (a) or otherwise, show that a ln a b ln b c ln c
abc ln a ln b ln c . 3 (6 marks)
3.
(04I05) Let n be a positive integer. (a) Let a > 0 . (i)
If k is a positive integer, prove that a + ak 1 + ak + 1 .
(ii)
Prove that (1 + a)n 2n 1(1 + an) .
(b) Let x and y be positive real numbers. Using (a)(ii), or otherwise,
xn y n x y prove that . 2 2 n
(7 marks)
P. 35
HKAL Pure Mathematics Past Paper Topic: Inequalities
4.
(02I07) (a) Let m and k be positive integers and k m . Prove that
m(m 1) (m k 1) (m 1)m (m k 2) for k > 1 . mk (m 1)k
Prove that the above inequality does not hold when k = 1 . (b) Let m be a positive integer. Using (a) or otherwise, prove that (1
1 m 1 m1 ) (1 ) . m m 1 (6 marks)
Type 2: Differentiation 5.
(90II01) By differentiating the function
ln x , or otherwise, prove that if x
e a b , then ab ba . (5 marks) 6.
(97II02) By considering the function f ( x) xe
x
, or otherwise, show that if
1 a b , then aeb bea . (4 marks) 7.
(01I03) (a) Let 0 < < 1 . Show that x (1 ) x for all x > 0 .
(b) Let a , b , p and q be positive numbers with 1 p
1 q
that a b
1 1 1 . Prove p q
a b . p q (5 marks)
P. 36
HKAL Pure Mathematics Past Paper Topic: Inequalities
8.
(98I06) Suppose 0 < p < 1 . (a) Let f (x) = x p px + p 1 for x > 0 . Find the absolute maximum value of f (x) . (b) Show that for all positive numbers a and b ,
a pb1 p pa (1 p)b . (6 marks) 9.
(03II02) 1
(a) Let f ( x) x x for all x 1 . Find the greatest value of f (x) . (b) Using (a) or otherwise, find a positive integer m , such that 1 m
m n
1 n
for all positive integers n . (7 marks)
10.
(00II02) Show that for x > 0 , x 1 + ln x . Find the necessary and sufficient condition for the equality to hold. (5 marks)
11.
(96I04) (a) Show that f ( x)
x is an increasing function (1, ) . 1 x
(b) Using (a), or otherwise, show that
rs 1 r s
r 1 r
s 1 s
for any
real numbers r and s . (5 marks)
P. 37
HKAL Pure Mathematics Past Paper Topic: Inequalities
12.
(90I12) (a) Let p > 1 and f ( x) x px for all x > 0 . p
(i)
Find the absolute minimum of f (x) on the interval (0, ) .
(ii)
Deduce that x 1 p( x 1) for all x > 0 . p
(4 marks) (b) (i)
Let , , and be positive numbers such that
1
1
1 and 1 .
By taking x = and respectively, prove that, for
p 1 ,
p1 p p 1 p 1 , where the equality holds if and only if 1 . (ii)
Deduce that, if a , b , c and d are positive and p > 1 , then
ab a
p 1
a b c b p
p 1
d p (c d ) p . (4 marks)
(c) Suppose
ai i 1, 2,
and
bi i 1, 2,
are two sequences of positive
numbers and p > 1 . 1
1
n p n p By considering a a j p and b j p , j 1 j 1 1
1
1
n p n p n p p prove that ai p bi p ai bi , i 1 i 1 i 1 where the equality holds if and only if
a a1 a2 a n . b1 b2 bn b (7 marks)
P. 38
HKAL Pure Mathematics Past Paper Topic: Inequalities
Type 3: Absolute Values 13.
(90I06) Solve the inequality x 1 x 2 2 . (5 marks)
14.
(93I07) Find all ( x, y) in R2 satisfying the following two conditions:
2 x 1 y 1 y x 3 . (6 marks) 15.
(03I01) (a) Solve the inequality | x | 6 3 , where x is a real number. (b) Using the result of (a), or otherwise, solve the inequality
|1 2 y | 6 3 , where y is a real number. (6 marks) Type 4: A.M. G.M 16.
(97I05) Let a , b and c be positive numbers. 3 (a) Using A.M. G.M. , show that (1 a)(1 b)(1 c) (1 3 abc ) .
Under what conditions on a , b and c will the equality hold?
a 0 0 (b) Let P be a 3 3 real matrix such that Q PQ 0 b 0 for 0 0 c 1
some 3 3 real matrix Q . 1 3
1 3
Using (a), show that [det( I P)] 1 [det P]
. (7 marks)
P. 39
HKAL Pure Mathematics Past Paper Topic: Inequalities
17.
(92I07)
Crn 1 (a) Prove that r , where n , r are positive integers and n r . n r! (b) If a1 , a2 , , an are positive real numbers and s a1 a2 an , using “A.M. G.M. ” and (a), or otherwise, prove that
(1 a1 )(1 a2 ) (1 an ) 1 s (c) Let cn
n
1
(1 2 k 1
sequence
cn
k
s 2 s3 sn . 2! 3! n!
) . Using (b) or otherwise, show that the
converges. (8 marks)
18.
(97I09) Let x1 , x2 , y1 , y2 , z1 and z2 be positive numbers such that
x1 y1 z12 0 and x2 y2 z22 0 . (a) Let D1 = x1 y1 z12 and D2 = x2 y2 z22 . Using A.M. G.M. , show that (i)
y2 y D1 1 D2 2 D1D2 , y1 y2
(ii)
y2 y D1 1 D2 x1 y2 x2 y1 2 z1 z2 , y1 y2
(iii)
( x1 x2 )( y1 y2 ) ( z1 z2 )2 4 D1D2 . (9 marks)
(b) Show that
8 1 1 , and if 2 2 ( x1 x2 )( y1 y2 ) ( z1 z2 ) x1 y1 z1 x2 y2 z2 2
the equality holds, then x1 = x2 , y1 = y2 and z1 = z2 . (6 marks)
P. 40
HKAL Pure Mathematics Past Paper Topic: Inequalities
19.
(01I14) (a) If a , b are two real numbers such that a 1 b , show that
a b ab 1 and the equality holds if and only if a = 1 and b = 1 . (3 marks) (b) Show by induction that if x1 , x2 , ... , xn are n (n 2) positive real numbers such that x1 x2 xn 1 , then x1 x2 xn n and the equality holds if and only if x1 x2 xn 1 . (6 marks) (c) Let a1 , a2 , ... , an be n (n 2) positive real numbers. Using (b) or otherwise, show that
a1 a2 an n a1a2 an and the equality n
holds if and only if a1 a2 an . (3 marks) (d) For u 0 and n = 2, 3, 4, … , using the identity
(1 u)n 1 u[(1 u)n1 (1 u)n2 1] or otherwise, show that (1 u )n 1 nu (1 u )
n 1 2
and the equality holds if and only if u = 0 . (3 marks)
P. 41
HKAL Pure Mathematics Past Paper Topic: Inequalities
20.
(03I10) (a) Let a and b be non-negative real numbers. Prove that
(a b)n a n na n1b
for all n = 2, 3, 4, … .
Write down a necessary and sufficient condition for the equality to hold. (3 marks) (b) Let {a1 , a2 , a3 ,...} be a sequence of positive real numbers satisfying
a1 a2 a3 . For any positive integer n , define
An
1 a1 a2 a3 an and Gn (a1a2 a3 an ) n . n
(i)
Prove that Ak 1 Ak for all k = 1, 2, 3, … .
(ii)
k 1 k Using (a), prove that Ak 1 Ak ak 1 for all k = 1, 2, 3, … .
Hence prove that An Gn and
An Gn if and only if a1 a2 a3 an for all n = 1, 2, 3, … . (8 marks) (c) Let n be a positive integer. Using (b), prove that n
n 2 n 1 n 1 . n 1 n 1 Hence deduce that 1 n 1
n 1
n
1 1 . n (4 marks)
P. 42
HKAL Pure Mathematics Past Paper Topic: Inequalities
21.
(05I11) (a) For any positive integer n , prove that
t n 1 n(t 1) for all t > 0 . (3 marks) (b) (i)
Let a , b and c be positive real numbers. 3
abc in (a), prove that ab abc 3 2 a b abc ( ab ) . 3 3 2
By putting n = 3 and t
(ii)
Let y1 , y2 ,..., yk 1 be positive real numbers, where k is a positive integer. Using (a), prove that
yk 1 (k 1)Gk 1 k Gk , where Gk (iii)
k
y1 y2 yk and Gk 1 k 1 y1 y2 yk 1 .
Using mathematical induction and (b)(ii), prove that
x1 x2 xn n x1 x2 xn n for any n positive real numbers x1 , x2 ,..., xn . (8 marks) (c) Let n be a positive integer. Using (b)(iii), prove that
nn (1)(3)(5)(2n 1) . Hence prove that (n n) (2n)! 2
n
(4 marks)
P. 43
HKAL Pure Mathematics Past Paper Topic: Inequalities
Type 5: Cauchy-Schwarz’s Inequality 22.
(93I01) Prove the following Schwarz’s inequality: 2
n n 2 n 2 a b i i ai bi , i 1 i 1 i 1 where ai , bi R and n N . Hence, or otherwise, prove that
1 n 1 n 2 ai ai . n i 1 n i 1 (5 marks)
23.
(06I06) Let a , b and c be positive real number such that a2 + b2 + c2 = 3 . (a) Using Cauchy-Schwarz’s inequality, prove that (i)
a+b+c3 ,
(ii)
a3 + b3 + c3 3 .
(b) For every n = 2 , 3 , 4 … , let P(n) be the statement an + bn + cn 3 . Prove that for any integer k 2 , (i)
if P(k) is true, then P(2k) is true;
(ii)
if P(k) is true, then P(2k 1) is true. (7 marks)
P. 44
HKAL Pure Mathematics Past Paper Topic: Inequalities
24.
(99I12) (a) Let ai , bi be real numbers where i = 1, 2, … , n . By considering the function f (t )
n
(a t b ) i 1
i
2
i
, or otherwise, prove Schwarz’s
2
n n n inequality ai bi ai 2 bi 2 . i 1 i 1 i 1 (4 marks) (b) Let x1 , x2 ,, xn be positive real numbers. Show that 2
n k 1 n k 2 n k xi xi xi i 1 i 1 i 1 for any non-negative integer k . (3 marks) (c) Let x1 , x2 ,, xn be positive real numbers such that Prove by induction on p that
n
n
i 1
i 1
n
x i 1
i
1 .
xi p n xi p1 for any
non-negative integer p . (5 marks) (d) Let y1 , y2 ,, yn be positive real numbers. Show that n n n p p 1 yi yi n yi i 1 i 1 i 1
for any non-negative integer p . (3 marks)
P. 45
HKAL Pure Mathematics Past Paper Topic: Inequalities
25.
(02I10) (a) Let a1 , a2 ,..., an be real numbers and b1 , b2 ,...., bn be non-zero real numbers. By considering inequality
n i 1
i1 aibi n
holds if and only if
(ai x bi )2 , or otherwise, prove Schwarz’s
2
n
a2
i 1 i
b , and that the equality n
2
i 1 i
a a1 a2 n . b1 b2 bn (6 marks)
(b) (i)
n xi Prove that i 1 n
2
n
x2
i 1 i
, where x1 , x2 ,..., xn are real
n
numbers. (ii)
Prove that
i1 i xi n
2
n
i 1
i
n i 1
i xi 2
, where
x1 , x2 ,..., xn are real numbers and 1 , 2 ,..., n are positive numbers. Find a necessary and sufficient condition for the equality to hold. (iii)
Using (b)(ii) or otherwise, prove that 2
yn yn 2 y12 y2 2 y1 y2 t t 2 t n t t 2 t n , where y1 , y2 ,..., yn are real numbers, not all zero, and t 2 . (9 marks)
P. 46
HKAL Pure Mathematics Past Paper Topic: Inequalities
26.
(91I08) (a) Let ak , bk ( k = 1, 2, … , n ) be non-zero real numbers. (i)
Prove the Schwarz’s inequality 2
n 2 n 2 n ak bk ak bk . k 1 k 1 k 1 (ii)
If p
bk q for k = 1, 2, … , n , prove that ak
pqak 2 ( p q)ak bk bk 2 0 for k = 1, 2, … , n . Deduce that ( p q)
n
k 1
(iii)
n
a b b k k
k 1
k
n
2
pq ak 2 . k 1
If
0 m ak M
and
0 m bk M for k = 1, 2, … , n ,
prove, by using (ii) or otherwise, that 2
n 2 n 2 1 M m 2 n ak bk ( ) ak bk . k 1 k 1 4 m M k 1 (10 marks) (b) Using (a) or otherwise, show that
1 1 n 1 169 1 n (n )2 (1 k )2 (1 k 1 )2 (n )2 . 9 3 k 1 3 3 k 1 144 (5 marks)
P. 47
HKAL Pure Mathematics Past Paper Topic: Inequalities
Type 6: Method of Difference 27.
(00I11) (a) By considering the derivative of f (x) = (1 + x)α 1 αx , show that (1 x) 1 x for α > 1 , x 1 and x 0 .
(4 marks) (b) Let k and m be positive integers. Show that
1 1 1 k
(i)
m 1 m
m 1 1 1 1 m k k
m 1 m
1 ,
m 1 1 m 1 m 1 m mm1 m m m m m k ( k 1) k ( k 1) k . m 1 m 1
(ii)
(6 marks) (c) Using (b) or otherwise, show that 1
1
1
1
3
2 12 2 2 3 2 n 2 2 1 2 1 . 3 3 3 n n2 1
Hence or otherwise, find lim
1
1
1
12 2 2 3 2 n 2
n
n
3 2
. (5 marks)
P. 48
HKAL Pure Mathematics Past Paper Topic: Inequalities
Type 7: Others 28.
(95I13) Let a and b be positive numbers. (a) Prove that
a abb abba where, if the equality holds, then a = b . (4 marks) (b) Using (a), or otherwise, prove that
ab 2
a b
abba
where, if the equality holds, then a = b . (3 marks) (c) Show that x x (1 x)1 x
1 for 0 < x < 1 2
where, if the equality holds, then x
ab Deduce that a b 2
1 . 2
a b
a b
where, if the equality holds, then a = b . (8 marks)
P. 49
HKAL Pure Mathematics Past Paper Topic: Inequalities
29.
(96I13)
a1m a2 m a a2 (a) Prove that 1 2 2 m
where a1 , a2 are positive and m is a positive integer. (4 marks)
a1m a2 m a3m a4 m a1 a2 a3 a4 (b) Prove that 4 4 m
where a1 , a2 , a3 , a4 are positive and m is a positive integer. (3 marks) (c) For n = 1, 2, 3, … , let P(n) be the statement
a1m a2 m an m a1 a2 an n n m
where a1 , a2 , … , an are positive and m is a positive integer. (i)
Prove that for h = 0, 1, 2, … , if P(2h) is true, then P(2h + 1) is true.
(ii)
Prove that for k = 1, 2, 3, … . if P(k + 1) is true, then P(k) is true.
(iii)
Hence prove that P(n) is true for all positive integers n . (8 marks)
P. 50
HKAL Pure Mathematics Past Paper Topic: Inequalities
30.
(06I10) (a) By differentiating f (x) = x ln x x , prove that x ln x x + 1 0 for all x 0 . (4 marks) (b) Let a be a positive real number. Define g(x) =
ax 1 for all x > x
0 . Prove that g is increasing. (3 marks) (c) Let p and q be real numbers such that p > q > 0 . (i)
Suppose that a1 , a2 ,…, an are positive real numbers satisfying
n
a k 1
k
q
=n .
Using (b), prove that
n
a k 1
(ii)
k
p
n .
Suppose that b1 , b2 ,…, bn are positive real numbers. 1
1
1 n p 1 n q Using (c)(i), prove that bk p bk q . n k 1 n k 1 p
q
1 n 1p 1 n 1q Hence prove that bk bk . n k 1 n k 1 (8 marks)
P. 51
HKAL Pure Mathematics Past Paper Topic: Inequalities
31.
(04I10) Let n be a positive integer. (a) Suppose 0 < p < 1 . (i)
By considering the function f (x) = x p px on (0, ) , or otherwise, prove that x px 1 p for all x > 0 . p
(ii)
p 1 p
Using (a)(i), or otherwise, prove that a b
pa (1 p)b
for all a , b > 0 . (iii)
Let a1 , a2 , a3 ,..., an and b1 , b2 , b3 ,..., bn be positive real numbers. Using (a)(ii), or otherwise, prove that p
n
a i 1
i
p
1 p i
b
1 p
n n ai bi i 1 i 1
. (9 marks)
(b) Suppose 0 < s < 2 . Let x1 , x2 , x3 ,..., xn and y1 , y2 , y3 ,..., yn be positive real numbers. Prove that 1
1
(i)
n 2 n 2 xi yi xi s yi 2 s xi 2 s yi s , i 1 i 1 i 1
(ii)
n s 2 s n 2 s s n 2 n 2 xi yi xi yi xi yi . i 1 i 1 i 1 i 1
n
(6 marks)
P. 52
HKAL Pure Mathematics Past Paper Topic: Inequalities
32.
(97I13)
1 n
n Let (1 )
n
T r 0
, where Tn,r is the (r + 1)-th term in the binomial
n,r
1 n
expansion of (1 ) n in ascending powers of
1 . n
(a) Show that Tn,0 = Tn,1 = 1 and
Tn,r
1 1 2 r 1 (1 )(1 ) (1 ) for r 2 . r! n n n (2 marks)
(b) For any fixed r , show that the sequences Tr ,r , Tr 1,r , Tr 2,r , … is bounded above and monotonic increasing. Find lim Tn ,r in terms of r . n
(4 marks) (c) Show that (i)
Tn ,r 1 Tn ,r
r for r 1 ,
1
(ii)
Tn,k
(iii)
Tn,r 1 (r 1) Tn,k for r 1 .
r
k r 1
Tn,r 1 for n k r 1 , n
k r
(7 marks) (d) Show that
n
T k r
n,k
1 for r 2 . (r 1)(r 1)! (2 marks)
P. 53
HKAL Pure Mathematics Past Paper Topic: Inequalities
33.
(94II14) (a) f (x) is a continuously differentiable and strictly increasing function on [0, c] such that f (0) = 0 . Let b [ 0, f (c) ] . Define g(t ) tb (i)
t 0
f ( x) dx , t [0, c] .
Determine the interval on which g(t) is strictly increasing and the interval on which g(t) is strictly decreasing. Hence 1
show that g(t ) g(f (b)) for t [0, c] . (ii)
Using the substitution y = f (x) and integration by parts, show that
(iii)
b 0
f 1 ( y) dy g(f 1 (b)) .
If a [0, c] , prove the inequality
a 0
b
f ( x) dx f 1 ( x) dx ab . 0
(10 marks) (b) If a, b, p, q are positive numbers and
1 1 1 , prove that p q
1 p 1 q a b ab . p q (5 marks)
P. 54
HKAL Pure Mathematics Past Paper Topic: Complex Number
Contents: Classification
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
21.
Chronological Order
Type 1: Algebraic Problems
90I05
Q.
9
Set 1: Properties 91I05
90I10
Q.
18
91I05
Q.
1
93I04 94I06 95I07 96I03 97I01 92I08 06I11 Set 2: Identities \ 90I05 02I06 98I13 04I12 Set 3: nth of Unity 05I06
91I09 92I08 92I11 93I04 93I11 93I13 94I06 94I11 95I07 95I11 96I03 96I12 97I01 98I04
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
19 7 30 2 22 17 3 20 4 27 5 28 6 23
98I13 99I05 99I13 00I04 00I13 01I08 01I11 02I02 02I06
Q. Q. Q. Q. Q. Q. Q. Q.
11 21 29 24 14 25 16 26
Q.
10
03I12
Q.
15
04I12
Q.
12
05I06 06I11
Q. Q.
13 8
00I13 03I12 01I11 93I13 Set4: Mapping 90I10 91I09 94I11 Type 2: Geometric Problems Set 1: Triangles 99I05
22. 93I11 Set 2: Straight Lines / 23. Circles 98I04 24. 00I04 25. 01I08 26. 02I02 27. 95I11
P. 55
HKAL Pure Mathematics Past Paper Topic: Complex Number
28. 96I12 29. 99I13 Set 3: Others 30. 92I11
P. 56
HKAL Pure Mathematics Past Paper Topic: Complex Number
1.
Type 1: Algebraic Problems Set 1: Properties (91I05) Let u , v be non-zero complex numbers. (a) Show that
uv uv 0 if and only if
u ik for some k R . v
(b) If uv uv 0 , what is the relationship between arg u and arg v ? (6 marks) 2.
(93I04) (a) If 1 z 2 z , find Re z . (b) Find all z C such that 1 2 and | z | z z i ( z z ) 2 1 z 2 z .
(5 marks) 3.
(94I06) Let Arg z denote the principal value of the argument of the complex number z ( Arg z ) . (a) If z 0 and z z 0 , show that Arg z (b) If z1 , z2 0 and z1 z2 z1 z2 , show that hence find all possible values of Arg
2
.
z1 z1 0 and z2 z2
z1 . z2 (5 marks)
4.
(95I07) Let a C and a 0 .
z 1 (a) Show that if z z a , then Re . a 2 (b) If z z a a , express z in terms of a . (6 marks)
P. 57
HKAL Pure Mathematics Past Paper Topic: Complex Number
5.
(96I03) Consider the equation z 3 az 2 az 1 0 ……… (*) where a is real.
(a) Find a real root of (*) . (b) Find the range of values of a such that (*) has non-real roots.
6.
(c) Show that all the non-real roots of (*) lies on the unit circle in the complex plane. (5 marks) (97I01) Let z1 , z2 , … , zn be arbitrary complex numbers. (a) Prove that z1 z1 z2 z2 z1 z2 z1 z2 . (b) Using (a), or otherwise, show that z1 z2 zn Re( z1 z2 z2 z3 zn1 zn zn z1 ) . 2
2
2
(4 marks) 7.
(92I08) Let u , v C . (a) Show that |u| + |v| |u + v| . (3 marks) (b) Suppose uv R . Prove that (i) there exist real numbers and , not both zero, such that u v 0 . (ii)
u v u v u v
if uv 0 , if uv 0 . (6 marks)
(c) Suppose uv R . Given z C , show that there exist unique , R such that z u v . (6 marks)
P. 58
HKAL Pure Mathematics Past Paper Topic: Complex Number
8.
(06I11) Let 0 < < . (a) Solve the equation z2 = cos + i sin . (2 marks) (b) Let u1 and u2 be the roots of the equation (u + 1)2 = cos
+ i sin , where Im(u1) < 0 . (i)
Find u1 and u2 .
(ii)
Prove that
u2 i tan . u1 4 n
u Hence, find all the integers n for which 2 is a real u1 number.
Prove that u112 = 212cos2 (cos 3 + i sin 3) . 4 Hence, find all the values of for which u112 u212 is
(iii)
a real number. (13 marks) 9.
Set 2: Identities \ (90I05) (a) Show that cos5 16cos5 20cos3 5cos . (b) Using (a), or otherwise, solve 16cos4 20cos2 5 0 for values of between 0 and 2 . Hence find the value of
cos 2
10
cos 2
3 . 10 (7 marks)
P. 59
HKAL Pure Mathematics Past Paper Topic: Complex Number
10. (02I06) For k = 1, 2, 3 , let zk cos k i sin k be complex numbers, where 1 2 3 2 . (a) Evaluate z1 z2 z3 .
1 1 1 1 (b) Prove that cos k ( zk ) and cos 2 k ( zk 2 2 ) . 2 zk 2 zk Hence or otherwise, prove that
cos 21 cos 22 cos 23 4cos 1 cos 2 cos 3 1 . (6 marks) 11. (98I13) Let r and be real numbers. (a) By considering z r (cos i sin ) , or otherwise, simplify r cos i sin . 1 r cos i r sin (4 marks) (b) For any positive integer n , show that n
r sin i cos n n n i sin n . cos 2 2 1 r sin i r cos (3 marks) (c) Find r and , with r 0 , such that 3
r sin i cos 3 i . For such r and , sketch the 2 1 r sin i r cos points representing z r (cos i sin ) on an Argand diagram. (6 marks) (d) Determine with reasons whether there exist r and , with 3
r sin i cos r 0 , such that 3 i . 1 r sin i r cos (2 marks)
P. 60
HKAL Pure Mathematics Past Paper Topic: Complex Number
12. (04I12) Let n be a positive integer. (a) Assume that is not an integral multiple of . (i)
Prove that
n (cos i sin ) 1 2 cos (n 1) i sin (n 1) . cos i sin 1 2 2 sin 2 sin
n
(ii)
Using the identity
zn 1 z z for z 1 , or z 1 k 1 n
k
otherwise, prove that
n
cos k k 1
n
sin k k 1
(iii)
sin
sin
n (n 1) cos 2 2 and sin 2
n (n 1) sin 2 2 . sin 2
Using (a)(ii), or otherwise, prove that n sin n cos(n 1) cos 2 (k ) . 2 2sin k 1 (9 marks)
(b) For n > 1 , evaluate (i)
n
sin k 1
(ii)
2
k , n
k k cos sin n n k 1 n
2
. (6 marks)
P. 61
HKAL Pure Mathematics Past Paper Topic: Complex Number
Set 3: nth of Unity 13. (05I06) Let be a real number. (a) Solve the quadratic equation z2 + 2z cos 6 + 1 = 0 . (b) Using the result of (a), express x6 2 x3 cos 6 1 as a product of quadratic polynomials with real coefficients. (7 marks) 14. (00I13) Let n = 2, 3, 4, … . (a) Evaluate lim x 1
x2n 1 . x2 1
(2 marks) (b) Find all the complex roots of x2n 1 = 0 . Hence or otherwise, show that x2n 1 can be factorized as ( x 2 1)( x 2 2 x cos
n
1)( x 2 2 x cos
2 n
1) ( x 2 2 x cos
(n 1) n
1) .
(6 marks) (c) Using (b) or otherwise, show that lim x 1
x2n 1 2 (n 1) . 22 n2 sin 2 sin 2 sin 2 2 x 1 2n 2n 2n
(4 marks) (d) Using (a) and (c), or otherwise, show that 1 2 (n 1) lim sin sin sin n 2n 2n 2n n
1
n 1 . 2
(3 marks)
P. 62
HKAL Pure Mathematics Past Paper Topic: Complex Number
15. (03I12) Let n be a positive integer. (a) (i) (ii)
Find all roots of z 2n 1 0 . By factorizing z 2n 1 0 into a product of quadratic factors with real coefficients, or otherwise, prove that
1 n1 1 (2k 1) z 2cos for all z 0 . n z z 2n k 0 (7 marks) (b) Using (a), or otherwise, prove that zn
(i)
n 1
cos cos k 0
(ii)
n 1
cos k 1
(2k 1) 2n
cos n n1 for any R , 2
(2k 1) 2 n . 4n 2 (8 marks)
P. 63
HKAL Pure Mathematics Past Paper Topic: Complex Number
16. (01I11) (a) Show that
1 cos i sin . i cot 1 cos i sin 2 (3 marks)
(b) Let n be a positive integer. Show that all the roots of equation
( z 1)n ( z 1)n 0 ……… (*) can be written as i k , where k R , k = 0, 1, … , n 1 . (4 marks) (c) If i k ( k = 0, 1, … , n 1 ) are the roots of (*) in (b), using the relations between the roots and coefficients, show that n 1
k 0
2 k
n(n 1) . (5 marks)
(d) Let P0 , P1 ,..., Pn1 be the n points in an Argand plane representing the roots of (*) in (b), and O be the origin. Q is the point representing r (cos i sin ) where r 0 and R . If dk is the distance between Pk and Q , show that n 1
d k 0
2 k
is independent of . (3 marks)
P. 64
HKAL Pure Mathematics Past Paper Topic: Complex Number
17. (93I13) For any n = 1, 2, … , the sets Gn and Hn are defined by
Gn {z C : z n 1} , H n {z C : z n 1} . Let p , q be any two positive integers. (a) Show that (i) Gp H p , (ii) Gp H p G2 p . (2 marks) (b) Show that if p is odd and q is even, then H p H q . (2 marks) (c) Suppose p = mq where m is an integer. Show that (i) Gq G p ; (ii) if m is odd, then H q H p ; (iii) if m is even, then H q Gp . (5 marks) (d) For any S , T C , define ST by
ST z C : z st for some s S and t T . Show that (i) GpGp H p H p Gp , (ii) Gp H p H pGp H p . (6 marks)
P. 65
HKAL Pure Mathematics Past Paper Topic: Complex Number
Set4: Mapping 18. (90I10)
1 . z (a) If z r (cos i sin ) and f ( z ) u iv where r 0 and , u, v R , express u and v in terms of r and . Let f : C\{0} C be defined by f ( z ) z
(2 marks) (b) Find and sketch the image of each of the following circles under f : (i)
z 1 ;
(ii)
z a , 0 < a < 1 . (4 marks)
(c) Show that f is surjective but not injective. (4 marks) (d) Let E z C \{0}: z 1 and f E : E C be defined by f E ( z ) f ( z) for all z E . Show that f E is injective but not surjective. (5 marks)
P. 66
HKAL Pure Mathematics Past Paper Topic: Complex Number
19. (91I09) A mapping f : C C is said to be real linear if
f ( z1 z2 ) f ( z1 ) f ( z2 ) for all , R , z1 , z2 C . Let : C C be a real linear mapping. (a) Prove that (i)
( z) ( z) for all R , z C .
(ii)
(0) 0 . (3 marks)
(b) Let : C C be a real linear mapping. Prove that if (1) = (1) and (i ) = (i ) , then = . (2 marks) (c) If, furthermore, is non-constant such that
( z1 z2 ) ( z1 ) ( z2 ) for all z1 , z2 C , show that: (i)
(1) = 1 and hence (x) = x for all x R .
(ii)
Either (z) = z for all z C or ( z ) z for all z C . (10 marks)
P. 67
HKAL Pure Mathematics Past Paper Topic: Complex Number
20. (94I11) A function f : C C is said to be real linear if
f ( z1 z2 ) f ( z1 ) f ( z2 ) for all , R and z1 , z2 C . (a) Suppose f is a real linear function. Show that (i)
if z = 0 whenever f (z) = 0 , then f is injective;
(ii)
if f (i) = i f (1) and f (i) 0 , then f is bijective. (4 marks)
(b) Suppose , C and g( z ) z z for all z C . Show that (i)
g is real linear;
(ii)
g is injective if and only if .
(8 marks) (c) If f is a real linear function, find a, b C such that
f ( z ) az bz for all z C . (3 marks) Type 2: Geometric Problems Set 1: Triangles 21. (99I05) A , B and C are three points in the Argand diagram representing the complex numbers z1 , z2 and z3 respectively. If z1 = 0 , z2 = 1 + i and ABC is equilateral , find z3 . (5 marks)
P. 68
HKAL Pure Mathematics Past Paper Topic: Complex Number
22. (93I11) Let Z1 , Z2 and Z3 be 3 distinct points representing the complex numbers z1 , z2 and z3 respectively. (a) Suppose W1 , W2 and W3 are 3 distinct points representing the complex numbers w1 , w2 and w3 respectively. Prove that Z1Z2Z3 is similar to W1W2W3 (vertices anticlockwise) if and only if z3 z1 w3 w1 . z2 z1 w2 w1 (4 marks) (b) Using (a), or otherwise, show that Z1Z2Z3 (vertices anticlockwise) is equilateral if and only if
z1 z2 2 z3 0 2 where cos 3
2 i sin . 3 (5 marks)
(c) A point representing a + i b is said to be an integral point if a and b are integers. Using (b), or otherwise, show that no triangle with distinct integral points as vertices can be equilateral. (6 marks) Set 2: Straight Lines / Circles 23. (98I04) Let z be a complex number satisfying 2 z 2i z i . (a) Show that the locus of z on an Argand diagram is a circle. Find its centre and radius. (b) Let S z C : 2 z 2i z i . Draw and shade the region which represents S on an Argand diagram. Hence find z0 S such that z0 z for all z S . (6 marks)
P. 69
HKAL Pure Mathematics Past Paper Topic: Complex Number
24. (00I04) Consider the circle
zz (2 3i) z (2 3i) z 12
(zC)
………(*) .
Rewrite (*) in the form of z a r where a C and r > 0 . Hence or otherwise, find the shortest distance between the point 4 5i and the circle. (5 marks) 25. (01I08) Let L be the straight line z (4 4i) z and C be the circle
z 1 . (a) Sketch L on an Argand diagram. (b) Let P , Q be points on L and C respectively such that PQ is equal to the shortest distance between L and C . Find PQ and the complex numbers representing P and Q . (6 marks) 26. (02I02) (a) Express
z 1 2 in the form of | z c | r , where c and r z4
are constants.
z 1 2 in the (b) Shade the region represented by z C : z4 Argand plane. (5 marks)
P. 70
HKAL Pure Mathematics Past Paper Topic: Complex Number
27. (95I11) Let P , Q be two points on a circle with centre C such that P , Q , C are non-collinear and taken anti-clockwise. PCQ and M is the mid-point of PQ . Let zP , zQ , zC and zM be the complex numbers represented by P , Q , C and M respectively. (a) Show that zC zM i( zM zP ) cot
2
. (5 marks)
(b) Express zC and the radius r of the circle in terms of zP , zQ and . (4 marks) (c) (i)
Show that any circle in the complex plane can be represented by an equation of the form zz az bz c 0 where a, b C and c R .
(ii)
Let C: zz az bz c 0 be a circle passing through the points representing 1 i and i . If the chord joining these two points subtends an angle
at the centre, 3
find the values of a , b and c . (6 marks)
P. 71
HKAL Pure Mathematics Past Paper Topic: Complex Number
28. (96I12) (a) Let a C and b 0 . Show that the equation zz az az b
(zC)
can be written in the form of z a r where r 0 . (4 marks) (b) Let A and B be two points on the complex plane representing 2 3i and 1 2i respectively. P , representing the complex number z , is a moving point so that PA 2 PB . Show that the equation of the locus of P is a circle with equation
C: zz iz iz 3 . Find its radius and centre. (5 marks) (c) Let Q , representing , be a point on the circle C in (b). (i)
i Show that the circle z 1 touches C at Q i
externally. (ii)
For any given r > 0 , write down the equations of the two circles with radius r which touch C at Q . (6 marks)
P. 72
HKAL Pure Mathematics Past Paper Topic: Complex Number
29. (99I13) Let be a complex number and u , v be variable complex numbers satisfying
u u and v v vv respectively. Let L be the locus of u and C be the locus of v . (a) Show that (i)
the equation of L can be written as u u ,
(ii)
the equation of C can be written as v . (4 marks)
(b) For = 2 + i , sketch L and C on an Argand diagram. (4 marks) (c) (i)
Let z
1 . Show that z satisfies z z zz for u
some constant . Hence sketch the locus of z on an Argand diagram for 2 i . (ii)
Let z
1 . Sketch the locus of z on an Argand v
diagram for 2 i . (7 marks)
P. 73
HKAL Pure Mathematics Past Paper Topic: Complex Number
Set 3: Others 30. (92I11) Let a be a positive real number and n a positive integer. (a) Solve the quadratic equation y 2 2 ya n cos n a 2n 0 where
R . Hence show that the polynomial x2n 2 xn a n cos n a 2n can be n 1 2r 2 factorized as x 2 2 xa cos a . n r 0 (6 marks) (b) Let P0 , P1 , P2 , … Pn be the n points in the Argand plane representing the nth roots of an , arranged anti-clockwise, with P0 on the positive real axis. Let Q be the point representing x(cos i sin ) where x 0 . For r 0, 1, 2, , n 1 , denote the length of the segment QPr by
dr . (i)
Show that
n 1
d r 0
(ii)
x 2 n 2 x n a n cos n a 2 n .
If Q lies on the positive real axis, show that n 1
d r 0
(iii)
2 r
r
xn an .
If OQ bisects P0OP1 , where O is the origin, show that
n 1
d r 0
r
xn an . (9 marks)
P. 74
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
Contents: Classification
Chronological Order
Type 1: General Properties
90I01
Q.
19
1.
91I01
90I08
Q.
42
2.
93I06
91I01
Q.
1
3.
96I01
91I03
Q.
15
4.
97I07
91I10
Q.
12
5.
00I09
92I01
Q.
20
6.
94I08
92I03
Q.
37
7.
02I12
92I09
Q.
9
8.
06I08
92I13
Q.
10
9.
92I09
92II10
Q.
54
10.
92I13
93I03
Q.
21
11.
95I08
93I06
Q.
2
12.
91I10
94I01
Q.
38
Type 2: Systems of Linear Equations
94I02
Q.
18
SECTION A
94I08
Q.
6
13.
97I03
94I09
Q.
34
14.
96I05
95I01
Q.
35
15.
91I03
95I08
Q.
11
16.
98I01
95I09
Q.
22
17.
99I01
96I01
Q.
3
18.
94I02
96I05
Q.
14
19.
90I01
96I08
Q.
39
20.
92I01
96I09
Q.
33
21.
93I03
97I03
Q.
13
SECTION B
97I07
Q.
4
22.
95I09
97I08
Q.
23
23.
97I08
97I10
Q.
46
24.
00I08
98I01
Q.
16
25.
01I09
98I02
Q.
47
26.
03I07
98I08
Q.
30
27.
04I07
98I09
Q.
40
28.
05I07
99I01
Q.
17
29.
06I07
99I06
Q.
53
30.
98I08
99I08
Q.
32
31.
02I08
99I09
Q.
43
32.
99I08
00I01
Q.
36
P. 75
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
33.
96I09
00I06
Q.
48
34.
94I09
00I08
Q.
24
Type 3: Power of a Matrix
00I09
Q.
5
35.
95I01
01I07
Q.
49
36.
00I01
01I09
Q.
25
37.
92I03
02I03
Q.
50
38.
94I01
02I08
Q.
31
39.
96I08
02I12
Q.
7
40.
98I09
03I07
Q.
26
41.
03I08
03I08
Q.
41
42.
90I08
04I03
Q.
51
43.
99I09
04I07
Q.
27
44.
04I08
04I08
Q.
44
45.
05I08
05I05
Q.
52
46.
97I10
05I07
Q.
28
Type 4: Transformation in Cartesian
05I08
Q.
45
06I07
Q.
29
06I08
Q.
8
Plane 47.
98I02
48.
00I06
49.
01I07
50.
02I03
51.
04I03
52.
05I05
53.
99I06
54.
92II10
P. 76
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
Type 1: General Properties 1.
(91I01) Factorize the determinant
a3 a 1
b3 b 1
c3 c . 1 (4 marks)
2.
(93I06) (a) Show that if A is a 3 3 matrix such that At A , then
det A 0 . 1 2 74 1 67 , use (a), or otherwise, to show (b) Given that B 2 74 67 1 det( I B) 0 . Hence deduce that det( I B ) 0 . 4
(7 marks) 3.
(96I01)
0 0 2 Let A 1 2 1 . 1 0 3 (a) Evaluate A3 5 A2 8 A 4I . (b) Hence, or otherwise, find A1 . (6 marks)
P. 77
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
4.
(97I07) (a) Let A be a 3 3 non-singular matrix. Show that
det( A1 xI )
x3 det( A x 1I ) . det A
0 1 0 (b) Let A 0 0 1 . 4 17 8 (i)
Show that 4 is a root of det(A xI ) = 0 and hence find the other roots in surd form.
(ii)
Solve det(A1 xI ) = 0 . (7 marks)
5.
(00I09) n n n 1 (a) Show that Cr Cr 1 Cr 1 where n , r are positive integers and
n r 1 . (2 marks) (b) Let A , B be two square matrices of the same order. If AB = BA , show by induction that for any positive integer n , n
( A B)n Crn An r B r
………. (*),
r 0
where A0 and B 0 are by definition the identity matrix I . Would (*) still be valid if AB BA ? Justify your answer. (6 marks)
cos sin
(c) Let A =
(i)
sin where θ is real. cos cos n sin n
Show that An =
sin n for all positive integers cos n
n . (ii)
Using (*) and the substitution B = A1 , show that n
Crn cos(n 2r ) 2n cosn and r 0
n
C r 0
n r
sin(n 2r ) 0 .
Hence or otherwise, express cos5θ in terms of cos 5θ , cos 3θ and cos θ . (7 marks)
P. 78
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
6.
(94I08)
a b c Let M c a b , where a , b and c are non-negative real b c a numbers. (a) Show that det( M )
1 (a b c)[(a b) 2 (b c) 2 (c a) 2 ] and 2
0 det(M ) (a b c)3 . (4 marks)
an (b) Let M cn b n n
bn an cn
cn bn for any positive integer n , show that an , an
bn and cn are non-negative real numbers satisfying
an bn cn (a b c)n . (4 marks) (c) If a + b + c = 1 and at least two of a , b and c are non-zero, show that (i)
lim det( M n ) 0 ,
(ii)
lim(an bn ) 0 and lim(an cn ) 0 ,
(iii)
lim an
n
n
n
n
1 . 3 (7 marks)
P. 79
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
7.
(02I12) (a) Let A be a 3 3 matrix such that
A3 A2 A I 0 , where I is the 3 3 identity matrix. (i)
Prove that A has an inverse, and find A1 in terms of A .
(ii)
Prove that A4 = I .
(iii)
Prove that ( A ) ( A ) A I 0 .
(iv)
Find a 3 3 invertible matrix B such that
1 3
1 2
1
B3 B 2 B I 0 . (6 marks)
1 1 1 (b) Let X 1 1 0 . 1 0 1 (i)
Using (a)(i) or otherwise, find X 1 .
(ii)
Let n be a positive integer. Find X n .
(iii)
Find two 3 3 matrices Y and Z , other than X , such that
Y 3 Y 2 Y I 0 and Z 3 Z 2 Z I 0 . (9 marks)
P. 80
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
8.
(06I08)
m m , where m > 0 . m m
Let M =
(a) Evaluate
M2
. (1 mark)
a b be a non-zero real matrix such that MX = XM . c d
(b) Let X = (i)
Prove that c = b and d = a .
(ii)
Prove that X is a non-singular matrix.
(iii)
Suppose X 6 X 1 =
1 0 . 0 1
(1)
Find X .
(2)
If a > 0 and (M k X )2 = M 2 , express k in terms of m . (10 marks)
(c) Using the result of (b)(iii)(2), find two real matrices P and Q , other than M and M , such that P Q M 4
4
4
. (4 marks)
P. 81
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
9.
(92I09)
cos sin
(a) Let A
sin . cos
cos n sin n
sin n for cos n
Prove by mathematical induction that An
n 1, 2, . (3 marks)
a b : a, b R and n be a positive integer. b a
(b) Let M (i)
For any X , Y M , show that (I)
(ii)
XY M ,
(II)
XY = YX ,
(III)
if X
0 0 1 1 , then X exists and X M . 0 0
For any X M , show that there exist r 0 and R such
cos sin
that X r
sin . cos 1 0 . 0 1
Hence find all X M such that X n (iii)
If Y , B M and Y n = Bn , show that there exists X M
1 0 and Y = BX . 0 1
such that X n
n
1 2 Hence find all Y M such that Y . 2 1 n
(12 marks)
P. 82
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
10.
(92I13)
a11 a21
Let M be the set of all 2 2 matrices. For any A
a12 M , a22
define tr( A) a11 a22 . (a) Show that for any A , B , C M and , R , (i)
tr( A B) tr( A) tr( B) ,
(ii)
tr( AB) tr( BA) ,
(iii)
the equality “ tr( ABC ) tr( BAC ) ” is not necessarily true. (5 marks)
(b) Let A M . (i)
Show that A tr( A) A det( A) I , where I is the 2 2 2
identity matrix. (ii)
If tr( A ) 0 and tr( A) 0 , use (a) and (b)(i) to show that 2
A is singular and A2 = 0 . (5 marks) (c) Let S , T M such that (ST TS )S S (ST TS ) . Using (a) and (b) or otherwise, show that ( ST TS ) 0 . 2
(5 marks)
P. 83
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
11.
(95I08) Let Mmn be the set of all m n matrices.
a1 b1
(a) Let A
(i)
a2 M 22 . b2 u1 s1 u2
Show that if A
s2 , where u1 , u2 , s1 , s2 R , then
det A 0 . (ii)
Conversely, show that if det A = 0 , then A = BC for some
B M 21 and C M12 . (5 marks)
a1 a2 (b) Let D b1 b2 c c 1 2
(i)
a3 b3 M 33 . c3
u1 Show that if D u2 u 3
v1 s s v2 1 2 t t v3 1 2
s3 , where t3
ui , vi , si , ti R ( i = 1, 2, 3 ) , then det D = 0 . (ii)
Suppose there are , R such that ci ai bi for
i 1, 2, 3 . Find S M 32 and T M 23 such that D ST . (iii)
Show that if det D = 0 , then D = PQ for some P M 32 and Q M 23 . (10 marks)
P. 84
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
12.
(91I10) Let M be the set of all 3 1 real matrices.
u1 x1 For any two 3 1 matrices u u2 and x x2 , u x 3 3 u2 x3 x2u3 we define u x u3 x1 x3u1 . u x xu 1 2 1 2 (a) Show that for any u, x, y M , and any , R , (i)
u ( x y) (u x) (u y) ,
(ii)
u x ( x u) . (3 marks)
(b) Show that if u x 0 for all x M , then u = 0 . Deduce that if u x v x for all x M , then u = v . (6 marks) (c) Let A be a 3 3 real matrix and u M such that
Ae1 u e1 , Ae2 u e2 and Ae3 u e3 1 0 0 where e1 0 , e2 1 , e3 0 . 0 0 1 Show that Ax u x for all x M . (4 marks)
p (d) Let u q . Find the matrix A such that Ax u x for all r
xM . (2 marks)
P. 85
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
Type 2: Systems of Linear Equations SECTION A 13.
(97I03) Suppose the system of linear equations
x ky y (*): x ky
0 z 0 z 0
has nontrivial solution. (a) Show that satisfies the equation 2 + k k = 0 . (b) If the quadratic equation in in (a) has equal roots, find k . Solve (*) for each of these values of k . (6 marks) 14.
(96I05)
Z (a) Solve Z
Y Y
X X
a b for X , Y and Z . c
(b) If a + b c > 0 , b + c a > 0 and c + a b > 0 ,
xy solve xy
xz xz
a yz b for x , y and z . yz c (6 marks)
15.
(91I03) Consider the following system of linear equations:
z 1 x 2 y y 2z 2 . x y q2 z q Determine all values of q for each of the following cases: (a) The system has no solution. (b) The system has infinitely many solutions. (4 marks)
P. 86
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
16.
(98I01) Consider the system of linear equations
2 x (*): x kx
y 2z 0 (k 1) z 0 . y 4z 0
Suppose (*) has infinitely many solutions. (a) Find k . (b) Solve (*) . (6 marks) 17.
(99I01) Suppose the system of linear equations
y z 0 x z 0 (*): x y x y z 0 has non-trivial solutions. (a) Find all the values of . (b) Solve (*) for each of the values of obtained in (a). (6 marks) 18.
(94I02) Consider the following system of linear equations:
z x 4 x 3 y (*) 3x 4 y 7 z y . x 7 y 6z z Suppose is an integer and (*) has non-trivial solution. Find and solve (*) . (6 marks)
P. 87
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
19.
(90I01) Consider the following system of linear equations:
y z 1 3x (*) 2 x 4 y 5 z 1 , 4 x 2 y 7 z c where c R . Suppose (*) is consistent. Find c and solve (*) . (4 marks) 20.
(92I01) Consider the following system of linear equations:
x (t 3) y 5 z 3 9 y 15 z s (*) 3x 2x ty 10 z 6 (a) If (*) is consistent, find s and t . (b) Solve (*) when it is consistent. (6 marks) 21.
(93I03) Suppose the following system of linear equations is consistent:
ax bx (*) cx x
by cz cy ax ay bz y z
1 1 , where a , b , c R . 1 3
(a) Show that a + b + c = 1 . (b) Show that (*) has a unique solution if and only if a , b and c are not all equal. (c) If a = b = c , solve (*) . (6 marks)
P. 88
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
SECTION B 22.
(95I09) Consider the following systems of linear equations
2x 2 y z k (S): hx 3 y z 0 3x hy z 0 and
6x hx (T): 3x 5 x
6 y 3z 3y z hy z 2 y 6z
2 0 . 0 h
(a) Show that (S) has a unique solution if and only if h2 9 . Solve (S) in this case. (3 marks) (b) For each of the following cases, find the value(s) of k for which (S) is consistent, and solve (S) : (i)
h=3 ,
(ii)
h = 3 . (7 marks)
(c) Find the values of h for which (T) is consistent. Solve (T) for each of these values of h . (5 marks)
P. 89
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
23.
(97I08) Consider the following two systems of linear equations:
2z 0 (a 1) x 2 y x ay 2z 0 (S): 3x y (a 7) z 0
and
2z 6 (a 1) x 2 y x ay 2 z 5b 1 . (T): 3x y (a 7) z 1 b (a) If (S) has infinitely many solutions, find all the values of a . Solve (S) for each of these values of a . (7 marks) (b) For the smallest value of a found in (a), find the values of b so that (T) is consistent. Solve (T) for these values of a and b . (4 marks) (c) Solve the system of equations
x x 3x 3x
2y 2 z
6
2y 2 z
6
y 9 z 2 4y z 11
.
(4 marks)
P. 90
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
24.
(00I08) Consider the system of linear equations
y z a x y 2 z b where R . (S): 2 x x (2 3) y 2 z c (a) Show that (S) has a unique solution if and only if 2 . Solve (S) for = 1 . (7 marks) (b) Let = 2 . (i)
Find the conditions on a , b and c so that (S) has infinitely many solutions.
(ii)
Solve (S) when a = 1 , b = 2 and c = 3 . (4 marks)
(c) Consider the system of linear equations
y z 3 x (T): 2 x 2 y 2 z 2 x y 4z
5 0 2 0 1 0
where R .
Using the results in (b), or otherwise, solve (T) . (4 marks)
P. 91
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
25.
(01I09) Consider the system of linear equations
z k x y y z 1 where , k R . (S): x 3x y 2 z 1 (a) Show that (S) has a unique solution if and only if 0 and
2 . (2 marks) (b) For each of the following cases, determine the value(s) of k for which (S) is consistent. Solve (S) in each case. (i) (ii) (iii)
0 and 2 , 0 , 2 . (8 marks)
x (c) If some solution of (x , y , z) of 3x
z 0 y z 1 y 2 z 1
satisfies ( x p) y z 1 , find the range of values of p . 2
2
2
(5 marks)
P. 92
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
26.
(03I07) (a) Consider the system of linear equations in x , y , z
ay z 0 x y az 2a , (E): 2 x 2 x 2a y (a 3) z 2a where a R . (i)
Find the range of values of a for which (E) has a unique solution. Solve (E) when (E) has a unique solution.
(ii)
Solve (E) for (1)
a=1 ,
(2)
a = 4 . (10 marks)
(b) Suppose (x , y , z) satisfies
y z 0 x y z 2 . 2x x 2 y 2 z 2 Find the least value of 24 x 3 y 2 z and the corresponding 2
2
values of x , y , z . (5 marks)
P. 93
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
27.
(04I07) (a) Consider the system of linear equations
x (a 2) y az 1 2 y 4z 1 , (E): x ax y 3z b where a , b R . (i)
Prove that (E) has a unique solution if and only if a 2 and
a 4 . Solve (E) in this case. (ii)
For each of the following cases, determine the value(s) of b for which (E) is consistent, and solve (E) for such value(s) of b . (1)
a=2 ,
(2)
a=4 . (10 marks)
2z 1 x (b) If all solutions (x , y , z) of x 2 y 4 z 1 satisfy 2 x y 3z 2
k ( x 2 3) yz , find the range of values of k . (5 marks)
P. 94
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
28.
(05I07) (a) Consider the system of linear equations in x , y , z
ay z b x (a 1) z 0 , (E): 2 x (a 3) y 2 3x a y (4a 1) z b where a , b R . (i)
Find the range of values of a for which (E) has a unique solution. Solve (E) when (E) has a unique solution.
(ii)
For each of the following cases, find the value(s) of b for which (E) is consistent, and solve (E) for such value(s) of b . (1)
a=1 ,
(2)
a = 2 . (10 marks)
z b x 2y y 3z 0 (b) Suppose that a real solution of 2 x 3x 4 y 7 z b satisfies x y z b 3 , where b R . Find the range of 2
2
2
values of b . (5 marks)
P. 95
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
29.
(06I07) Consider the system of linear equations in x , y , z
ay z 4 x (E): x (2 a) y (3b 1) z 3 , 2 x (a 1) y (b 1) z 7 where a , b R . (a) Prove that (E) has a unique solution if and only if a 1 and b 0 . Solve (E) in this case. (6 marks) (b) (i)
For a = 1 , find the value(s) of b for which (E) is
consistent, and solve (E) for such value(s) of b . (ii)
Is there a real solution (x , y , z) of
x y z 4 2 x 2 y z 6 4 x 4 y 3z 14 satisfying x2 2y2 z = 14 ? Explain your answer. (7 marks) (c) Is (E) consistent for b = 0 ? Explain your answer. (2 marks)
P. 96
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
30.
(98I08) Consider the system of linear equations in
ax y bz 1 bz 1 . (E): x ay x y abz b (a) Show that (E) has a unique solution if and only if a 2 , a 1 and b 0 . Solve (E) in this case. (7 marks) (b) For each of the following cases, determine the value(s) of b for which (E) is consistent. Solve (E) in each case. (i)
a=2,
(ii)
a=1 . (6 marks)
(c) Determine whether (E) is consistent or not for b = 0 . (2 marks)
P. 97
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
31.
(02I08) (a) Consider the system of linear equations in x , y , z
z 0 ax 2 y y 2 z b , where a , b R . (S): x y az b (i)
Show that (S) has a unique solution if and only if a 2 1 . Solve (S) in this case.
(ii)
For each of the following cases, determine the value(s) of b for which (S) is consistent, and solve (S) for such value(s) of b . (1)
a=1 ,
(2)
a = 1 . (9 marks)
(b) Consider the system of linear equations in x , y , z
ax 2 y x y (T): y 5 x 2 y
z 2z az z
0 1 , where a R . 1 a
Find all the values of a for which (T) is consistent. Solve (T) for each of these values of a . (6 marks)
P. 98
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
32.
(99I08) Consider the system of linear equations
z x y y ( 2) z 7 where R . (E): 3x x y z 3 (a) Show that (E) has a unique solution if and only if 1 . (3 marks) (b) Solve (E) for (i)
1 ,
(ii)
1 ,
(iii)
1 . (8 marks)
(c) Find the conditions on a , b , c and d so that the system of linear equations
x 3x x ax
y y y by
z 1 3z 7 z 3 cz d
is consistent. (4 marks)
P. 99
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
33.
(96I09) Consider the system of linear equations
z 3 x 2 y . y 2z 4 x
(*):
(a) Solve (*) . (3 marks) (b) Find the solutions of (*) that satisfy xy + yz + zx = 2 . (4 marks) (c) Find all possible values of a and ( a, R ) so that
z 3 x 2y y 2z 4 x ax y z is solvable. (4 marks) (d) Using (b), or otherwise, find all possible values of a and ( a, R ) so that
x x xy ax
2y z y 2z yz zx y z
3 4 2
has at least one solution. (4 marks)
P. 100
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
34.
(94I09) (a) Consider
(I)
a11 x a12 y a13 z 0 a21 x a22 y a23 z 0 a x a y a z 0 32 33 31
(II)
a11 x a12 y a13 a21 x a22 y a23 a x a y a 32 33 31
and
(i)
0 0 . 0
Show that if (I) has a unique solution, then (II) has no solution.
(ii)
Show that (u, v) is a solution of (II) if and only if (ut, vt, t) are solutions of (I) for all t R .
(iii)
If (II) has no solution and (I) has nontrivial solutions, what can you say about the solutions of (I) ? (5 marks)
(b) Consider
(III)
y z 0 (3 k ) x 7 x (5 k ) y z 0 6 x 6 y (k 2) z 0
(IV)
y 1 0 (3 k ) x 7 x (5 k ) y 1 0 . 6 x 6 y (k 2) 0
and
(i)
Find the values of k for which (III) has non-trivial solutions.
(ii)
Find the values of k for which (IV) is consistent. Solve (IV) for each of these values of k .
(iii)
Solve (III) for each k such that (III) has non-trivial solutions. (10 marks)
P. 101
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
Type 3: Power of a Matrix 35.
(95I01)
a 1 where a , b R and a b . 0 b n a n bn a Prove that An a b for all positive integers n . b n 0
(a) Let A
1 2 (b) Hence, or otherwise, evaluate 0 3
95
. (6 marks)
36.
(00I01)
1 0 0 Let M = b a where b2 + ac = 1 . Show by induction that c b
M
2n
1 0 0 n[ (1 b) a] 1 0 for all positive integers n . n[ c (1 b)] 0 1
0 0 1 Hence or otherwise, evaluate 2 3 2 1 4 3
2000
.
(5 marks) 37.
(92I03)
1 0 2 0 , B . 1 3 1
Let A
a 0 , find , a and b . 0 b
(a) If B1 exists and B 1 AB (b) Hence find A100 .
(7 marks)
P. 102
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
38.
(94I01)
3 8 2 4 . and P 1 1 1 5
Let A
(a) Find P 1 AP . (b) Find An , where n is a positive integer. (6 marks) 39.
(96I08) (a) Solve the equation
1 3 1 0 det 0 2 2 0 1
……… (*) . (3 marks)
(b) Let 1 , 2 ( 1 2 ) be the roots of (*) . Find two non-zero vectors (x1 , y1) and (x2, y2) such that
xi 1 3 xi y i y , 2 2 i i
i = 1, 2.
[Modified]
x1 y1
Let P
x2 1 . Show that P is non-singular and find P . y2
1 3 P . 2 2
Evaluate P 1
(9 marks) 1996
1 3 (c) Evaluate 2 2
. (3 marks)
P. 103
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
40.
(98I09)
a b where a, b, c, d R , a 0 and det A 0 . c d
Let A =
a b for some k R . ka ka
(a) Show that A =
(3 marks)
1 0 such that PA = r 1 0
(b) Find P in the form of
, R .
for some 0
1 s such that 0 1
If a + d 0 , find Q in the form of
0 PAP 1Q for some R . 0 0
(5 marks)
3 7 1 0 S for some R . 6 14 0 0
(c) Find S such that S
n
3 7 Hence, or otherwise, evaluate where n is a positive 6 14 integer. (7 marks)
P. 104
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
41.
(03I08)
2 3
(a) If det
3 0 , find the two values of . (2 marks)
(b) Let 1 and 2 be the values obtained in (a), where 1 2 . Find 1 and 2 such that
2 1 3
3 cos 1 0 , 0 1 < . 1 sin 1 0
2 2 3
3 cos 2 0 , 0 2 < . 2 sin 2 0
cos 1 cos 2 n . Evaluate P , where n is a positive sin sin 1 2
Let P integer.
2 3
3 d1 P is a matrix of the form 0 0
Prove that P 1
0 . d2 (8 marks)
2 (c) Evaluate 3
n
3 , where n is a positive integer. 0 (5 marks)
P. 105
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
42.
(90I08) (a) Let X and Y be two square matrices such that XY = YX . Prove that (i)
( X Y )2 X 2 2 XY Y 2 ,
(ii)
( X Y )n Crn X n rY r for n = 3, 4, 5, … .
n
r 0
(Note: For any square matrix A , define A0 = I .) (3 marks)
0 2 4 (b) By using (a)(ii) and considering 0 0 3 , or otherwise, find 0 0 0 100
1 2 4 0 1 3 0 0 1
.
(4 marks) (c) If X and Y are square matrices, (i)
prove that ( X Y ) X 2 XY Y
(ii)
prove that ( X Y ) X 3 X Y 3 XY Y
2
3
2
3
2
2
implies XY YX ;
2
3
does NOT imply
XY YX . (Hint:
Consider a particular X and Y ,
1 0 b 0 , Y .) 1 0 0 0
e.g. X
(8 marks)
P. 106
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
43.
(99I09) (a) Let A and B be two square matrices of the same order. If
AB BA 0 , show that (A + B)n = A n + B n for any positive integer n . (4 marks)
a b p q where a , b are not both zero. If B , 0 0 r s show that AB BA 0 if and only if p = r = 0 and aq + bs = 0 .
(b) Let A
(4 marks)
x 0
(c) Let C
y where x , z are non-zero and distinct. Find z
non-zero matrices D and E such that C = D + E and DE = ED =0 . (3 marks)
2 5 (d) Evaluate 0 1
99
. (4 marks)
P. 107
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
44.
(04I08)
k k , where , , k R with . k k
Let A
Define X
1 1 ( A I ) , where I is the 2 2 ( A I ) and Y
identity matrix. (a) Evaluate XY , YX , X + Y , X 2 and Y 2 . (4 marks) (b) Prove that A X Y for all positive integers n . n
n
n
(4 marks)
5 4 (c) Evaluate 2 3
2004
. (4 marks)
(d) If and are non-zero real numbers, guess an expression for
A1 in terms of , , X and Y , and verify it. (3 marks)
P. 108
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
45.
(05I08)
p q , where p , q , r , s R . r s
(a) Let M (i)
Suppose det M = 0 . Prove that M
(ii)
n 1
( p s)n M for any positive integer n .
Suppose qr > 0 . Let and be the roots of the quadratic equation x2 (p + s) x + det M = 0 . Denote the 2 2 identity matrix by I . (1)
Prove that and are two distinct real numbers.
(2)
Prove that M 2 ( + ) M + I =
(3)
Define A = M I and B = M I .
0 0 . 0 0
0 0 and det A = det B = 0 . 0 0 Find real numbers and , in terms of and , such that M = A + B . Prove that AB = BA =
(11 marks) n
1 2 (b) Evaluate , where n is a positive integer. 4 3 Candidates may use the fact, without proof, that if X and Y are 2 2 matrices satisfying XY = YX =
0 0 n n n , then ( X Y ) X Y for any positive 0 0 integer n . (4 marks)
P. 109
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
46.
(97I10) (a) (i)
cos sin
Let S
sin : R . cos
Show that for any matrices A and B in S , AB is also in S . (ii)
cos sin
sin . cos
Let T( ) Prove that
T( )
n
T(n ) for any positive integer n . (4 marks)
a b 2 2 where a, b R and a + b 0 . b a
(b) Let M (i)
Show that M = k T( ) for some real numbers k and . Express k , cos and sin in terms of a and b .
(ii)
If a 0 , prove that there exists a positive integer n such
p 0 ) if and only 0 q
that M n is diagonal ( i.e. of the form if (iii)
1
tan 1
b is rational. a
If a = 0 , find all positive integers n such that M n is diagonal. (11 marks)
Type 4: Transformation in Cartesian Plane 47.
(98I02)
x' a b x a b 2 for any (x, y) R , then is said to be y' c d y c d
If
the matrix representation of the transformation which transforms (x, y) to ( x' , y' ) . Find the matrix representation of (a) the transformation which transforms any point (x, y) to (x, y) , (b) the transformation which transforms any point (x, y) to (y, x) . (4 marks)
P. 110
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
48.
(00I06) A transformation T in R2 transforms a vector x to another vector
cos 3 y Ax b where A sin 3
sin
3 3 and b . 1 cos 3
2 0
(a) Find y when x . (b) Describe the geometric meaning of the transformation T . (c) Find a vector c such that y = A(x + c) . (7 marks) 49.
(01I07) A 2 2 matrix M is the matrix representation of a transformation T in R2 . T transforms (1, 0) and (0, 1) to (1, 1) and (1, 1) respectively. (a) Find M .
0 a b 0 c d
(b) Find > 0 such that M can be decomposed as where
a b 1 . c d
Hence describe the geometric meaning of T . (5 marks) 50.
(02I03) (a) Write down the matrix A representing the rotation in the Cartesian plane anticlockwise about the origin by 45 . (b) Write down the matrix B representing the enlargement in the Cartesian plane with scale factor
2 .
x y
(c) Let X and V = BAX , where A and B are the matrices
1 0 V 4 , express y in terms of 0 1
defined in (a) and (b). If V t x .
(5 marks)
P. 111
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
51.
(04I03) (a) Let R be the matrix representing the rotation in the Cartesian plane anticlockwise about the origin by 60 . (i)
Write down R and R 6 .
(ii)
Let A =
2 0
1 1 . Verify that A RA is a matrix in which all 3
the elements are integers. (b) Using the results of (a), or otherwise, find a 2 2 matrix M , in which all the elements are integers, such that M 3 I but M I ,
1 0 . 0 1
where I
(7 marks) 52.
(05I05) Let T1 be the transformation which transforms a vector x to a vector
y = Ax , where A = (a) (i)
3 2 1 2
1 2 . 3 2 0 2
Find y when x = .
(ii)
Describe the geometric meaning of the transformation T1 .
(iii)
Find A2005 .
(b) For every integer n greater than 1 , let Tn be the transformation which transforms a vector x to a vector y = Anx . Is there a positive integer m such that the transformation Tm
0
2 ? Explain your answer. 2
transforms to 2
(7 marks)
P. 112
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
53.
(99I06) It is given that the matrix representing the reflection in the line
cos y (tan ) x is sin 2
sin 2 . cos 2
Let T be the reflection in the line y =
1 x . 2
(a) Find the matrix representation of T . (b) The point (4, 7) is transformed by T to another point (x1, y1) . Find x1 and y1 .
(c) The point (4, 10) is reflected in the line y
1 x 3 to another 2
point ( x2 , y2 ) . Find x2 and y2 . (7 marks)
P. 113
HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices
54.
(92II10) Let be a Cartesian coordinate system on a plane and be another Cartesian coordinate system with the same origin, obtained from by an anti-clockwise rotation through an angle . Suppose (x, y) and (x, y) are the coordinates of an arbitrary point P with respect to and respectively.
x y
x' . y'
(a) Let V , V'
cos sin
sin . cos
(i)
Show that V MV' , where M
(ii)
If the equation of a conic section in the coordinate system
a h , C (c ) , h b
is given by V t AV C , where A
a, b, h, c R , show that this conic section is represented in the coordinate system by
V' t A' V' C , where A is a 2 2 matrix such that det A det A' . Furthermore, show that can be chosen such that A is a diagonal matrix. (10 marks) (b) The equation of a conic section (H) in is given by
7 x2 2hxy 13 y 2 16 . Find h if (H) is (i)
an ellipse,
(ii)
a hyperbola,
(iii)
a pair of straight lines,
(iv)
given by x' 4 y' 4 in . 2
2
(5 marks)
P. 114
HKAL Pure Mathematics Past Paper Topic: Functions
Contents:
1.
Classification
Chronological Order
Type 1: Bisection / Even / Odd / Periodic
92I06
Q.
7
93I09
Q.
10
Functions 95I04
2.
99II06
94II07
Q.
8
3.
02II05
95I04
Q.
1
4.
03II03
97II10
Q.
11
5.
06II02
98II10
Q.
13
6.
04II11
99II06
Q.
2
Type 2: Inverse Function
99II10
Q.
9
7.
92I06
01II13
Q.
12
8.
94II07
02II05
Q.
3
9.
99II10
03II03
Q.
4
Type 3: Others
04II11
Q.
6
10.
93I09
06II02
Q.
5
11.
97II10
12.
01II13
13.
98II10
P. 115
HKAL Pure Mathematics Past Paper Topic: Functions
Type 1: Bisection / Even / Odd / Periodic Functions 1.
(95I04) Let f : [1, 1] [0, ] , f (x) = arc cos x and g : R R , g(x) = f (cos x) . (a) Show that g(x) is even and periodic. (b) Find g(x) for x [0, ] . Hence sketch the graph of g(x) for x [2, 2 ] . (5 marks)
2.
(99II06) (a) Suppose f : R R is a function satisfying f (a x) f (a x) and
f (b x) f (b x) for all x , where a , b are constants and a > b . Let w = 2(a b) . Show that w is a period of f , i.e.,
f ( x w) f ( x) for all x R . (b) Suppose g : R R is a periodic function with period T > 0 satisfying g( x) g( x) for all x . Show that there exists c with
0 c T such that g(c + x) = g(c x) for all x . (6 marks) 3.
(02II05) (a) If the function g: R R is both even and odd, show that g(x) = 0 for all x R . (b) For any function f : R R , define
1 1 F( x) [f ( x) f ( x)] and G( x) [f ( x) f ( x)] . 2 2 (i)
Show that F is an even function and G is an odd function.
(ii)
If f ( x) M( x) N( x) for all x R , where M is even and N is odd, show that M(x) = F(x) and N(x) = G(x) for all
xR . (6 marks)
P. 116
HKAL Pure Mathematics Past Paper Topic: Functions
4.
(03II03)
1 when x 0 , Let f : R R be defined by f ( x) 0 when x 0 , 1 when x 0 . (a) Prove that f is an odd function. (b) Is f an injective function? Explain your answer briefly. (c) Sketch the graph of y = f (x + 1) . (d) Let g(x) = f (x + 1) + f (x 1) for all x R . Sketch that graph of y = g(x) . Write down the value(s) of x at which g(x) is discontinuous. (6 marks) 5.
(06II02) Let f : R R be defined by
1 when x is an even number, f ( x) 2 when x is not an even number. (a) (i)
Sketch the graph of y = f (x) for 4 x 4 .
(ii)
Is f a periodic function? Explain your answer.
(b) Let g : R R be defined by g(x) = f (x + 1) + f (x) . (i)
Sketch the graph of y = g(x) for 4 x 4 .
(ii)
Is g an injective function? Explain your answer. (7 marks)
P. 117
HKAL Pure Mathematics Past Paper Topic: Functions
6.
(04II11) For any real number x , let [x] denote the greatest integer not greater than x . Let f : R R be defined by
1 f ( x) 2 x [ x] 1 2 (a) (i)
when x is an integer, when x is not an integer.
Prove that f is a periodic function with period 1 .
(ii)
Sketch the graph of f (x) , where 2 x 3 .
(iii)
Write down all the real number(s) x at which f is discontinuous. (6 marks)
(b) Define F( x)
x 0
f (t ) dt for all real numbers x . x2 x . 2
(i)
If 0 x 1 , prove that F( x)
(ii)
Is F a periodic function? Explain your answer.
(iii)
Evaluate
0
F( x) dx . (9 marks)
Type 2: Inverse Function 7.
(92I06) Let f : R R be bijective and a1 a2 an , where n 2 . (a) Suppose f is strictly increasing. Prove that its inverse f 1 is also
1 n f (ak ) an . n k 1
strictly increasing and deduce that a1 f 1
(b) Define h( x) p f ( x) q , where p , q R and p 0 . Show that h 1 ( x) f 1 (
xq ) p
1 n 1 1 n and deduce that h h(ak ) f f ( ak ) . n k 1 n k 1 1
(5 marks)
P. 118
HKAL Pure Mathematics Past Paper Topic: Functions
8.
(94II07) Let f ( x)
x 1
sin(cos t ) dt , where x [0,
2
) .
(a) Show that f is injective. (b) If g is the inverse function of f , find g(0) . (6 marks) 9.
(99II10) (a) Let f : R R be a strictly increasing bijective function. (i)
Show that the inverse function f
1
is also strictly
increasing. (ii)
Let a < b and t1 , t2 , , tn [a, b] , n 2 .
1 n f (ti ) b . n i 1
Show that a f 1
Find a necessary and sufficient condition on t1 , t2 , , tn such
1 n f (ti ) b . n i 1
that a f 1
(8 marks) 1
(b) Let g : R R be defined by g( x) x 3 . (i)
Show that g is bijective and strictly increasing. 1 1 1 23 n 3 Hence show that 1 n
(ii)
3
n for n 2 .
Find the area enclosed by graphs of y = g(x) and
y g 1 ( x) . (7 marks)
P. 119
HKAL Pure Mathematics Past Paper Topic: Functions
Type 3: Others 10.
(93I09) Let f : R R be a function such that f (x + y) = f (x) + f (y) for all
x, y R . (a) Show that (i)
f (0) = 0 ,
(ii)
f (x) = f (x) for all x R ,
(iii)
f (nx) = n f (x) for all n Z and x R . (5 marks)
(b) Show that if there exists K > 0 such that f (x) < K for all x R , then f ( x) 0 for all x R . (3 marks) (c) Suppose there exists K > 0 such that f (x) < K for all x [0, 1) . Let g(x) = f (x) f (1) x for all x R . Show that, for all x, y R , (i)
g(x + y) = g(x) + g(y) ,
(ii)
g(x + 1) = g(x) ,
(iii)
g(x) < K + | f (1) | .
Hence, or otherwise, show that f (x) = f (1) x for all x R . (7 marks)
P. 120
HKAL Pure Mathematics Past Paper Topic: Functions
11.
(97II10) Denote the open interval (0, ) by R+ . Let f : R+ R be a continuous function such that f (xy) = f (x) + f (y) for all x, y R+ . (a) Show that (i)
f (1) = 0 ,
(ii)
f (x1) = f (x) for all x R+ ,
(iii)
f (x n) = n f (x) for all n Z and x R+ . (6 marks)
(b) Show that f (x r) = r f (x) for all r Q and x R+ . (4 marks) (c) Show that f (x ) = f (x) for all R and x R+ . You may use the fact that for any R , there exists a sequence rn in Q such that lim rn . n
(3 marks) (d) If f (2) = 1 , show that f (x) = log2 x . (2 marks)
P. 121
HKAL Pure Mathematics Past Paper Topic: Functions
12.
(01II13) Let f : (0, ) (0, ) be a continuous function satisfying f[f ( x)] x and f (1 x)
f ( x) for all x . 1 f ( x)
(a) Show that for any n N and x R , f (n x)
f ( x) . 1 n f ( x) (3 marks)
(b) Define xn 1
f (1) n 1 for any n N . Show that f ( xn ) . 1 n f (1) n2
Hence, by considering lim xn , show that f (1) = 1 . n
Deduce that f (n)
1 1 and f n . n n (5 marks)
(c) For any q N and q 2 , let S(q) be the statement
p q for all p N with 0 < p < q .” q p
“f (i)
Show that S(2) is true.
(ii)
Assume that S(h) is true for 2 h q and h N . Use (a)
q 1 p for 0 < p < q + 1 . p q 1
to show that f
Hence deduce that S(q + 1) is true. (iii)
Use (a) to show that for any positive rational number x ,
f ( x)
1 . x (7 marks)
P. 122
HKAL Pure Mathematics Past Paper Topic: Functions
13.
(98II10) (a) Let f : R R be a continuous function. (i)
Show that
a 0
f (t b) dt
a b 0
b
f (t ) dt f (t ) dt for all 0
a, b R . (ii)
If f (x + y) = f (x) + f (y) for all x, y R , show that
x 0
x
f (t 1) dt f (1) x f (t ) dt for all x R . 0
Using (i), or otherwise, show that f (x) = f (1) x for all
xR . (8 marks) (b) Suppose g is a non-constant continuous function defined for all positive real numbers and g(xy) = g(x) + g(y) for all x, y > 0 . By considering the function f (t) = g(et ) for t R , show that
g( x) log a x for some a > 0 . (7 marks)
P. 123
HKAL Pure Mathematics Past Paper Topic: Sequences
Contents:
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
18. 19. 20. 21. 22. 23. 24. 25.
Classification
Chronological Order
Type 1: Method of Difference
90I02
Q. 6
90II02 91II02 92I10 94I04 94II05 Type 2: Partial Fractions 90I02 93I05 04I01 99I07 01I01 03I03 06I04 Type 3: Differentiation 95I05 05I03
90I11 90II02 90II11 91I02 91I07 91II02 91II08 92I10 92II13 93I05 93II06 94I04 94II05 95I05 95I12 95II10
Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.
21 1 27 32 22 2 30 3 33 7 18 4 5 13 29 17
Type 4: Sandwich Theorem 98II06 96II07 95II10 Type 5: Monotone Convergence Theorem SECTION A 93II06 01II03 04I02 SECTION B 90I11 91I07 01I10 02I13 05I09
96II07 96II13 98II06 99I07
Q. Q. Q. Q.
16 28 15 9
01I01
Q. 10
01I10
Q. 23
01II03
Q. 19
02I13 03I03 04I01 04I02 05I03 05I09 05II11 06I04
Q. Q. Q. Q. Q. Q. Q. Q.
06I05
Q. 26
26. 06I05 27. 90II11 28. 96II13
P. 124
24 11 8 20 14 25 31 12
HKAL Pure Mathematics Past Paper Topic: Sequences
29. 95I12 30. 91II08 31. 05II11 Type 6: Others 32. 91I02 33. 92II13
P. 125
HKAL Pure Mathematics Past Paper Topic: Sequences
1.
Type 1: Method of Difference (90II02)
Let n be a positive integer and x 0, . n 1 Show that sin x cot kx cot(k 1) x sin kx sin(k 1) x for all k 1, 2, 3, , n . Deduce that 1 1 1 sin nx . 2 sin x sin 2 x sin 2 x sin 3x sin nx sin(n 1) x sin x sin(n 1) x (5 marks) 2.
(91II02) Show that
(1 cos cos 2 cos n )sin
2
sin
(n 1) n cos . 2 2
Hence solve 1 cos cos 2 cos n 0 , 0 2 . (6 marks)
P. 126
HKAL Pure Mathematics Past Paper Topic: Sequences
3.
(92I10) Let
a1 , a2 ,
,
b1 , b2 ,
be two sequences of real numbers,
and b0 0 . (a) Show that
k
k 1
i 1
i 1
ai (bi bi1 ) ak bk (ai ai 1 )bi , k 2, 3, . (4 marks)
(b) Suppose
ai
is decreasing and bi K for all i , where K is
a constant. Show that
n
a (b b i 1
i
i 1
i
) K a1 2 ak , k = 1, 2, … . (6 marks)
(c) Using (b), or otherwise, show that for any positive integers n and p ,
n p
(1)i 3 . i 2n i n
(5 marks) 4.
(94I04)
a1 , a2 ,, an
and
b1 , b2 ,, bn
are two sequences of real
numbers. Define sk a1 a2 ak for k = 1, 2, … , n . (a) Prove that
n
a b k 1
k k
s1 (b1 b2 ) s2 (b2 b3 ) sn 1 (bn 1 bn ) snbn .
(b) If b1 b2 bn 0 and there are constants m and M such that
m sk M for k = 1, 2, … , n , n
prove that mb1 ak bk Mb1 . k 1
(5 marks)
P. 127
HKAL Pure Mathematics Past Paper Topic: Sequences
5.
(94II05) n For n = 1, 2, 3, … and R , let sn 3k 1 sin 3 k . 3 k 1
3 1 Using the identity sin 3 sin sin 3 , show that 4 4
sn
3n 1 sin n sin . 4 3 4
Hence, or otherwise, evaluate lim sn . n
(4 marks) 6.
Type 2: Partial Fractions (90I02) 1 (a) Resolve into partial fractions. x( x 1)( x 2) n
1 . n k 1 k ( k 1)( k 2)
(b) Evaluate lim
(6 marks) 7.
(93I05) Express
x4 in partial fractions. x 3x 2 2
Hence evaluate
1
k 1 k k 2
2
k 4 . 3k 2 (6 marks)
8.
(04I01) (a) Resolve
1 into partial fractions. (2 x 1)(2 x 1)(2 x 3)
(b) Prove that
n
1
1
1
1
(2k 1)(2k 1)(2k 3) 12 8(2n 1) 8(2n 3)
for
k 1
all positive integer n . Hence or otherwise, evaluate
1
(2k 1)(2k 1)(2k 3)
.
k 10
(6 marks)
P. 128
HKAL Pure Mathematics Past Paper Topic: Sequences
9.
(99I07) A sequence
a1
1 5
an
is defined as follows:
1 1 2n 5 an 1 an
and
(a) Show that an (b) Resolve
for n = 1, 2, 3, … .
1 for n = 1, 2, 3, … . n 4n 2
x2 into partial fractions. ( x 4 x) 2 2
n
Hence or otherwise, evaluate lim (k 2)ak 2 . n
k 1
(7 marks) 10. (01I01) (a) Resolve
8 into partial fractions. x( x 2)( x 2)
(b) Show that
2001
8
11
r (r 2)(r 2) 12
.
r 3
(5 marks) 11. (03I03) (a) Resolve (b) (i)
5x 3 into partial fractions. x( x 1)( x 3)
5k 3
n
3
k (k 1)(k 3) 2
Prove that
for any positive integer
k 1
n . (ii)
Evaluate
5k 3
k (k 1)(k 3)
.
k 1
(7 marks) 12. (06I04) (a) Resolve
(b) Express
9 x 36 into partial fractions. x( x 2)( x 3) n
9k 36
k (k 2)(k 3)
in the form A
k 1
B C D , n 1 n 2 n 3
where A , B , C and D are constants. (c) Is there a positive integer N such that
n
9k 36
k (k 2)(k 3) 8
?
k 1
Explain your answer. (7 marks) P. 129
HKAL Pure Mathematics Past Paper Topic: Sequences
Type 3: Differentiation 13. (95I05) (a) For x > 0 , prove that ln x x 1 where the equality holds if and only if x 1 . (b) Prove that ln
r 1 for r > 1 . r 1 r 1 n 1
1 for n = 2, 3, 4, … . k 1 k
Hence deduce that ln n
(7 marks) 14. (05I03) (a) By considering the function f (x) = x ln(x + 1) , or otherwise, prove that x ln(1 x) for all x > 1 . (b) Using (a), prove that the series
1
n
is divergent.
n 1
(7 marks) Type 4: Sandwich Theorem 15. (98II06) Let a1 = 2 , b1
3 2n 2n 1 an 1 , bn bn 1 for n 2 . and an 2n 1 2n 2
(a) Prove that an bn and anbn 2n 1 for n 1 . (b) Using (a), or otherwise, show that an 2 2n 1 for n 1 .
1 . n a n
Hence find lim
(7 marks) 16. (96II07) (a) Show that 1 ex e x 1 x for 0 < x 1 . (b) Using mathematical induction, or otherwise, prove that n 1 r xr ex n1 x x e (n 1)! r 0 r ! r 0 r ! n
for n = 0, 1, 2, … and x (0, 1] .
1 1 Hence show that lim 1 1 e . n 2 n! (7 marks)
P. 130
HKAL Pure Mathematics Past Paper Topic: Sequences
17. (95II10) For any > 0 , define a sequence of real numbers as follows:
a1 1 , an an 1
an 1
for n > 1 .
(a) Prove that (i)
an 2 an12 2 for n 2 ;
(ii)
an 2 2 2n 1 for n 1 . (2 marks)
(b) Using (a), show that for n 2 ,
2
n 1
an 2 2 2n 1 k 1
2 2k 1
. (3 marks)
(c) Prove that for k 1 , k 1 1 dx . 2 2 k 1 2k 1 2 x 1 (2 marks) an 2 exists and find the n n
(d) Using above results, show that lim limit.
an 2 exists. n n
State with reasons whether lim
(8 marks)
P. 131
HKAL Pure Mathematics Past Paper Topic: Sequences
Type 5: Monotone Convergence Theorem SECTION A 18. (93II06) (a) Show that if > , then n
1 m m 1 2
(b) Let un
. 1 1
nm , n = 1, 2, … . n m 1
Use (a), or otherwise, to show that
un un1 for n = 1, 2, … . Hence show that lim un exists. n
(7 marks) 19. (01II03) Let a1 = 1 and an 1
4 an 2 for n N . Show that 1 an 2 2
for n N . Hence show that
an
is convergent and find its limit. (6 marks)
20. (04I02) Let {an} be a sequence of positive real numbers, where
a1 1 and an
12an 1 12 , n = 2 , 3, 4 ,… . an 1 13
(a) Prove that an 3 for all positive integers n . (b) Prove that {an} is convergent and find its limit. (6 marks)
P. 132
HKAL Pure Mathematics Past Paper Topic: Sequences
SECTION B 21. (90I11) Let u0 and v0 be real numbers such that 0 v0 u0 . For n = 1, 2, … , define
un (a) (i) (ii)
2un 1vn 1 un 1 vn 1 , vn . un 1 vn 1 2 Show that un vn for n = 0, 1, 2, … . Deduce that
un
is monotonic decreasing and
vn
is
monotonic increasing. (iii)
Show that lim un and lim vn exist. n
n
(5 marks) (b) (i)
Prove that un vn
1 (u0 v0 ) for n = 0, 1, 2, … . 2n
(ii)
Prove that lim un lim vn .
(iii)
Evaluate lim(un vn ) and lim un .
n
n
n
n
(10 marks)
P. 133
HKAL Pure Mathematics Past Paper Topic: Sequences
22. (91I07) Given that
an
is an increasing sequence of positive numbers
and lim an L . Suppose sequences n
bn , cn
are defined such
that
b1 c1
1 a1 2
1 and bn (an 1 cn 1 ) , cn an1bn1 for n 2 . 2 (a) Show by induction that (i)
bn
(ii)
bn an and cn an for n 1 .
(b) Show that
and
bn
cn
and
are strictly increasing,
cn
are convergent.
Hence evaluate lim bn and lim cn . n
n
(7 marks)
P. 134
HKAL Pure Mathematics Past Paper Topic: Sequences
23. (01I10) a an 1 2bn 1 a 1 Let 1 and n , n = 2, 3, 4, … . b a b b 1 n n 1 n 1 1
(a) Show that for any positive integer n , (i)
an , bn > 0 and an 2 2bn 2 (1)n ;
(ii)
(1 2)n an bn 2 . (4 marks)
(b) For n = 1, 2, 3, … , define un
an . bn
un 2 . un 1
(i)
Show that un 1
(ii)
Show that u2 n1 2 and u2 n 2 .
(iii)
Show that un 2
3un 4 . 2un 3
Hence show that the sequence {u1 , u3 , u5 ,...} is strictly increasing and the sequence {u2 , u4 , u6 ,...} is strictly decreasing. (iv)
Show that the sequences {u1 , u3 , u5 ,...} and {u2 , u4 , u6 ,...} converge to the same limit. Find this limit. (11 marks)
P. 135
HKAL Pure Mathematics Past Paper Topic: Sequences
24. (02I13) Let {xn} be a sequence of real numbers such that x1 x2 and
3xn2 xn1 2 xn 0 for n = 1, 2, 3, … . (a) (i) (ii)
Show that for n 1 , xn 2 xn (1)n
2n 1 ( x1 x2 ) . 3n
Show that the sequence {x1 , x3 , x5 ,...} is strictly decreasing and that the sequence {x2 , x4 , x6 ,...} is strictly increasing. (5 marks)
(b) (i)
(ii)
For any positive integer n , show that x2 n x2 n1 . Show that the sequences {x1 , x3 , x5 ,...} and {x2 , x4 , x6 ,...} converge to the same limit. (6 marks)
(c) By considering
p n 1
( xn 2 xn ) or otherwise, find lim xn in n
terms of x1 and x2 . [You may use the fact, without proof, that from (b)(ii), lim xn n
exists.] (4 marks)
P. 136
HKAL Pure Mathematics Past Paper Topic: Sequences
25. (05I09) Let a1 and b1 be real numbers satisfying a1b1 > 0 . For each n = 1 , 2 , 3 , …. , define 2anbn a 2 bn 2 an 1 n and bn 1 . an bn an bn (a) Suppose a1 b1 > 0 . (i)
Prove that an bn for all n = 1 , 2 , 3 , … .
(ii)
Prove that the sequence {an} is monotonic decreasing and that the sequence {bn} is monotonic increasing.
(iii)
Prove that lim an and lim bn both exist.
(iv)
Prove that lim an lim bn .
(v)
Find lim(an bn ) and lim an in terms of a1 and b1 .
n
n
n
n
n
n
(12 marks) (b) Suppose a1 b1 < 0 . Do the limits of the sequences {an} and {bn} exist? Explain your answer. (3 marks) 26. (06I05) n
1 and yn = k 1 n k
For every positive integer n , define xn = n 1
1
nk
.
k 1
(a) Prove that the sequence {xn} is strictly increasing and that the sequence {yn} is strictly decreasing. (b) Prove that the sequence {xn} and {yn} converge to the same limit. (7 marks)
P. 137
HKAL Pure Mathematics Past Paper Topic: Sequences
27. (90II11) Let
an
be a sequence of real numbers such that 0 < a1 < 1
and an1 sin(an ) for all n = 1, 2, … . (a) Making use of the fact that sin x < x for 0 < x < 1 , show that lim an exists and find its value. n
(7 marks) (b) (i) (ii)
Evaluate lim x 0
x 2 sin 2 x . x 2 sin 2 x
1 1 Hence find lim 2 . 2 n a n 1 an
(5 marks) n
(c) It is known that if lim xn exists and equals L , then lim n
n
x i 1
i
n
also exists and equals L . Use this fact, or otherwise, to
show that lim nan 2 n
exists and find its value. (3 marks)
P. 138
HKAL Pure Mathematics Past Paper Topic: Sequences
28. (96II13) (a) Let x > 1 and define a sequence
an
an
by a1 x and
an 12 1 for n 2 . 2an 1
(i)
Show that an > 1 and an > an + 1 for all n .
(ii)
Show that lim an 1 . n
(8 marks) (b) Let f : [1, ) R be a continuous function satisfying x2 1 f ( x) f 2x
for all x 1 .
Using (a), show that f (x) = f (1) for all x 1 . (7 marks)
P. 139
HKAL Pure Mathematics Past Paper Topic: Sequences
29. (95I12) Let p > 0 and p 1 .
an
is a sequence of positive numbers
a0 2 defined by , n = 1, 2, 3, … . 1 1 an p n p an 1 (a) Prove that lim an 0 if the limit exists. n
(2 marks) (b) (i) (ii)
If 2 a0 a1 a2 , show that lim an does not exist. n
If ak 1 ak for some k 1 , show that an1 an for
n k and deduce that lim an 0 . n
(4 marks) (c) (i) (ii)
If 0 p 1 , show that lim an does not exist. n
If p 2 , show that lim an 0 . n
(4 marks) (d) Suppose 1 < p < 2 . (i)
Prove by mathematical induction that an
2 for n p 1
0 . (ii)
Prove that lim an 0 . n
(5 marks)
P. 140
HKAL Pure Mathematics Past Paper Topic: Sequences
30. (91II08) (a) Let f (x) = x ln(1 + x) . Prove that f (x) 0 for all x > 1 . (3 marks)
1 1 1 (b) Let an 1 ln(n 1) 2 3 n 1 1 1 and bn 1 ln n . 2 3 n Show that
an
is increasing and
Hence prove that
an
and
bn
bn
is decreasing.
are convergent and have
the same limit. (5 marks) (c) (i)
Using (b) or otherwise, show that 1 1 1 k 1 lim ln n kn 1 kn 2 kn n k where k is a positive integer.
(ii)
1 1 1 1 1 . Evaluate lim 1 n 2n 1 2n 2 3 4 (7 marks)
P. 141
HKAL Pure Mathematics Past Paper Topic: Sequences
31. (05II11) Let fn : R R be defined by f n ( x) cos x cos2 x cosn1 x , where n 1, 2, 3, . (a) (i)
(ii)
Prove that fn is strictly decreasing on 0, . 2 Prove that the equation f n ( x) = 1 has one and only one
root in
0, . 2 (4 marks) (b) For each n = 1, 2, 3,… , let n be the root of the equation
f n ( x) = 1 in 0, . 2 2 . 3
(i)
Prove that cos 1
(ii)
Is the sequence {n} monotonic? Explain your answer.
(iii)
Find lim cos n n
(iv)
Prove that the sequence {n} is convergent and find
n
its limit. (11 marks)
P. 142
HKAL Pure Mathematics Past Paper Topic: Sequences
Type 6: Others 32. (91I02) 1 Let f ( x) . ( x 1)(2 x) Express f (x) into partial fractions. Hence, or otherwise, determine ak and bk ( k 0, 1, 2, ) such that
f ( x) ak x k when x 1 k 1
bk when x 2 . k k 0 x
and f ( x)
(7 marks)
P. 143
HKAL Pure Mathematics Past Paper Topic: Sequences
33. (92II13) Suppose
ak
is a sequence of positive numbers such that
a0 a1 1 , and ak ak 1 ak 2 for k = 2, 3, … . n 1 1 Let x and Sn ( x) ak x k . 3 3 k 0
(a) For k = 0, 1, 2, … , prove that
ak 1 2ak , and deduce that ak 2k . Hence prove that Sn ( x) 3 for n = 0, 1, 2, … . (6 marks) (b) Prove that lim Sn ( x) exists and equals n
1 . 1 x x2
[Hint: Put y = x for the case when x < 0 .] (5 marks) (c) Evaluate (i)
(ii)
(iii)
1 ak 5 k 0
k
,
1 (1) ak 5 k 0
k
k
1 a2 k 25 k 0
,
k
. (4 marks)
P. 144
HKAL Pure Mathematics Past Paper Topic: Limit
Contents: Classification
Chronological Order
Type 1: L’Hospital’s Rule
90II04a
(without Diff. of Integrals)
91II05
Q.
1
Q.
14
1.
90II04a
2.
92II01a
92II01a
Q.
2
3.
93II01
92II01b
Q.
21
4.
94II01
93II01
Q.
3
5.
01II01a
94II01
Q.
4
6.
03II01a
95II01a
Q.
13
7.
05II01a
96II06
Q.
20
8.
98II01
97II05a
Q.
10
9.
99II01
97II05b
Q.
15
10.
97II05a
98II01
Q.
8
11.
02II06a
99II01
Q.
9
12.
04II01a
00II05
Q.
16
13.
95II01a
01II01a
Q.
5
14.
91II05
01II01b
Q.
22
Type 2: L’Hospital’s Rule
02II06a
Q.
11
(with Diff. of Integrals)
02II06b
Q.
19
15.
97II05b
16.
00II05
03II01a
Q.
6
17.
05II01b
03II01b
Q.
23
18.
06II01a
04II01a
Q.
12
19.
02II06b
04II01b
Q.
24
20.
96II06
05II01a
Q.
7
Type 3: Sandwich Theorem
05II01b
Q.
17
21.
92II01b
06II01a
Q.
18
22.
01II01b
06II01b
Q.
25
23.
03II01b
24.
04II01b
25.
06II01b
P. 145
HKAL Pure Mathematics Past Paper Topic: Limit
Type 1: L’Hospital’s Rule (without Diff. of Integrals) 1.
(90II04a)
1 1 . x tan x
Evaluate lim x 0
(3 marks) 2.
(92II01a)
tan x x . x 0 x 2 sin x
Evaluate lim
(3 marks) 3.
(93II01) Evaluate
x
2 , cos x
(a) lim x
2
(1 mx)n (1 nx)m , where m, n 2 . x 0 x2
(b) lim
(5 marks) 4.
(94II01) Evaluate (a) lim x 1
1 x , 1 5 x
(b) lim (sec x tan x) . x
2
(4 marks) 5.
(01II01a) Find
e lim x 0
x
e x
2
1 cos 2 x
. (4 marks)
6.
(03II01a) Evaluate
1 1 . sin x tan x
Evaluate lim x 0
(3 marks)
P. 146
HKAL Pure Mathematics Past Paper Topic: Limit
7.
(05II01a) Let f (x) = x x for all x > 0 . Prove that f '( x) x (1 ln x) . x
xx x . x 1 ln x x 1
Hence evaluate lim
(5 marks) 8.
(98II01) Evaluate
e x 1 sin x (a) lim , x 0 x 1
3e x 2 x (b) lim . x 0 5 (6 marks) 9.
(99II01)
x sin x . x 0 1 cos x
(a) Evaluate lim
(b) Using (a) or otherwise, evaluate lim (1 cos x) x 0
1 ln x
. (6 marks)
10.
(97II05a) Evaluate lim (sin x) x . x 0
(4 marks) 11.
(02II06a)
1 Find lim x 0 x
sin x
. (3 marks)
12.
(04II01a) 1
Evaluate lim (tan 3x cos 4 x) x . x 0
(3 marks)
P. 147
HKAL Pure Mathematics Past Paper Topic: Limit
13.
(95II01a) 1
ax bx 1 x (a) Evaluate lim where a, b > 0 . n 3 (3 marks) 14.
(91II05) 1
ax 1 x Find lim for each of the following cases: x a 1 (a) 0 < a < 1 , (b) a > 1 (6 marks) Type 2: L’Hospital’s Rule (with Diff. of Integrals 15.
(97II05b)
1 3 x
Evaluate lim x 0
x 0
et dt 2
1 . x2 (3 marks)
16.
(00II05) Let k be a positive integer. Evaluate (a)
d x cos t 2dt , dx 0
(b)
d y2 k cos t 2dt , 0 dy
(c) lim y 0
1 y 2k
y2 k
cos t 2dt .
0
(6 marks) 17.
(05II01b) Evaluate
lim
x 0
t sin(sin t ) dt x3
x 0
. (2 marks)
18.
(06II01a) Evaluate
lim x 0
3 x 1 1
t 5 t 3 1 dt ln( x 1)
. (4 marks)
P. 148
HKAL Pure Mathematics Past Paper Topic: Limit
19.
(02II06b)
sin t Let f (t ) t 1 Find
lim
x 0
if t 0 , if t 0 .
f (t ) dt x x3
x 0
. (3 marks)
20.
(96II06) Let f (x) be a function with continuous second order derivative.
1 f (t ) dt ( x a)(f ( x) f (a)) f ''(a) a 2 (a) Prove that lim . 3 x a ( x a) 12
x
(b) Suppose there exists a constant K such that , for all x and a ,
1 f (t ) dt ( x a)(f ( x) f (a)) K ( x a) 4 . a 2 x
Show that f is a polynomial of degree not greater than 1 . (6 marks) Type 3: Sandwich Theorem 21.
(92II01b) Prove that | x sin
1 | | x | for all x 0 . x
1 1 sin x . Hence evaluate lim x x 0 1 1 sin x x (3 marks) 22.
(01II01b) Prove that lim x 2 cos x 0
1 0 . x (2 marks)
23.
(03II01b)
x cos x . x x cos x
Evaluate lim
(3 marks)
P. 149
HKAL Pure Mathematics Past Paper Topic: Limit
24.
(04II01b) Evaluate lim (cos 2004 x cos x ) . x
(4 marks) 25.
(06II01b) Evaluate lim sin x sin x 0
1 . x (2 marks)
P. 150
HKAL Pure Mathematics Past Paper Topic: Differentiation
Contents: Classification
Chronological Order
Type 1: Differentiability
90II12
Q.
33
1.
99II05
90II13
Q.
14
2.
00II08
91II07
Q.
9
3.
02II02
91II10
Q.
34
4.
04II02
91II13
Q.
16
5.
96II04
92II07
Q.
17
6.
05II02
92II08
Q.
31
7.
97II06
93II07
Q.
13
8.
01II06
93II08
Q.
35
9.
91II07
93II13
Q.
49
Type 2: Higher Derivatives
94II10
Q.
38
10.
95II06
94II12
Q.
22
11.
96II01
95II06
Q.
10
12.
99II04
95II09
Q.
27
13.
93II07
96II01
Q.
11
14.
90II13
96II04
Q.
5
15.
06II09
96II08
Q.
25
16.
91II13
96II10
Q.
41
Type 3: Functional Equations
97II06
Q.
7
17.
92II07
97II08
Q.
39
18.
02II03
98II08
Q.
36
19.
03II08
98II13
Q.
48
20.
05II08
99II04
Q.
12
21.
00II04
99II05
Q.
1
22.
94II12
99II08
Q.
32
Type 4: Curve Sketching
99II13
Q.
42
Set 1
00II04
Q.
21
23.
06II07
00II08
Q.
2
24.
02II08
00II09
Q.
26
25.
96II08
00II12
Q.
46
26.
00II09
00II14
Q.
40
27.
95II09
01II06
Q.
8
28.
03II07
01II08
Q.
37
29.
04II07
01II11
Q.
44
30.
05II07
02II02
Q.
3
Set 2
02II03
Q.
18
P. 151
HKAL Pure Mathematics Past Paper Topic: Differentiation
31. \ 92II08
02II08
Q.
24
32.
99II08
02II12
Q.
47
Set 3
03II07
Q.
28
33.
90II12
03II08
Q.
19
34.
91II10
04II02
Q.
4
35.
93II08
04II07
Q.
29
36.
98II08
04II12
Q.
43
37.
01II08
05II02
Q.
6
38.
94II10
05II07
Q.
30
39.
97II08
05II08
Q.
20
Type 5: Monotoncity
06II07
Q.
23
00II14
06II09
Q.
15
Type 6: Mean Value Theorem
06II11
Q.
45
40.
Set 1 41.
96II10
42.
99II13 Set 2
43.
04II12
44.
01II11
45.
06II11 Set 3
46.
00II12
47.
02II12 Set 4
48.
98II13
49.
93II13
P. 152
HKAL Pure Mathematics Past Paper Topic: Differentiation
Type 1: Differentiability 1.
(99II05)
1 2 x cos Let f ( x) x ax b
if x 0 , if x 0
be differentiable at x = 0 . Find a and b . (6 marks) 2.
(00II08)
x 2 bx c if x 0 , Let f ( x) sin x 2 x if x 0 . x (a) If f is continuous at x = 0 , find c . (b) If f (0) exists, find b . (6 marks) 3.
(02II02)
if x , ax 2 Let f ( x) ebx sin x if x . 2 b a , show that e2 . 2 2 Furthermore, if f is differentiable at , find the values of a and b . 2
If f is continuous at
(5 marks) 4.
(04II02)
x 2 ax b when x 1 , Let f ( x) sin x when x 1 . If f is differentiable at 1 , find a and b . (6 marks)
P. 153
HKAL Pure Mathematics Past Paper Topic: Differentiation
5.
(96II04)
1 3 x sin Let f ( x) x 0
for x 0
.
for x 0
(a) Evaluate f (x) for x 0 . (b) Prove that f (0) exists. (c) Is f (x) continuous at x = 0 ? Explain. (6 marks) 6.
(05II02) Let a be a constant and f : R R be defined by
x2 x a when x , f ( x) 2 a cos x when x . It is known that f is continuous everywhere. (a) Prove that a
4
.
(b) Prove that f is differentiable at . (c) Is f continuous at ? Explain your answer. (7 marks) 7.
(97II06) Let f ( x) x x . (a) Find f (x) for x > 0 and x < 0 respectively. (b) Prove that f (0) exists. (c) Prove that f (x) is continuous at x = 0 . (6 marks)
8.
(01II06)
x3 12 x when x 2 , Let f : R R be defined by f ( x) x x 2 x when x 2 . e e (a) Show that lim f' ( x) lim f' ( x) 0 . x 2
x 2
(b) Is f differentiable at x = 2 ? Explain your answer. (5 marks)
P. 154
HKAL Pure Mathematics Past Paper Topic: Differentiation
9.
(91II07) Let f : R R . (a) If c R and f ( x) f (c) ( x c) 2 for all x R , prove that
f' (c) 0 . (b) If f ( x) f ( y) ( x y) 2 for all x , y R , prove that f is a constant function. (5 marks) Type 2: Higher Derivatives 10.
(95II06)
x 1 Let r be a real number. Define y for x > 1 . x 1 r
(a) Show that
dy 2ry . dx x 2 1
(b) For n = 1, 2, 3 … , show that
( x2 1) y ( n1) 2(nx r ) y ( n) (n2 n) y ( n1) 0 , (0) where y y and y ( k )
dk y for k 1 . dx k (5 marks)
11.
(96II01) Let f ( x) x e
n ax
Evaluate f
(2 n )
where a is real and n is a positive integer.
(0) . (4 marks)
12.
(99II04) Let f ( x)
2x . x 1 2
(a) Resolve f (x) into partial fractions. (b) Find f
(n)
(0) , where n = 1, 2, 3, … . (5 marks)
P. 155
HKAL Pure Mathematics Past Paper Topic: Differentiation
13.
(93II07) Let n be a positive integer and u(x) be a function such that
u' ( x), u'' ( x), , u ( n ) ( x) exist. (a) Given that y( x) u( x)e
qx
, where q is a real number, express
y( n ) ( x) in terms of u( x), u' ( x), u'' ( x), , u ( n) ( x) . (b) By putting u( x) e
px
, where p is a real number, use (a) to prove
n the Binomial Theorem, i.e. ( p q)
n
C r 0
n r
p r q n r . (6 marks)
P. 156
HKAL Pure Mathematics Past Paper Topic: Differentiation
14.
(90II13) Let f ( x)
1 1 x2
for all x R .
Let f ( n ) denote the nth derivative of f for n = 1, 2, … , and f (0) f . (a) Prove that
(1 x2 )f' ( x) x f ( x) 0 . Deduce that
(1 x2 )f ( n1) ( x) (2n 1) x f ( n) ( x) n2 f ( n1) ( x) 0 for n = 1, 2, … . (2 marks) (b) Define Pn ( x) (1 x 2 ) (i)
n
1 2
f ( n ) ( x) for n = 0, 1, 2, … .
For n = 0, 1, 2, … , prove that
Pn1 ( x) (1 x2 )Pn' ( x) (2n 1) x Pn ( x) . Deduce that Pn(x) is a polynomial of degree n with leading coefficient (1) n ! . n
(ii)
For n = 1, 2, … , show that
Pn1 ( x) (2n 1) x Pn ( x) n2 (1 x 2 )Pn1 ( x) 0 and find Pn(0) . (iii)
For n = 1, 2, … , prove that
Pn' ( x) n2 Pn1 ( x) and deduce that , for r = 1, 2, … , n ,
Pn( r ) ( x) (1)r (n(n 1)(n r 1))2 Pnr ( x) . (iv)
Hence show that Pn ( x) is either an odd function or an even function for n 1, 2, ... . (Note:
Pn ( r ) (0) is the coefficient of xr in Pn(x) .) r! (13 marks)
P. 157
HKAL Pure Mathematics Past Paper Topic: Differentiation
15.
(06II09) (a) Prove that lim x 0
xn e
1 x
0 for any positive integer n . (3 marks)
0 (b) Let f ( x) 1 x e
when x 0 , when x 0 .
(i)
Find f (x) for all x 0 .
(ii)
Prove that f (0) = 0 . Hence prove that f (x) is continuous at x = 0 . 1
(iii)
1 for any x
For any x > 0 , prove that f (n)(x) = e x p n
positive integer n , where p n (t ) is a polynomial in t . (iv)
Prove that f (n)(0) = 0 for any positive integer n . (12 marks)
P. 158
HKAL Pure Mathematics Past Paper Topic: Differentiation
16.
(91II13) 2 n (n) For n = 0, 1, 2, … , let R n ( x) ( x 1) and Pn ( x) R n ( x) .
(a) (i)
Show that Pn(x) is a polynomial in x of degree n ,
n 0, 1, 2, . (ii)
Prove by induction that every polynomial of degree n can be expressed as
n
P ( x) j 0
j
j
where j R . (4 marks)
(b) Prove by induction that
(1 x2 )R (nk 2) ( x) 2 x(n k 1)R (nk 1) ( x) (k 1)(2n k )R (nk ) ( x) 0 for k = 0, 1, 2, … . Hence deduce that
[(1 x2 )Pn' ( x)]' n(n 1)Pn ( x) ……… (*) (5 marks) (c) (i)
Using (*) or otherwise, show that if m 0 , n 0 , then 1
1
1
1
n(n 1) Pm ( x)Pn ( x) dx (1 x 2 ) Pn' ( x)Pm' ( x) dx . (ii)
Deduce that if m , n are distinct non-negative integers, then
1
P ( x)Pn ( x) dx 0 .
1 m
(6 marks)
P. 159
HKAL Pure Mathematics Past Paper Topic: Differentiation
Type 3: Functional Equations 17.
(92II07) Let f be a differentiable function such that
f ( x y) f ( x) f ( y) 3xy( x y) for all x , y R .
f ( h) . h 0 h
(a) Show that f' (0) lim
(b) Hence, or otherwise, show that for all x R , f (x) = f (0) + 3x2 , and deduce that f (x) = f (0) x + x3 . (6 marks) 18.
(02II03) Let f : R R be a continuous function satisfying the following conditions:
f ( x) 1 1 ; x 0 x
(i)
lim
(ii)
f (x + y) = f (x) f (y)
for all x and y .
(a) Prove that f (x) exists and f (x) = f (x) for every x R . (b) By considering the derivative of
f ( x) x , show that f ( x) e . x e (5 marks)
P. 160
HKAL Pure Mathematics Past Paper Topic: Differentiation
19.
(03II08) Let f : R R be a non-constant function satisfying the following conditions: (1)
f (x + y) = f (x) + f (y) + f (x)f (y) for all x , y R .
(2)
lim
(a) (i) (ii)
f ( h) a , where a R . h 0 h
Prove that f (0) 1 f ( x) 0 for all x R . Prove that f (0) = 0 and that f (x) 1 for all x R . (5 marks)
x x , or otherwise, prove that 2 2
(b) By considering f ( x) f
f ( x) 1 for all x R . (2 marks) (c) Prove that f is differentiable everywhere and that
f '( x) a 1 f ( x) for all x R . Deduce that a 0 . (4 marks) (d) By considering the derivative of the function ln 1 f ( x) , prove that f ( x) e 1 for all x R . ax
(4 marks)
P. 161
HKAL Pure Mathematics Past Paper Topic: Differentiation
20.
(05II08) Let f : R R be a function satisfying the following conditions: (1)
f ( x y) e x f ( y) e y f ( x) for all x , y R ;
(2)
lim
f ( h) = 2005 . h 0 h
(a) Find f (0) . (1 mark) (b) Find lim f (h) . Hence prove that f is a continuous function. h 0
(4 marks) (c) (i)
Prove that f is differentiable everywhere and that
f '( x) 2005e x f ( x) for all x R .
(ii)
Let n be a positive integer. Using (c)(i), find f
(n)
(0) .
(6 marks)
f ( x) , find f (x) . ex
(d) By considering the derivative of the function
(4 marks) 21.
(00II04) Let f and g be differentiable functions defined on R and satisfying the following conditions: A.
f (x) = g(x) for x R ;
B.
g(x) = f (x)
C.
f (0) = 0 and g(0) = 1 .
for x R ;
By differentiating h( x) [f ( x) sin x] [g( x) cos x] , or otherwise, 2
2
show that f (x) = sin x and g(x) = cos x for x R . (5 marks)
P. 162
HKAL Pure Mathematics Past Paper Topic: Differentiation
22.
(94II12) Let f : R R be a continuously differentiable function satisfying the following conditions for all x R . A.
f (x) > 0 ;
B.
f (x + 1) = f (x) ;
C.
x x 1 f ( )f ( ) f ( x) . 4 4
Define g( x)
d ln f ( x) for x R . dx
(a) Show that for all x R , (i)
f (x + 1) = f (x) ;
(ii)
g(x + 1) = g(x) ;
(iii)
1 x x 1 [g( ) g( )] g( x) . 4 4 4 (8 marks)
(b) Let M be a constant such that g( x) M for all x [0,1] . (i)
Using (a), or otherwise, show that g( x)
M for all 2
xR . Hence deduce that g( x) 0 for all x R . (ii)
Show that f (x) = 1 for all x R . (7 marks)
P. 163
HKAL Pure Mathematics Past Paper Topic: Differentiation
Type 4: Curve Sketching Set 1 23.
(06II07)
x2 x 6 Let f ( x) x6
(x 6) .
(a) Find f (x) and f (x) . (2 marks) (b) Solve each of the following inequalities: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find the relative extreme point(s) of the graph of y = f (x) . (2 marks) (d) Find the asymptote(s) of the graph of y = f (x) . (3 marks) (e) Sketch the graph of y = f (x) . (2 marks) (f) Find the area of the region bounded by the graph of y = f (x) and the x-axis. (3 marks)
P. 164
HKAL Pure Mathematics Past Paper Topic: Differentiation
24.
(02II08) Let f ( x) x 2
8 x 1
(x 1) .
(a) Find f (x) and f (x) . (2 marks) (b) Determine the range of values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (1 marks) (e) Sketch the graph of f (x) . (2 marks) (f) Let g(x) = f (|x|)
(x 1) .
(i)
Is g(x) differentiable at x = 0 ? Why?
(ii)
Sketch the graph of g(x) . (5 marks)
P. 165
HKAL Pure Mathematics Past Paper Topic: Differentiation
25.
(96II08) Let f ( x)
( x 1)3 . ( x 1) 2
(a) Find f (x) and f (x) for x 1 . (2 marks) (b) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (2 marks) (e) Sketch the graph of f (x) . (2 marks) (f) Let g( x) f ( x) . Does g(1) exist? Find the asymptote(s) and sketch the graph of g(x) . (4 marks)
P. 166
HKAL Pure Mathematics Past Paper Topic: Differentiation
26.
(00II09) Let f ( x)
x . (1 x 2 )2
(a) Find f (x) and f (x) . (2 marks) (b) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) > 0 . (3 marks)
(c) Find all relative extreme points, points of inflexion and asymptotes of y f ( x) . (4 marks) (d) Sketch the graph of f (x) . (3 marks) (e) Let g( x) f ( x) . (i)
Does g(0) exist? Why?
(ii)
Sketch the graph of g(x) . (3 marks)
P. 167
HKAL Pure Mathematics Past Paper Topic: Differentiation
27.
(95II09) Let f ( x)
| x| , where x 1 . ( x 1) 2
(a) (i)
Find f (x) and f (x) for x > 0 .
(ii)
Find f (x) and f (x) for x < 0 .
(iii)
Show that f (0) does not exist. (4 marks)
(b) Determine the values of x for each of the following cases: (i)
f (x) < 0 ,
(ii)
f (x) > 0 ,
(iii)
f (x) < 0 ,
(iv)
f (x) > 0 . (4 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of and sketch the graph of f (x) . (4 marks)
P. 168
HKAL Pure Mathematics Past Paper Topic: Differentiation
28.
(03II07) Let f ( x)
x | x 1| x2
(a) (i)
Find f (x) for x 1 .
( x 2 ) .
(ii)
Is f differentiable at 1 ? Explain your answer.
(iii)
Find f (x) for x 1 . (4 marks)
(b) Determine the range of values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of the graph of f (x) . (3 marks) (e) Sketch the graph of f (x) . (2 marks)
P. 169
HKAL Pure Mathematics Past Paper Topic: Differentiation
29.
(04II07)
| x | x3 Let f ( x) 3 x 2 (a) (i)
(x 3 2) .
Find f (x) and f (x) for x > 0 .
(ii)
Write down f (x) and f (x) for x < 0 .
(iii)
Prove that f (0) exists.
(iv)
Does f (0) exist? Explain your answer. (5 marks)
(b) Determine the range of values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find the relative extreme point(s) and point(s) of inflexion f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (3 marks) (e) Sketch the graph of f (x) . (2 marks)
P. 170
HKAL Pure Mathematics Past Paper Topic: Differentiation
30.
(05II07) Define f ( x) x | x | (a) (i) (ii)
x for x < 2 or x > 0 . x2
Find f (x) and f (x) for x > 0 . Write down f (x) and f (x) for x < 2 . (4 marks)
(b) Solve each of the following inequalities: (i)
f (x) > 0 ,
(ii)
f (x) > 0 . (3 marks)
(c) Find the relative extreme point(s) of the graph of y = f (x) . (2 marks) (d) Find the asymptote(s) of the graph of y = f (x) . (4 marks) (e) Sketch the graph of y = f (x) . (2 marks)
P. 171
HKAL Pure Mathematics Past Paper Topic: Differentiation
Set 2 31. \ (92II08) Let f ( x) xe x for x R . 2
(a) Find f (x) and f (x) . (2 marks) (b) Determine the values of x for each of the following cases: (i)
f (x) = 0 ,
(ii)
f (x) > 0 ,
(iii)
f (x) < 0 ,
(iv)
f (x) = 0 ,
(v)
f (x) > 0 ,
(vi)
f (x) < 0 . (3 marks)
(c) Find all relative extrema and points of inflexion of f (x) . (3 marks) (d) Find the asymptote of the graph of f (x) . (1 mark) (e) Sketch the graph of f (x) . (3 marks) (f) Hence sketch the curve x y ( x y )e
1 ( x y )2 2
. (3 marks)
P. 172
HKAL Pure Mathematics Past Paper Topic: Differentiation
32.
(99II08) 1
Let f ( x) xe x for x 0 . (a) Find lim f ( x) and show that f (x) as x 0 + . x 0
(3 marks) (b) Find f (x) and f (x) for x 0 . (2 marks) (c) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) > 0 . (3 marks)
(d) Find all relative extrema of f (x) . (2 marks) (e) Find all asymptotes of the graph of f (x) . (3 marks) (f) Sketch the graph of f (x) . (2 marks)
P. 173
HKAL Pure Mathematics Past Paper Topic: Differentiation
Set 3 33.
(90II12) 2
Let f ( x) (2 x 1) x 3 for x R . (a) Find f (x) and f (x) for x 0 . (2 marks) (b) Show that f (0) does not exist. (1 mark) (c) Determine those values of x such that (i)
f (x) = 0 ,
(ii)
f (x) > 0 ,
(iii)
f (x) < 0 ,
(iv)
f (x) = 0 ,
(v)
f (x) > 0 ,
(vi)
f (x) < 0 . (3 marks)
(d) Find the relative extrema and points of inflexion of the function. (4 marks) (e) Show that the graph of the function has no asymptotes. (2 marks) (f) Using the result of (a), (b), (c), (d) and (e), sketch the graph of the function. (3 marks)
P. 174
HKAL Pure Mathematics Past Paper Topic: Differentiation
34.
(91II10) Let f ( x)
3
x3 x 2 x 1 .
(a) Find the roots of f (x) = 0 . (1 mark) (b) Find f (x) for x 1, 1 . Show that both f (1) and f (1) do not exist. (3 marks) (c) Determine the values of x for each of the following cases: (i)
f (x) = 0 ,
(ii)
f (x) > 0 ,
(iii)
f (x) < 0 , (3 marks)
(d) Find the relative minimum, the relative maximum and the point of inflexion of f (x) . (4 marks) (e) Find the asymptote of the graph of f (x) . (2 marks) (f) Sketch the graph of f (x) . (2 marks)
P. 175
HKAL Pure Mathematics Past Paper Topic: Differentiation
35.
(93II08) Let f ( x)
3
x 2 x3 .
(a) Find f (x) and f (x) . (2 marks) (b) Show that both f (0) and f (1) do not exist. (2 marks) (c) Determine the sets of values of x such that: (i)
f (x) = 0 ,
(ii)
f (x) > 0 ,
(iii)
f (x) < 0 ,
(iv)
f (x) = 0 ,
(v)
f (x) > 0 ,
(vi)
f (x) < 0 . (3 marks)
(d) Find the relative extreme point(s) and the point(s) of inflexion on the curve y f ( x) . (3 marks) (e) Find the asymptote(s) of the curve y = f (x) . (3 marks) (f) Sketch the curve y = f (x) . (2 marks)
P. 176
HKAL Pure Mathematics Past Paper Topic: Differentiation
36.
(98II08) 1
2
Let f ( x) x 3 ( x 1) 3 . (a) (i) (ii)
Find f (x) for x 1, 0 . Show that f'' ( x)
2 5 3
9 x ( x 1)
4 3
for x 1, 0 .
(2 marks) (b) Determine with reasons whether f (1) and f (0) exist or not. (2 marks) (c) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(d) Find all relative extrema and points of inflexion of f (x) . (3 marks) (e) Find all asymptotes of the graph of f (x) . (2 marks) (f) Sketch the graph of f (x) . (3 marks)
P. 177
HKAL Pure Mathematics Past Paper Topic: Differentiation
37.
(01II08) 2
1
Let f ( x) x 3 (6 x) 3 . (a) (i)
Find f (x) for x 0, 6 .
(ii)
Show that f (0) and f (6) do not exist.
(iii)
Show that f ''( x)
8 4 3
x (6 x)
5 3
for x 0, 6 .
(4 marks) (b) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (3 marks)
(c) Find all relative extreme points and points of inflexion of f (x) . (3 marks) (d) Find all asymptotes of the graph of f (x) . (2 marks) (e) Sketch the graph of f (x) . (3 marks)
P. 178
HKAL Pure Mathematics Past Paper Topic: Differentiation
38.
(94II10) Let f ( x)
x2 ,xR . x2 1
(a) (i)
Evaluate f (x) for x 0 .
3
Prove that f (0) does not exist. (ii)
Determine those values of x for which f (x) > 0 and those values of x for which f (x) < 0 .
(iii)
Find the relative extreme points of f (x) . (8 marks)
(b) (i)
Evaluate f (x) for x 0 . Hence determine the points of inflexion of f (x) .
(ii)
Find the asymptote of the graph of f (x) . (4 marks)
(c) Using the above results, sketch the graph of f (x) . (3 marks)
P. 179
HKAL Pure Mathematics Past Paper Topic: Differentiation
39.
(97II08) 2
x3 Let f ( x) x 1 (a) (i)
(x 1) .
Find f (x) for x 1, 0 . Does f (0) exist? Explain.
(ii)
Show that f ''( x)
2(2 x 2 8 x 1) 4 3
9 x ( x 1)
for x 1, 0 .
3
(4 marks) (b) Determine the values of x for each of the following cases: (i)
f (x) > 0 ,
(ii)
f (x) < 0 ,
(iii)
f (x) > 0 ,
(iv)
f (x) < 0 . (4 marks)
(c) Find the relative extrema and points of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of the graph of f (x) . (1 marks) (e) Sketch the graph of f (x) . (3 marks)
P. 180
HKAL Pure Mathematics Past Paper Topic: Differentiation
Type 5: Monotoncity 40.
(00II14) 1 x x x a b Let a > b > 0 and define f ( x) 2 ab
(a) (i)
for x 0 , for x 0 .
Evaluate lim f ( x) . x 0
Hence show that f is continuous at x = 0 . (ii)
Show that lim f ( x) a . x
(6 marks) (b) Let h(t ) (1 t )ln(1 t ) (1 t )ln(1 t ) for 0 t < 1 and
g( x) ln f ( x) for x 0 . (i)
Show that h(t) > h(0) for 0 < t < 1 .
(ii)
For x > 0 , let t
ax bx , Show that 0 < t < 1 and ax bx
a x ln a x b x ln b x 2 h(t ) 2 ln x x x x a b a b (iii)
.
Show that for x > 0 ,
x 2g' ( x)
a x ln a x b x ln b x 2 ln x . x x x a b a b
Hence deduce that f (x) is strictly increasing on [0, ) . (9 marks)
P. 181
HKAL Pure Mathematics Past Paper Topic: Differentiation
Type 6: Mean Value Theorem Set 1 41.
(96II10) (a) Let f (x) be a function such that f (x) is strictly decreasing for
x0 . (i)
Using Mean Value Theorem, or otherwise, show that
f '(k 1) f (k 1) f (k ) f '(k ) for k 1 . (ii)
Using (i), show that for any integer n 2,
f '(2) f '(3) f '(n) f (n) f (1) f '(1) f '(2) f '(n 1) . (5 marks) (b) Define H n 1 (i)
1 1 1 for any positive integer n . 2 3 n
Using (a), or otherwise, show that
H n 1 ln n H n
1 n
for n 2 .
Hn . ln n
Hence, evaluate lim n
(ii)
Define n H n ln n . Show that
n
is a decreasing sequence and lim n n
exists. (10 marks)
P. 182
HKAL Pure Mathematics Past Paper Topic: Differentiation
42.
(99II13) Let f (x) be a differentiable function on R such that f '( x) f ( x) for all x R . (a) Suppose a 0 and f (a) = 0 . Let x (a, a + 1) . (i)
Using Mean Value Theorem or otherwise, show that there exists 1 (a, x) such that f ( x) f (1 ) ( x a) .
(ii)
Using (a)(i) or otherwise, show that for each n = 1, 2, 3, … , there exists n (a, x) such that f ( x) f (n ) ( x a)n .
(iii)
Using (a)(ii) or otherwise, show that f (x) = 0 for all
x [a, a 1] . You may use the fact that there is M > 0 such that f ( x) M for all x [a, a 1] . (9 marks) (b) Suppose f (0) = 0 . (i)
Using (a) or otherwise, show that f (x) = 0 for all x [0, ) .
(ii)
Show that f (x) = 0 for all x R . (6 marks)
P. 183
HKAL Pure Mathematics Past Paper Topic: Differentiation
Set 2 43.
(04II12) (a) Let f : R R be a twice differentiable function. Assume that a and b are two distinct real numbers. (i)
Find a constant k ( independent of x ) such that the function h( x) f ( x) f (b) f '( x)( x b) k ( x b)
2
satisfies
h(a) = 0 . Also find h(b) . (ii)
Let I be the open interval with end points a and b . Using Mean Value Theorem and (a)(i), prove that there exists a real number c I such that
f (b) f (a) f '(a)(b a)
f'' (c) (b a) 2 . 2 (7 marks)
(b) Let g : R R be a twice differentiable function. Assume that there exists a real number (0, 1) such that g( x) g( ) 1 for all x (0,1) . (i)
Using (a)(ii), prove that there exists a real number (0,1) such that g(1) 1
(ii)
g'' ( ) (1 ) 2 . 2
If g'' ( x) 2 for all x (0,1) , prove that g(0) g(1) 1 . (8 marks)
P. 184
HKAL Pure Mathematics Past Paper Topic: Differentiation
44.
(01II11) (a) Let f and g be real-valued functions continuous on [a, b] and differentiable in (a, b) . (i)
By considering the function
h( x) f ( x)[g(b) g(a)] g( x)[f (b) f (a)] on [a, b] , or otherwise, show that there is c (a, b) such that
f '(c)[g(b) g(a)] g' (c)[f (b) f (a)] . (ii)
Suppose g(x) > 0 for all x (a, b) . Show that
g( x) g(a) 0 for any x (a, b) . If, in addition ,
P( x)
f '( x ) is increasing on (a, b) , show that g' ( x )
f ( x) f ( a) is also increasing on (a, b) . g( x) g(a) (9 marks)
e x cos x 1 if x 0, , (b) Let Q( x) sin x cos x 1 4 1 if x 0 .
Show that Q is continuous at x = 0 and increasing on 0, . 4
, 4
Hence or otherwise, deduce that for x 0,
x 0
Q(t ) dt x . (6 marks)
P. 185
HKAL Pure Mathematics Past Paper Topic: Differentiation
45.
(06II11) (a) Let f : R R and g : R R be continuous on [a, b] and differentiable in (a, b) , where a < b . Suppose that g(a) g(b) and g(x) 0 for all x (a, b) . Define
h( x) f ( x) f (a)
f (b) f (a) g( x) g(a) for all x R . g(b) g(a)
(i)
Find h(a) and h(b) .
(ii)
Using Mean Value Theorem, prove that there exists (a, b) such that
f '( ) f (b) f (a) . g' ( ) g(b) g(a) (5 marks)
(b) Let u : R R be twice differentiable. For each x R , let F : R R and G : R R be defined by
( x t )2 . For each c x , 2 F' ( ) F(c) prove that there exists I such that and G' ( ) G(c)
F(t ) u( x) u(t ) u' (t )( x t ) and G(t )
u( x) u(c) u' (c)( x c)
u'' ( ) ( x c) 2 , where I is the open 2
interval with end points c and x . (5 marks) (c) Let v : R R be twice differentiable . It is given that
v( x) 2006 . x 0 x
lim (i)
Prove that v(0) = 0 . Hence find v(0) .
(ii)
Suppose that v(x) 2 for all x R . Prove that v(x) 2006x + x2 for all x R . (5 marks)
P. 186
HKAL Pure Mathematics Past Paper Topic: Differentiation
Set 3 46.
(00II12) (a) Let f be a real-valued function defined on an open interval I , and f ''( x) 0 for x I . (i)
Let a, b, c I with a < c < b . Using Mean Value Theorem or otherwise, show that Hence show that f (c)
(ii)
f (c) f (a) f (b) f (c) . ca bc bc ca f ( a) f (b) . ba ba
Let a, b I with a < b and (0, 1) , show that
a a (1 )b b . Hence show that f[ a (1 )b] f (a) (1 )f (b) . (8 marks) (b) Let 0 < a < b . Using (a)(ii) or otherwise, show that (i)
if p > 1 and 0 < < 1 , then
[ a (1 )b] p a p (1 )b p ;
(ii)
if 0 < < 1 , then a (1 )b a b
1
. (7 marks)
P. 187
HKAL Pure Mathematics Past Paper Topic: Differentiation
47.
(02II12) (a) Let g(x) be a function continuous on [a, b] , differentiable in (a, b) , with g(x) decreasing on (a, b) and g(a) = g(b) = 0 . Using Mean Value Theorem, show that there exists c (a, b) such that g is increasing on (a, c) and decreasing on (c, b) . Hence show that g(x) 0 for all x [a, b] . (5 marks) (b) Let f be a twice differentiable function and f (x) 0 on an open interval I . Suppose a , b , x I with a < x < b . By considering the function
g( x) (b x)f (a) ( x a)f (b) (b a)f ( x) or otherwise, show that
f ( x)
bx xa f ( a) f (b) . ba ba
Hence, or otherwise, prove that f (1 x1 2 x2 ) 1f ( x1 ) 2f ( x2 ) for all x1 , x2 I , where 1 , 2 0 with 1 + 2 = 1 . (5 marks) (c) Let x1 and x2 be positive numbers. (i)
If 1 , 2 0 with 1 + 2 = 1 , prove that
1 x1 2 x2 x1 x2 . 1
(ii)
2
If 1 , 2 are positive numbers, prove that
1 x1 2 x2 1 2
1 2
x11 x2 2 . (5 marks)
P. 188
HKAL Pure Mathematics Past Paper Topic: Differentiation
Set 4 48.
(98II13)
1
Let I 0, and g( x) cos x cos3 x , where x I . 3 3 Let x0 I and define xn g( xn1 ) for n = 1, 2, 3, … . (a) Show that the equation x = g(x) has exactly one root in I . (3 marks) (b) Show that xn I for all n . (3 marks) (c) Show that g' ( x)
3 for all x I . 4 (2 marks)
(d) Let be the root of x = g(x) mentioned in (a).
3 xn 1 for n = 1, 2, 3, … . 4
(i)
Show that xn
(ii)
Show that {xn} converges and lim xn .
(iii)
If x0
n
6
xn
, find a positive integer n such that
1 . 100 (7 marks)
P. 189
HKAL Pure Mathematics Past Paper Topic: Differentiation
49.
(93II13) Let a , b R and a < b . Let f (x) be a differentiable function on R such that f (a) 0 , f (b) 0 and f (x) is strictly decreasing. (a) Show that f (a) > 0 . (2 marks) (b) Let x0 = a and x1 x0
f ( x0 ) . f' ( x0 )
Show that a < x1 < b , f (x1) < 0 and f (x1) > 0 . (6 marks) (c) Let x0 = a and xn xn 1
f ( xn 1 ) for n = 1, 2, 3, … . f' ( xn 1 )
Show that a < xn < b , f (xn) < 0 and f (xn) > 0 for n = 1, 2, … . (4 marks) (d) Show that lim xn exists and is a zero of f (x) . n
(3 marks)
P. 190
HKAL Pure Mathematics Past Paper Topic: Integration
Contents: Classification
Chronological Order
Type 1: Evaluation of Integral
90II03
Q.
17
Set 1: Basic
90II04b
Q.
14
1.
97II04
90II05
Q.
29
2.
94II02
90II07
Q.
22
3.
02II01
90II08
Q.
69
4.
01II12
91II03
Q.
39
Set 2: Integration by Parts
91II04
Q.
19
5.
93II05
91II06
Q.
30
6.
96II05
91II09
Q.
63
7.
97II01
91II12
Q.
46
8.
01II02
92II04
Q.
12
9.
04II03
92II05
Q.
27
10.
05II04
92II09
Q.
50
11.
00II01
92II11
Q.
32
12.
92II04
92II12
Q.
65
13.
02II09
93II05
Q.
5
Set 3: Change of Variables
93II11
Q.
72
14.
90II04b
93II12
Q.
55
15.
95II02
94II02
Q.
2
16.
99II02
94II06
Q.
34
17.
90II03
94II08
Q.
68
18.
96II03
94II11
Q.
44
19.
91II04
94II13
Q.
66
20.
95II11
95II01b
Q.
26
Type 2: Evaluation of Infinite Series
95II02
Q.
15
21.
98II03
95II04
Q.
35
22.
90II07
95II07
Q.
31
23.
03II04
95II08
Q.
64
24.
04II04
95II11
Q.
20
25.
06II04
95II13
Q.
73
26.
95II01b
96II02
Q.
67
27.
92II05
96II03
Q.
18
28.
00II06
96II05
Q.
6
P. 191
HKAL Pure Mathematics Past Paper Topic: Integration
Type 3: Differentiation of Integrals
96II12
Q.
47
Set 1
97II01
Q.
7
29.
90II05
97II04
Q.
1
30.
91II06
97II09
Q.
48
31.
95II07
97II11
Q.
38
32.
92II11
97II13
Q.
61
33.
03II09
98II02
Q.
36
Set 2
98II03
Q.
21
94II06
98II05
Q.
53
Set 3
98II09
Q.
45
35.
95II04
98II11
Q.
74
36.
98II02
99II02
Q.
16
37.
01II05
99II09
Q.
57
38.
97II11
99II11
Q.
70
Type 4: Area / Volume
00II01
Q.
11
39.
91II03
00II06
Q.
28
40.
01II07
00II11
Q.
54
41.
06II05
00II13
Q.
71
Type 5: Reduction Formula
01II02
Q.
8
Set 1
01II05
Q.
37
42.
06II03
01II07
Q.
40
43.
04II08
01II10
Q.
49
Set 2
01II12
Q.
4
44.
94II11
02II01
Q.
3
45.
98II09
02II09
Q.
13
46.
91II12
02II10
Q.
75
47.
96II12
02II13
Q.
56
48.
97II09
03II04
Q.
23
49.
01II10
03II09
Q.
33
Set 3
03II10
Q.
62
50.
92II09
03II12
Q.
51
51.
03II12
04II03
Q.
9
52.
05II10
04II04
Q.
24
Type 6: Convergence of Sequence /
04II08
Q.
43
Series
04II10
Q.
58
34.
Set 1 53.
98II05
05II04
Q.
10
54.
00II11
05II10
Q.
52
P. 192
HKAL Pure Mathematics Past Paper Topic: Integration
Set 2
05II12
Q.
59
55.
93II12
06II03
Q.
42
56.
02II13
06II04
Q.
25
57.
99II09
06II05
Q.
41
58.
04II10
06II08
Q.
60
59.
05II12
60.
06II08 Set 3
61.
97II13
62.
03II10 Set 4
63.
91II09
64.
95II08
65.
92II12
66.
94II13 Type 7: Comparison of Integrals Set 1
67.
96II02
68.
94II08
69.
90II08
70.
99II11
71.
00II13 Set 2
72.
93II11
73.
95II13 Type 8: Others
74.
98II11
75.
02II10
P. 193
HKAL Pure Mathematics Past Paper Topic: Integration
Type 1: Evaluation of Integral Set 1: Basic 1.
(97II04) Show that (sin 2 x sin 4 x sin 2nx)sin x sin nx sin(n 1) x .
Hence, or otherwise, evaluate
2
4
sin 6 x sin 7 x dx . sin x (6 marks)
2.
(94II02) Evaluate
tan
(a)
(b)
3
x dx ,
x2 x 2 dx . x( x 2) 2 (6 marks)
3.
(02II01)
3 x
(1 x)(1 x ) dx
Find the indefinite integral
2
Hence evaluate the improper integral
0
.
3 x dx . (1 x)(1 x 2 )
Out of Syllabus: Improper Integral You may assume the definition:
0
u
h( x) dx lim h( x) dx . u
0
(6 marks)
P. 194
HKAL Pure Mathematics Past Paper Topic: Integration
4.
(01II12) (a) Evaluate
1 dx , x 1
(i)
x
(ii)
x2 1 ( x2 x 1)( x2 x 1) dx .
2
(6 marks)
x 6 r 5 (b) For n = 1, 2, 3, … and 0 x < 1 , define g n ( x) and r 1 6r 5 n
x 6 r 1 h n ( x) . For any fixed x [0, 1) , r 1 6r 1 n
(i)
show that the sequences {g n ( x)} and {h n ( x)} are increasing;
(ii)
deduce that lim g n ( x) and lim h n ( x) exist. n
n
(3 marks)
(c) For n = 1, 2, 3, … and 0 x < 1 , define f n ( x)
x 6 r 5 x 6 r 1 6r 1 r 1 6r 5 n
and let f ( x) lim f n ( x) . n
(i)
For any fixed x (0, 1) . evaluate f n' ( x) and show that
lim f n' ( x) exists. n
(ii)
Assuming that f '( x) lim f n' ( x) for any fixed x (0, 1) and n
f is continuous on [0, 1) , show that f '( x)
1 x2 1 x2 x4
for
x (0,1) . Hence find f (x) . (6 marks)
P. 195
HKAL Pure Mathematics Past Paper Topic: Integration
Set 2: Integration by Parts 5.
(93II05) Evaluate
e
2x
(sin x cos x)2 dx . (7 marks)
6.
(96II05) (a) Evaluate (i)
x ln x dx
(ii)
;
ln x dx . x
x 2 ln x (b) Consider the curve y where 1 x e . 2 4 [Out of Syllabus: Surface Area] Find the area of the surface generated by rotating the curve about the x-axis. (6 marks) 7.
(97II01) (a) Show that
d x 1 tan . dx 2 1 cos x
(b) Using (a), or otherwise, find
x sin x
1 cos x dx
. (5 marks)
8.
(01II02) Evaluate (a)
x3 1 x2 dx ,
(b)
x
2
tan 1 x dx . (5 marks)
9.
(04II03) (a) Evaluate
sec d 3
. (3 marks)
(b) [Out of syllabus: Arc Length] Consider the curve C : x2 = 2y , 0 x 1 . Find the length of C . (4 marks)
P. 196
HKAL Pure Mathematics Past Paper Topic: Integration
10.
(05II04) (a) (i) (ii)
Evaluate
Prove that
1 x dx .
1 2 0
arcsin x 2 dx 2 6 4 . 6 1 x (4 marks)
(b) [Out of syllabus: Surface Area] 11.
(00II01) Evaluate
x cos x dx
Hence evaluate
2 0
.
x cos x dx . (4 marks)
12.
(92II04) Evaluate
2 0
xe
x 1
dx . (4 marks)
P. 197
HKAL Pure Mathematics Past Paper Topic: Integration
13.
(02II09) (a) Find
e
x
sin x dx .
(b) Let f : R [0, ) be a periodic function with period T . (i)
Prove that
b kT a kT
b
e x f ( x) dx e kT e x f ( x) dx for any positive a
integer k . (ii)
Let I n
nT 0
e x f ( x) dx .
Prove that I n (iii)
1 e nT I1 for any positive integer n . 1 e T
If is a positive number and n is a positive integer such that nT (n 1)T , prove that 1 e nT 1 e ( n1)T x I1 e f ( x) dx I1 . 0 1 e T 1 e T
Hence find the improper integral
0
e x f ( x) dx in terms of
I1 and T . (9 marks) (c) Using the result of (a) and (b)(iii), evaluate
0
e x | sin x | dx . (3 marks)
Out of Syllabus: Improper Integral You may assume the definition:
0
u
h( x) dx lim h( x) dx . u
0
Set 3: Change of Variables 14.
(90II04b) Evaluate
dx x 4x 2 2
. (3 marks)
P. 198
HKAL Pure Mathematics Past Paper Topic: Integration
15.
(95II02) (a) Using the substitution x sin 2 ( 0
2
) , prove that
f ( x) dx 2 f (sin 2 ) d . x(1 x)
(b) Hence, or otherwise, evaluate
dx and x(1 x)
x dx . 1 x (5 marks)
16.
(99II02) (a) Let f be a continuous function. Show that
0
f ( x) dx f ( x) dx . 0
(b) Evaluate
0
x sin x dx . 1 cos 2 x (6 marks)
17.
(90II03) Suppose f (x) and g(x) are real-valued continuous functions on
[0, a] satisfying the conditions that f ( x) f (a x) and g( x) g(a x) K where K is a constant. Show that
a 0
f ( x)g( x) dx
a 1 K f ( x) dx . 0 2
Hence, or otherwise, evaluate
0
x sin x cos 4 x dx . (5 marks)
18.
(96II03) (a) Suppose f (x) is continuous on [0, a] . Show that
a 0
a
f ( x) dx f (a x) dx . 0
Further, if f ( x) f (a x) K for all x [0, a] , where K is a constant, prove that (i) (ii)
a K 2 f( ) ; 2
a 0
a f ( x) dx a f ( ) . 2
(b) Hence, or otherwise, evaluate
2 0
1 e
sin x
1
dx . (6 marks)
P. 199
HKAL Pure Mathematics Past Paper Topic: Integration
19.
(91II04) Evaluate the following definite integral:
2 0
d . 1 2sin (6 marks)
20.
(95II11) (a) Evaluate
2 0
d . 2sin cos 2 (5 marks)
(b) Let f ( ) a sin b cos c and
g( ) A sin B cos C where A , B are not both zero. Show that there exist real numbers p , q and r such that
f ( ) p g( ) q g' ( ) r for all real numbers . (5 marks) (c) Hence, or otherwise, evaluate
2 0
7sin 4cos 3 d . 2sin cos 2 (5 marks)
Type 2: Evaluation of Infinite Series 21.
(98II03) Evaluate
ln(1 x) dx
.
Hence, or otherwise, find lim n
n
1
k
n ln(1 n )
.
k 1
(5 marks) 22.
(90II07) (a) Evaluate
(b) Let un
ln(1 x ) dx 2
.
1 2n 1 2n 2 1 k2 2 n ( n k ) . Prove that ln u ln 1 2 . n n4 k 1 k 1 n n
Hence, or otherwise, find the value of lim un . n
(8 marks)
P. 200
HKAL Pure Mathematics Past Paper Topic: Integration
23.
(03II04) Using the substitution t tan
x , evaluate 2
Hence, or otherwise, evaluate lim
n
dx
2 cos x
n
k 1
1 k n 2 cos( ) 2n
. .
(7 marks) 24.
(04II04) Using the substitution u
1 , prove that x
2 1 2
ln x dx 0 . 1 x2
1 k ln 2( ) 2 2n . Hence, or otherwise, evaluate lim n 1 k k 1 2n 1 ( ) 2 2 2n 3n
(6 marks) 25.
(06II04) (a) Using the substitution t 1 x 2 , find
x3 1 x2
dx .
1 n k3 . n n3 k 1 n2 k 2
(b) Evaluate lim
(7 marks) 26.
(95II01b) By considering a suitable definite integral, evaluate
12 22 n2 lim 3 3 3 3 3 . n n 1 n 2 n n3 (3 marks) 27.
(92II05)
2n 2 k 2 . 3 3 n k 1 n k
Using a definite integral, or otherwise, evaluate lim
n
(7 marks) 28.
(00II06)
1 1 1 . nn n 1 n 2
Use a suitable integral to evaluate lim n
(4 marks)
P. 201
HKAL Pure Mathematics Past Paper Topic: Integration
Type 3: Differentiation of Integrals Set 1 29.
(90II05)
d xn f (t ) dt , where f is continuous and n is a positive dx 0
(a) Evaluate integer. (b) If F( x)
x2 x
3
et dt , find F(1) . 2
(6 marks) 30.
(91II06) (a) Evaluate
2 d u ( 2)t dt . du 0
(b) Define F( x)
sec x tan x
( 2)t dt for 2
2
x
2
.
Solve F' ( x) 0 . (6 marks) 31.
(95II07) Let f : R (1, ) be a differentiable function. (a) Differentiate ln[1 f ( x)] . (b) If f ( x) x3
x 0
3t 2 f (t ) dt for all x R , by considering f (x) , find
f ( x) . (6 marks)
P. 202
HKAL Pure Mathematics Past Paper Topic: Integration
32.
(92II11) (a) Let f (x) be a polynomial and n a positive integer such that
deg f ( x) n . Prove that for any a R , if f (a) f '(a) f ''(a) f
( n 1)
( a) 0 ,
then f (x) is divisible by (x a)n . (8 marks) (b) Let p(x) , q(x) , r(x) and s(x) be polynomials and
F( x)
p(t)r(t) dt q(t)s(t) dt p(t)q(t) dt r(t)s(t) dt . x
1
x
x
1
1
x
1
Prove that if deg F(x) 4 , then F(x) is divisible by (x 1)4 . (7 marks)
P. 203
HKAL Pure Mathematics Past Paper Topic: Integration
33.
(03II09) (a) Let u : R R be a twice differentiable function satisfying the following conditions: (1)
u(x) = u(x)
(2)
u(0) = 0 ,
(3)
u(0) = 1 .
for all x R .
Define v(x) = u(x) sin x for all x R . By differentiating w( x) v( x) v' ( x) , prove that u( x) sin x 2
2
for all x R . (5 marks) (b) Let f : R R and g : R R be continuous functions such that x
x
0
0
f ( x) e x et g (t ) dt and g( x) e x e x et f (t ) dt for all x R . (i)
Prove that f (x) + f (x) = g(x) for all x R .
(ii)
(1)
Prove that f (x) + 2f (x) + 2f (x) = 0 for all x R .
(2)
Let h( x) e f ( x) for all x R . x
Prove that h(x) = h(x) . Using (a), find f (x) . (iii)
Find g(x) . (10 marks)
Set 2 34.
(94II06) Let f : R R be a continuous function. Show that
If
1 0
1 0
x
x f (t ) dt f (t ) dt for all x R . 0
f ( xt ) dt 0 for all x R , show that f (x) = 0 for all x R . (5 marks)
P. 204
HKAL Pure Mathematics Past Paper Topic: Integration
Set 3 35.
(95II04) For x 0 , define F( x)
x 0
sin t dt . t 1
(a) Find the value of x0 for which F( x) F( x0 ) for all x [0, 2 ] . (b) By considering F(0) and F(2 ) , show that F( x) 0 for all
x (0, 2 ) . (7 marks) 36.
(98II02) Let f : R R be a continuous periodic function with period T . (a) Evaluate
d dx
x T 0
x
f (t ) dt f (t ) dt 0
.
(b) Using (a), or otherwise, show that
x T x
T
f (t ) dt f (t ) dt for all x . 0
(4 marks) 37.
(01II05) Let f be a real-valued function continuous on [0, 1] and differentiable in (0, 1) . Suppose f satisfies A.
f (0) = 0 ,
B.
f (1) =
C.
0 < f (t) < 1 for t (0, 1) .
Define F( x) 2
x 0
1 , 2
f (t ) dt f ( x) for x [0, 1] . 2
(a) Show that F(x) > 0 for x (0, 1) . (b) Show that
1 0
f (t ) dt
1 . 8 (6 marks)
P. 205
HKAL Pure Mathematics Past Paper Topic: Integration
38.
(97II11) Define F( x) (a) (i)
x 1
1 1 t3
dt for any x 1 .
Show that F(x) is strictly increasing.
(ii)
Show that 0 < F(x) < 2 for any x > 1 .
(iii)
Find an x0 such that F(x0) > 1 . (6 marks)
(b) Suppose G(u) is a function on (0, 1) such that F(G(u)) = u .
3 [G(u )]2 . 2
(i)
Show that G' (u) 1 [G(u)]3 and G'' (u )
(ii)
Show that G'' (u) G' (u) .
(iii)
Does the graph of G(u) have any points of inflexion? Explain. (9 marks)
Type 4: Area / Volume 39.
(91II03) Consider the curve
x sin 3 t 0 t , . 3 2 y cos t (a) [Out of Syllabus: Arc Length] Find the length of the curve. (b) Find the area bounded by the curve, the x-axis and the y-axis. (7 marks) 40.
(01II07) The figure shows the curve
x cos3 t , 0 t 2 . : 3 y sin t , (a) [Out of syllabus: Arc Length] (b) Find the area enclosed by . (4 marks)
P. 206
HKAL Pure Mathematics Past Paper Topic: Integration
41.
(06II05) (a) Find
ln y dy
.
(b) Find the volume of the solid of revolution generated by revolving the region bounded by the curve y 2 x and the straight line y = 2
2 about the y-axis. (6 marks) Type 5: Reduction Formula Set 1 42.
(06II03) For any positive integers m and n , define Im, n =
(a) Prove that Im + 2, n + 2
1 1 = n 1 2
mn
4 0
sin m d . cos n
m 1 Im, n . n 1
(b) Using the substitution u = cos , evaluate I3, 1 . (c) Using the results of (a) and (b), evaluate I7, 5 . (7 marks) 43.
(04II08) (a) For any non-negative integers m and n , define x
I m,n ( x) cos m cos n d for all x R . 0
Prove that I m1,n 1 ( x)
cos m1 x sin(n 1) x m 1 I m , n ( x) . mn2 mn2 (6 marks)
(b) Evaluate
2 0
cos 4 cos 3 d . (4 marks)
(c) Evaluate
2 0
sin 4 cos 3 d . (5 marks)
P. 207
HKAL Pure Mathematics Past Paper Topic: Integration
Set 2 44.
(94II11) For any non-negative integer n , let I n
(a) (i)
Show that
1 n 1 4
n 1
In
4 0
tan n x dx .
1 . n 1 4
You may assume without proof that
x tan x
4x
for x [0,
4
] .
(ii)
Using (i), or otherwise, evaluate lim I n .
(iii)
Show that I n I n 2
n
1 for n = 2, 3, 4, … . n 1 (8 marks)
(b) For n = 1, 2, 3, … , let an
(1) k 1 . k 1 2k 1 n
(i)
Using (a)(iii), or otherwise, express an in terms of I2n .
(ii)
Evaluate lim an . n
(7 marks)
P. 208
HKAL Pure Mathematics Past Paper Topic: Integration
45.
(98II09) Let I m (a) (i) (ii)
2 0
cos m t dt where m = 0, 1, 2, … .
Evaluate I0 and I1 . Show that I m
m 1 I m2 for m 2 . m
Hence, or otherwise, evaluate I 2n and I 2 n 1 for n 1 . (7 marks) (b) Show that I 2 n1 I 2 n I 2 n1 for n 1 . (2 marks) 2
1 2 4 6 (2n) (c) Let An where n = 0, 1, 2, … . 2n 1 1 3 5 (2n 1) (i)
Using (a) and (b) , show that
(ii)
Show that
(iii)
Evaluate lim
An
n
2n 1 An An . 2n 2
is monotonic increasing.
1 2 4 6 (2n) . 2n 1 1 3 5 (2n 1) (6 marks)
P. 209
HKAL Pure Mathematics Past Paper Topic: Integration
46.
(91II12) For any non-negative integer n , let I n
(a) (i)
(ii)
2 0
cos 2 n 1 x dx .
Evaluate I0 and express In in terms of I n 1 for n 1 .
Show by induction that I n
(n !) 2 22 n for n = 0, 1, 2, … . (2n 1)! (5 marks)
(n !) 2 2n 1 (b) For any non-negative integer m , let Sm . n 0 (2n 1)! m
(i)
Show that m 1
1 1 cos 2 x 2 Sm 2 2 cos x 0 1 1 cos 2 x 2 (ii)
Deduce that
(iii)
dx .
2 0
2 cos x 2 cos x dx m Sm 2 dx . 0 1 1 2 2 2 1 cos x 1 cos x 2 2
Show that
(n !) 2 2n 1 . n 0 (2n 1)!
(10 marks)
P. 210
HKAL Pure Mathematics Past Paper Topic: Integration
47.
(96II12) For non-negative integers k and m , define F(k , m)
1 0
u k (1 u 2 ) m du .
(a) Show that (i)
F(k , 0)
1 ; k 1
(ii)
F(k , m)
2m F(k 2, m 1) for m 1 . k 1 (4 marks)
2m (m!) (b) Show that F(k , m) . (k 1)(k 3) (k 2m 1) (4 marks) (c) Using (b), prove that
2 0
cos 2 m1 d
[2m (m!)]2 . (2m 1)! (4 marks)
(d) Show that F(k , m)
(1)r Crm . r 0 2r k 1 m
(3 marks)
P. 211
HKAL Pure Mathematics Past Paper Topic: Integration
48.
(97II09) Let m , n be non-negative integers. (a) Define I m,n
1 0
x m (1 x)n dx .
Show that (m 1) I m,n nI m1,n1 for n 1 . Hence, or otherwise, show that I m,n
m !n ! . (m n 1)! (7 marks)
(b) Let f be a real-valued function with continuous derivatives on [0, 1] up to order 2n , where n 1 , and f
(k )
(0) f ( k ) (1) 0 for
k 0,1, ..., 2n 1 . (i)
Show that
1 0
1
f (2 n ) ( x)g( x)dx (1) n (2n)! f ( x)dx , where 0
g( x) x n (1 x)n . (ii)
Suppose there is a constant M such that f (2 n ) ( x) M for all x [0,1] . Using (i), or otherwise, deduce that
(n !)2 M 0 f ( x) dx (2n)!(2n 1)! . 1
(8 marks)
P. 212
HKAL Pure Mathematics Past Paper Topic: Integration
49.
(01II10) (a) Prove by induction that lim x(ln x)n 0 for any non-negative x 0
integer n . (3 marks) (b) Let n be a positive integer.
(ln x) dx x(ln x)
n (ln x)n1 dx .
(i)
Show that
(ii)
Show that the improper integral (i.e. lim h 0
n
1 h
n
1 0
ln x dx is convergent
ln x dx exists) and find its value.
Hence deduce that the improper integral
1 0
(ln x) n dx is
convergent and find its value. (8 marks) (c) Let n be a positive integer and be a positive real number. For
0 h 1 , show that
1 h
x 1 (ln x)n dx
1
n 1
Hence show that the improper integral
1 h
1 h
(ln x)n dx .
x 1 (ln x)n dx is
convergent and find its value. (4 marks) Out of Syllabus: Improper Integral You may assume the definition:
1 0
1
h( x) dx lim h( x) dx if h(0) is undefined. u 0
u
P. 213
HKAL Pure Mathematics Past Paper Topic: Integration
Set 3 50.
(92II09) (a) Let g be a continuously differentiable function and p 1 . Prove that
x 0
x
( x t ) p g' (t ) dt x p g(0) p g(t )( x t ) p 1 dt for any 0
xR . (2 marks) (b) For any n = 1, 2, … , and x R , prove that
ex 1
x x x2 x n1 1 ( x t )n1 et dt . 1! 2! (n 1)! (n 1)! 0
Hence or otherwise, show that
1 1 1 1 3 . (e ) 2 1 e (2n)! (2n)! 2! 4! (7 marks) (c) (i)
Let f0 be a continuous function. For any n = 1, 2, … , and x
x R , define f n ( x) f n1 (t ) dt . 0
Prove that f n ( x)
(ii)
Evaluate
d100 dx100
x 0
x 1 ( x t ) n1 f 0 (t ) dt (n 1)! 0
( x t )99 sin(t 2 ) dt . (6 marks)
P. 214
HKAL Pure Mathematics Past Paper Topic: Integration
51.
(03II12) (a) Let f : (1, 1) R be a function with derivatives of any order. For each m 1, 2, 3,... and x (1, 1) , define
Im
x 1 ( x t )m1 f ( m ) (t ) dt . (m 1)! 0
(i)
f ( m ) (0) m Prove that I m1 I m x . m!
(ii)
Using (a)(i), prove that m 1
f ( k ) (0) k x Im k! k 0
f ( x)
where f (0) f . (6 marks)
1
(b) Define g( x)
1 x2
for all x (1, 1) . Let n be a positive
integer. (i)
Prove that (1 x )g' ( x) x g( x) 0 . 2
Hence deduce that
(1 x2 )g( n1) ( x) (2n 1) x g( n) ( x) n2g( n1) ( x) 0 where g
(0)
g . 2
(0) 0 and g
(ii)
Prove that g
(iii)
Using (a), prove that
(2 n1)
(2 n )
(2n)! (0) n . (2 )(n !)
n 1
x Ck2 k 2 k 1 x ( x t )2 n1 g (2 n ) (t ) dt . 2k 0 2 (2 n 1)! k 0
g( x)
(9 marks)
P. 215
HKAL Pure Mathematics Past Paper Topic: Integration
52.
(05II10) 2 n Define g 0 ( x) 1 and g n ( x) ( x 1) for every positive integer n .
(a) For every non-negative integer n , let I n
1 1
g n ( x) dx .
Express I n 1 in terms of I n .
Hence prove that I n 1
(1)n 1 22 n 3 (n 1)!
2
.
(2n 3)!
(5 marks) (k ) (k ) (b) Prove that g n1 (1) g n1 (1) 0 for all k = 0, 1,…, n ,
where n is a non-negative integer. (3 marks) (c) For every non-negative integer n , let h n ( x) (i)
g n ( n ) ( x) . 2n n !
Using (b), prove that
1 1
p( x)h n1 ( x) dx
1 (1)n1 p( n1) ( x)g n1 ( x) dx n 1 2 (n 1)! 1
for any polynomial p(x) . (ii)
Using (c)(i), or otherwise, evaluate
1 1
x h n ( x)h n1 ( x) dx (7 marks)
Note: For any function f , f
(0)
=f .
P. 216
HKAL Pure Mathematics Past Paper Topic: Integration
Type 6: Convergence of Sequence / Series Set 1 53.
(98II05)
(a) In the figure, using the fact that the shaded area is less than the area of the trapezium ACDB , or otherwise, show that
1 1 1 ln b ln a (b a)( ) . 2 a b (b) Using (a), or otherwise, show that ln n 1
1 1 1 n 1 for 2 3 n 2n
n 2, 3, 4,... . State with reasons whether lim(1 n
1 1 1 ) exists. 2 3 n (6 marks)
P. 217
HKAL Pure Mathematics Past Paper Topic: Integration
54.
(00II11) (a) In the figure, SR is tangent to the curve y = ln x at x = r , where
r 2 . By considering the area of PQRS , show that
1 2 1 r 2 r
ln x dx ln r .
Hence show that
n 3 2
1 ln x dx ln(n !) ln n for any integer n 2 . 2
(5 marks) (b) By considering the graph of y = ln x and a suitable trapezium, show that for r 2 , Hence show that
n 1
r r 1
ln x dx
1 ln(r 1) ln r . 2
1 ln x dx ln(n !) ln n for any integer n 2 . 2 (4 marks)
(c) Using integration by parts, find
ln x dx
. n
1 2 n
1 n e Using the results of (a) and (b), deduce that e n!
3
3 2 2e
for any integer n 2 . (6 marks)
P. 218
HKAL Pure Mathematics Past Paper Topic: Integration
Set 2 55.
(93II12) (a) Show that
1 (1)n t 2 n 2 4 n 1 2 n 2 1 t t (1) t 1 t2 1 t2 for all t R and n = 1, 2, 3, … . Deduce that
tan 1 x x
n 2n x (1) t x3 x5 (1)n1 2 n 1 x dt 0 3 5 2n 1 1 t2
for all x R and n = 1, 2, 3, … . (4 marks) (b) Using (a), or otherwise, show that
x3 x5 (1)n 1 2 n 1 x 2 n 1 tan x x x 3 5 2n 1 2n 1 1
for all x 0 and n = 1, 2, 3, … .
(1) k Hence find . k 0 2k 1
(6 marks) (c) Show that tan 1
1 1 tan 1 . 2 3 4
Deduce that
1 1 1 1 1 1 1 1 ( 1) n 1 1 1 3 3 5 5 2 n 1 2 n 1 4 2 3 3 2 3 5 2 3 2n 1 2 3
1 n 22 n 1
for n = 1, 2, 3, … . (5 marks)
P. 219
HKAL Pure Mathematics Past Paper Topic: Integration
56.
(02II13) (a) (i)
Let I n ( ) and
2
0
tan n u du , where n is a non-negative integer
2
.
tan 2 n 1 Show that I 2 n ( ) I 2 n 1 ( ) for all n 1 . 2n 1 (ii)
Using the substitution t = tan u , or otherwise, show that
t 2n x 2 n 1 x 2 n 3 x d t (1)n1 (1)n tan 1 x for any 0 1 t2 2n 1 2 n 3 1 x
positive integer n . (5 marks) (b) (i)
Let x 0 and n be a positive integer. Prove that 2n x t x 2 n1 x 2 n 1 dt . (2n 1)(1 x 2 ) 0 1 t 2 2n 1
(ii)
Using (a) or otherwise, show that
1 n (1) p 1 1 . 2(2n 1) 4 p 1 2 p 1 2n 1 (iii)
Suppose that tan show that
1 . Evaluate tan 2 and tan 4 , and 5
1 1 4 tan 1 tan 1 . 4 5 239
Hence prove that
4
(1) p 1 4 1 1 4 1 2 p 1 2 n 1 . 2 p 1 239 2392 n 1 (2n 1) 5 p 1 2 p 1 5 n
(10 marks)
P. 220
HKAL Pure Mathematics Past Paper Topic: Integration
57.
(99II09) Let n be a positive integer. (a) Show that
1 t 2n 2 2 n 2 for t 2 1 . 1 t t 2 2 1 t 1 t (2 marks)
(b) For 1 < x < 1 , show that
t 1 dt ln , 2 0 1 t 1 x2 x
(i)
(ii)
x2 x4 t 2 n 1 1 x2n d t ln . 0 1 t2 2n 1 x2 2 4 x
(7 marks)
(c) Show that 0 ln 3
k
1 8 9 8 2n 2 9 k 1 2k 9 n
1 8 Hence evaluate k 1 2k 9
n 1
.
k
. (6 marks)
P. 221
HKAL Pure Mathematics Past Paper Topic: Integration
58.
(04II10) (a) (i) (ii)
dx . 2 x x2
Prove that
(iii)
1
Evaluate
1 2 0
2 2 dx . 1 2 x x2
Using (a)(ii), deduce that
1 2 0
2 2 4 x 2 2 x2 dx . 1 x4 (6 marks)
(b) (i)
Let k be a non-negative integer. Prove that k
x 4 k 4 (1)n x 4 n n 0
1 x4k 4 1 x4
for all real numbers x . (ii)
Using (b)(i) and (a)(iii), or otherwise, prove that
1 n 2 2 1 . 4n 1 4 n 2 4 n 3
( 4 ) n 1
Candidates may use the fact, without proof, that for any given polynomial p(x) ,
lim k
1 2 0
x k p( x)dx 0 . (9 marks)
P. 222
HKAL Pure Mathematics Past Paper Topic: Integration
59.
(05II12) (a) Let f : R R be a function with derivatives of any order. For each m = 0, 1, 2, … and x R , define
f ( k ) (0) k E m ( x ) f ( x) x , where f (0) f . k! k 0 m
(i)
Using integration by parts, prove that
1 x ( x t ) m f ( m1) (t ) dt . 0 m!
E m ( x) (ii)
Suppose that there is a constant C such that f ( k ) ( x) C for all k = 0, 1, 2,… and x R . Using (a)(i), prove that
E m ( x)
C m 1 . x (m 1)! (6 marks)
(b) Let n be a non-negative integer. (i)
Using (a)(i), prove that
(1)k x 2 k 1 (1)n 1 x ( x t )2 n1 sin t dt 0 (2n 1)! k 0 (2k 1)! n
sin x
for all x R . (ii)
Using (b)(i) and the identity sin 3 x
3 1 sin x sin 3x , 4 4
prove that
1 (1) k 1 1 k 1 1 1 1 9 sin 3 1 2 n 1 . 3 4 k 0 (2k 1)! 4(2n 2)! 3 n
(9 marks)
P. 223
HKAL Pure Mathematics Past Paper Topic: Integration
60.
(06II08) (a) (i)
For any two positive integers m and n , evaluate
(ii)
0
cos mx cos nx dx .
For any positive integer n , evaluate
0
0
cos nx dx and
cos 2 nx dx . (5 marks)
(b) Using integration by parts, prove that
0
x 2 cos nx dx
(1)2 2 for n2
any positive integer n . (3 marks) (c) Let N be a positive integer and f ( x) a0
a0 , a1 ,, aN are constants. It is given that
f ( x) x cos nx dx = 0
2
0
N
a m 1
m
cos mx , where
f ( x) x dx = 0
2
0
for all n =1, 2,…, N .
(i)
Find a0 .
(ii)
(1) n 4 Prove that an for all n =1, 2,…, N . n2
(iii)
For any positive integer k , let
Ik
0
and
f ( x) x cos( N k ) x dx 2
. Evaluate lim I k . k
(7 marks)
P. 224
HKAL Pure Mathematics Past Paper Topic: Integration
Set 3 61.
(97II13) Let f (x) be a decreasing continuous function on [1, ) , and f (x) > 0 for all x . For any positive integer n , define
an f (n) (a) (i)
n 1 n
f ( x) dx and cn a1 a2 an .
Show that 0 an f (n) f (n + 1) and cn f (1) . Hence deduce that lim cn exists. n
(ii)
Prove that 0 ck cn f (n + 1) for k > n . (5 marks)
(b) Let c lim cn . Show that n
(i)
0 c cn f (n + 1) , and
(ii)
f (1) f (2) f (n) f ( x) dx c n f (n)
n
1
for some n [0,1] . (5 marks) (c) Let
1 1 Sn 1 ln n and 2 n 1 1 1 1 1 Tn 1 . 2 3 4 2n 1 2 n
(i)
Using (b)(ii), or otherwise, show that lim Sn exists.
(ii)
Express Tn in terms of S2n and Sn .
n
Hence find lim Tn . n
(5 marks)
P. 225
HKAL Pure Mathematics Past Paper Topic: Integration
62.
(03II10) (a) Let f : [1, ) [0, ) be a continuous and decreasing function. For every n = 1, 2, 3, … . define Fn
(i)
Prove that f ( j 1)
j 1 j
n 1
n
f ( x) dx and Sn f ( j ) . j 1
f ( x) dx f ( j ) for all j = 1, 2, 3, … .
Hence deduce that Sn f (1) Fn Sn1 for all n = 2, 3, 4, … .
(ii)
Using (a)(i), prove that the series
f ( j)
is convergent if
j 1
and only if there is a constant K ( independent of n ) such that Fn K for all n = 1, 2, 3,… . (7 marks) (b) Using (a), prove that (i)
1
n3
n 1
(ii)
1
n
is convergent,
is divergent.
n 1
(4 marks) (c) Using (a) to determine whether
1
n(ln n) n2
2
is convergent. Explain
your answer. (4 marks)
P. 226
HKAL Pure Mathematics Past Paper Topic: Integration
Set 4 63.
(91II09) (a) Prove that lim n n 1 . n
(3 marks) (b) Show that n
ln1 ln 2 ln(n 1) ln x dx ln 2 ln 3 ln n 1
where n 2 . n n 1
Deduce that (n 1)! n e
n! . (7 marks) 1
(n !) n (c) Using (a) and (b), or otherwise, evaluate lim . n n (5 marks) 64.
(95II08) Let I k
(1)k (1 x) x3k 0 1 x3 dx , k = 0, 1, 2, … . 1
(a) Evaluate I0 . (4 marks) (b) Prove that
1 1 Ik . 3k 1 3k 1 (3 marks)
(c) Express I k 1 I k in terms of k . (3 marks) (d) For n = 0, 1, 2, … , let bn
(1)k . k 0 (3k 1)(3k 2) n
Using (a) and (c), express I n 1 in terms of bn . Hence use (b) to evaluate lim bn . n
(5 marks)
P. 227
HKAL Pure Mathematics Past Paper Topic: Integration
65.
(92II12) (a) For any x > 0 , by considering the integral
1 x 1
1 dt or otherwise, t
prove that
x ln(1 x) x , 1 x and deduce that
1 1 1 ln 1 . 1 x x x (3 marks) x
1 (b) For any x > 0 , define f ( x) 1 . Using (a) or otherwise, x prove that f is strictly increasing and 1 f ( x) e . (4 marks) (c) For x > 0 and n = 2, 3, … , define
Fn ( x) f ( x) f (n 1)
n x
1 dt , t f (t ) 2
x
1 where f ( x) 1 . x (i)
For each fixed n , prove that there exists a unique n R such that Fn ( n ) 0 . Does lim n exist? Explain. n
(ii)
For each fixed x , prove that lim Fn ( x) exists. n
(8 marks)
P. 228
HKAL Pure Mathematics Past Paper Topic: Integration
66.
(94II13) 1
n Let Ln 2 cos n x dx for any positive integer n . 2 1
(a) Show that Ln n . (3 marks) (b) For n = 1, 2, 3, … , let rn cos Find the values of x in [
1 . 2n
, ] such that cos x rn . 2 2 1
1 n Hence show that Ln rn . n (5 marks) (c) Show that 1
(i)
lim n n 1 , n
(ii) lim Ln 1 . n
(7 marks) Type 7: Comparison of Integrals Set 1 67.
(96II02)
nx n Let an dx for n = 1, 2, 3, … . 0 1 x2 1
(a) Show that
n 1 1 nx nx n d x a n 0 2 0 2 dx for n = 1, 2, 3, … . 1
(b) Evaluate lim an . n
(5 marks)
P. 229
HKAL Pure Mathematics Past Paper Topic: Integration
68.
(94II08) (a) Show that for any a, y R , e e e ( y a) . y
a
a
(b) By taking y = x2 in the inequality in (a), prove that
1 0
1 3
e dx e . x2
(6 marks) 69.
(90II08) (a) Let I n
x n 1 0 (1 x)2 dx for n = 0, 1, 2, … . 1
(i)
Find I0 .
(ii)
Prove that lim I n 0 . n
(5 marks) (b) (i)
Prove that n n 1 ( x) m 1 ( x) n (1)i j x d x 0 1 x i 1 j 1 i j 1 x 1
for any positive integers m and n . (ii)
(1)i j Hence evaluate lim . n i 1 j 1 i j n
n
(10 marks)
P. 230
HKAL Pure Mathematics Past Paper Topic: Integration
70.
(99II11) (a) For n = 0, 1, 2, … and y 0 , define In ( y )
y 0
t n et dt .
n y Prove that In ( y) y e n In1 ( y) for n 1 and y 0 .
Hence deduce that In ( y) n! for n 0 and y 0 . (5 marks) (b) Let n be a positive integer. (i)
By considering g( x) n ln(n x) n ln(n x) 2 x for
0 x n , show that (n x)n e( n x ) (n x)n e( n x ) for
0 xn . (ii)
Use (b)(i) to show that Hence deduce that
2n 0
2n n
n
u n eu du u n eu du . 0
n
t n et dt 2e n (n t )n et dt . 0
(8 marks) (c) Using the above results or otherwise, show that
1 n en n t for all positive integers n . ( n t ) e d t n! 0 2 (2 marks)
P. 231
HKAL Pure Mathematics Past Paper Topic: Integration
71.
(00II13)
Let n be a positive integer. Define f n
(a) (i)
( x)
x 0 1 0
(1 t 4 ) n dt (1 t 4 ) n dt
.
Show that fn(x) is an odd function.
(ii)
Find f n' ( x) and f n'' ( x) .
(iii)
Sketch the graph of fn(x) for 1 x 1 . (7 marks)
(b) Using the facts A.
t 3 (1 t 4 )n (1 t 4 )n for 0 t 1 and
B.
(1 t 4 )n
t3 (1 t 4 )n for 0 < x t 1 , 3 x
or otherwise, show that 0 1 fn(x)
(1 x 4 ) n 1 for 0 x 1 . x3 (5 marks)
(c) For each x [1, 1] , let g( x) lim f n ( x) . Evaluate g(x) when n
0 x 1 and when x = 0 respectively. Sketch the graph of g(x) for 1 x 1 . (3 marks)
P. 232
HKAL Pure Mathematics Past Paper Topic: Integration
Set 2 72.
(93II11) (a) Let f and g be real-valued functions defined on (a, ) where
a 0 , and f be twice differentiable satisfying the following conditions: A.
g is decreasing,
B.
g(t) 0 and f (t) 0 for all t (a, ) ,
C.
lim g(t )f' (t ) 0 .
(i)
t
Using Mean Value Theorem to show that
f (n) f' (n)(t n) f (t ) f (n) f' (n 1)(t n) for all t [n, n 1] , where n is a positive integer greater than a . (ii)
Hence, or otherwise, show that
n 1 n
f (n) f (n 1) f' (n 1) f' (n) f (t ) dt , 2 2
where n is a positive integer greater than a . n 1 n2 f ( j ) f ( j 1) 2 Show that lim f (t ) dt g(n ) 0 . n n 2 j n 2
(iii)
(9 marks)
ln n n2 1 1 dt , or otherwise, n ln t n n 2
(b) Using (a) and the fact that lim
1 1 1 ln n . ln(n2 ) n2 ln(n 1) ln(n 2)
evaluate lim n
(6 marks)
P. 233
HKAL Pure Mathematics Past Paper Topic: Integration
73.
(95II13) (a) Suppose f (x) , g(x) are continuously differentiable functions such that f '( x) 0 for a x b . (i)
Let w( x)
(ii)
b a
x a
g(t ) dt . Show that b
b
a
a
f ( x)g( x) dx f (b) g( x) dx f '( x)w( x) dx .
Using the Theorem (*) below, show that
b a
b
c
c
a
f ( x)g( x) dx f (b) g( x) dx f (a) g( x) dx for some
c [a, b] . Theorem (*) : If w(x) , u(x) are continuous functions and u( x) 0 for a x b , then
b a
b
w( x)u( x) dx w(c) u( x) dx for some c [a, b] . a
(5 marks) (b) Let F(x) be a function with a continuous second derivative such that F'' ( x) 0 and F' ( x) m 0 for a x b . Using (a) with
f ( x)
b a
1 and g( x) F' ( x) cos F( x) , show that F' ( x)
cos F( x) dx
4 . m (5 marks)
(c) (i)
Show that
1 0
1
cos( x n ) dx cos( x n1 ) dx . 0
Hence show that lim n
(ii)
1 0
cos( x n ) dx exists.
Using (b), or otherwise, show that lim n
2 0
cos( x n ) dx exists. (5 marks)
P. 234
HKAL Pure Mathematics Past Paper Topic: Integration
Type 8: Others 74.
(98II11) (a) Let f be a non-negative continuous function on [a, b] . Define x
F( x) f (t ) dt for x [a, b] . 0
Show that F is an increasing function on [a, b] . Hence deduce that if
b a
f (t ) dt 0 , then f (x) = 0 for all
x [a, b] . (5 marks) (b) Let g be a continuous function on [a, b] . If
b a
g( x)u( x) dx 0 for
any continuous function u on [a, b] , show that g(x) = 0 for all
x [a, b] . (3 marks) (c) Let h be a continuous function on [a, b] . Define
A (i)
b 1 h(t ) dt . ba a
If v(x) = h(x) A for all x [a, b] , show that
(ii)
b a
If
v( x) dx 0 .
b a
h( x)w( x) dt 0 for any continuous function w on
[a, b] satisfying
b a
w( x) dt 0 , show that h(x) = A for all
x [a, b] . (7 marks)
P. 235
HKAL Pure Mathematics Past Paper Topic: Integration
75.
(02II10) Let f and g be continuous functions defined on [0, 1] such that f is decreasing and 0 g( x) 1 for all x [0, 1] . For x [0, 1] , define G( x)
( x)
G( x ) 0
(a) (i)
x 0
g(t ) dt and
x
f (t ) dt f (t )g(t ) dt . 0
Prove that G( x) x . Hence prove that ' ( x) 0 for all
x (0,1) . (ii)
Evaluate (0) and hence prove that
1 0
f (t )g(t ) dt
G(1) 0
f (t ) dt . (7 marks)
(b) Let H( x)
x 0
[1 g(t )]dt for all x [0, 1] .
(i)
Prove that G(1) + H(1) = 1 .
(ii)
Using (a)(ii), prove that
1 1G (1)
1
f (t ) dt f (t )g(t ) dt . 0
(5 marks) (c) Using the results of (a)(ii) and (b)(ii), prove that
1 n n 1
1
f (t ) dt f (t ) t n dt 0
Hence show that lim n
1 0
1 n 1 0
f (t ) dt , where n is a positive integer.
f (t ) t n dt 0 . (3 marks)
P. 236
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
Contents: Classification
Chronological Order
Type 1: Tangents
90II09
Q.
15
SECTION A
91II01
Q.
14
1.
96II09
92II06
Q.
11
2.
00II07
93II09
Q.
18
3.
03II05
93II10
Q.
25
4.
05II05
94II09
Q.
16
5.
06II06
95II03
Q.
8
SECTION B
96II09
Q.
1
6.
04II09
96II11
Q.
23
7.
05II09
97II03
Q.
12
Type 2: Relations of Roots
97II12
Q.
21
8.
95II03
98II12
Q.
24
9.
00II10
99II12
Q.
20
10.
06II10
00II07
Q.
2
Type 3: Locus
00II10
Q.
9
SECTION A
01II04
Q.
13
11.
92II06
01II09
Q.
22
12.
97II03
02II11
Q.
17
13.
01II04
03II05
Q.
3
14.
91II01
03II11
Q.
19
SECTION B
04II09
Q.
6
Set 1
05II05
Q.
4
15.
90II09
05II09
Q.
7
16.
94II09
06II06
Q.
5
17.
02II11
06II10
Q.
10
Set 2 18.
93II09
19.
03II11 Type 4: Others
20.
99II12
21.
97II12
22.
01II09
23.
96II11
24.
98II12
25.
93II10
P. 237
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
Type 1: Tangents SECTION A 1.
(96II09) Consider the curves
x2 y 2 1 8 2
E:
P: y kx 3 2
and (k > 0) .
A common tangent L of E and P touches E at (2, 1) . (a) (i)
Find the equation of L and the value of k .
(ii)
Determine the coordinates of the point at which L touches
P . (4 marks) (b) Find the area enclosed by L , P and the y-axis. (4 marks) (c) Find the equation of the remaining three common tangents of E and P . (7 marks) 2.
(00II07) The curve in the figure has parametric equations
x 2(t sin t ) , 0 t 2 . y 2(1 cos t )
(a) Find the equation of the tangent to the curve at the point where
t
2
.
(b) [Out of Syllabus] Find the arc length of the curve. (6 marks)
P. 238
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
3.
(03II05)
x2 y 2 Consider the hyperbola H : 2 2 1 , where a and b are positive a b constants, with its asymptotes L1 : y the point (a sec , b tan ) , where
b b x and L2 : y x . Let P be a a
. 2 2
(a) Prove that P lies on H . (b) The tangents to H at P cuts L1 and L2 at Q1 and Q2 respectively. (i)
Find the coordinates of Q1 and Q2 .
(ii)
Find the areas of OPQ1 and OQ2P , where O is the origin. Hence find Q1P : PQ2 . (7 marks)
4.
(05II05) Let P: y2 = 80x be a parabola. (a) Prove that the straight line y = mx + c is a tangent to P if and only if mc 20 .
x 3cos , 3 . 2 2 y sin ,
(b) Consider the curve :
Find the coordinates of the two points on at which the normals to are tangents to P . (7 marks)
P. 239
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
5.
(06II06)
x2 y 2 Let the equation of the ellipse E be 2 2 1 , where a and b are a b two distinct positive constants. The coordinates of the points P and Q are
(a cos , b sin ) and (a b) cos ,(a b)sin respectively, where
0
2
.
(a) Prove that (i)
P lies on E ,
(ii)
the straight line passing through P and Q is the normal to E at P .
(b) Let c be a constant such that the straight line x sin y cos = c is a tangent to E . Express the distance between P and Q in terms of c . (7 marks)
P. 240
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
SECTION B 6.
(04II09)
x2 y 2 1 , where a and b are two positive a 2 b2 constants with a > b . Let P be the point (a cos , b sin ) , where Consider the ellipse E :
0
2
.
(a) Prove that P lies on E . (1 mark) (b) Let L be the tangent to E at P . L cuts the x-axis and the y-axis at P1 and P2 respectively. Find (i)
the equation of L ,
(ii)
the coordinates of P1 and P2 . (4 marks)
2 2 2 2 2 2 (c) Consider the two circles C1 : x y a and C2 : x y b .
Also consider the two points P1 and P2 described in (b). For
k 1, 2 , let Lk be the tangents to Ck from Pk , with the point of contact Qk lying in the first quadrant. (i)
Prove that L1 is parallel to L2 .
(ii)
Find the coordinates of Q1 and Q2 .
(iii)
Let l be the straight line passing through Q1 and Q2 . Is l a common normal to C1 and C2 ? Explain your answer. (10 marks)
P. 241
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
7.
(05II09) Let H1 : xy 4 and H 2 : xy 1 be two hyperbolas, and L be the
2 t
tangent to H1 at P(2t , ) . (a) Find the equation of L . (2 marks) (b) Let A( ,
1
1 ) and B( , ) be the two distinct points where L
intersects H2 . (i)
Prove that + = 4t and = t 2 .
(ii)
Prove that the length of chord AB is
1 12 t 2 2 . t (6 marks)
(c) It is given that the tangents to H2 at A and B intersect at Q , where A and B are the points described in (b). (i)
Find the coordinates of Q in terms of t .
(ii)
Using (b)(ii), or otherwise, prove that the area of QAB is independent of t . (7 marks)
P. 242
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
Type 2: Relations of Roots 8.
(95II03) Consider the parabola y 4ax . 2
(a) Prove that the equation of the normal at P(at , 2at ) is 2
y tx 2at at 3 ……… (*) . 2 (b) Pi (ati , 2ati ) , i = 1, 2, 3, are three distinct points on the parabola.
Suppose the normals at these points are concurrent. By considering (*) as a cubic equation in t , or otherwise, show that
t1 t2 t3 0 . (5 marks)
P. 243
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
9.
(00II10) The equation of the parabola is y2 = 4ax . (a) Find the equation of the normal to at the point (at 2 , 2at) . Show that if this normal passes through the point (h, k) , then
at 3 (2a h)t k 0 . (4 marks) 2 (b) Suppose the normals to at three distinct points (at1 , 2at1 ) ,
(at2 2 , 2at2 ) and (at32 , 2at3 ) are concurrent. Using the result of (a), show that t1 t2 t3 0 . (2 marks) (c) If the circle x2 + y2 + 2gx + 2fy + c = 0 intersects at (as12 , 2as1 ) , (as2 2 , 2as2 ) , (as32 , 2as3 ) and (as4 2 , 2as4 ) , show that
s1 s2 s3 s4 0 . (4 marks) (d) A circle intersects at points A , B , C and D . Suppose A , B and C are distinct and the normals to at these three points are concurrent. (i)
Show that D is the origin.
(ii)
If A , B are symmetric about the x-axis, show that the circle touches at the origin. (5 marks)
P. 244
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
10.
(06II10) Let the equation of the parabola P be y2 = 4ax , where a is a non-zero constant. (a) Find the equation of the normal to P at the point (at2 , 2at) . (3 marks) (b) The normals to P at two distinct points (at12 , 2at1) and (at22 , 2at2) intersect at the point (h, k) . Let t3 = (t1 + t2) . (i)
Prove that the roots of the equation at 3 + (2a h)t k = 0 are t1 , t2 and t3 .
(ii)
Does the normal to P at the point (at32 , 2at3) pass through the point (h, k) ? Explain your answer.
(iii)
Express t1t2 t2t3 t3t1 and t1t2t3 in terms of a , h and k . (8 marks)
(c) Let A and B be two points on P at which the normals to P are perpendicular to each other. Using the results of (b)(iii), or otherwise, find the equation of the locus of the point of intersection of the two normals as A and B vary. (4 marks) Type 3: Locus SECTION A 11.
(92II06) Consider the line (L) : y 2a and the circle (C) : x y a 2
2
2
, where
a 0 . Let P be a variable point on (L) . If the tangents from P to (C) touch the circle (C) at points Q and R respectively, show that the mid-point of QR lies on a fixed circle, and find the centre and radius of the circle. (6 marks)
P. 245
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
12.
(97II03) Let P be the parabola y 2 = kx , where k is a non-zero constant. A(ks2 , ks) and B(kt 2 , kt) are two distinct points on P moving in such a way that the tangents drawn to P at A and B are perpendicular to each other. (a) Show that st
1 . 4
(b) If M is the mid-point of AB , show that M lies on a parabola and find the equation of this parabola. (5 marks) 13.
(01II04) Let P be the parabola y2 = 4ax where a is a non-zero constant, and
A(at12 , 2at1 ) , B(at22 , 2at2 ) be two distinct points on P . (a) Find the equation of chord AB . (b) If A and B move in such a way that chord AB always passes through (a, 0) , find the equation of the locus of the mid-point of AB . (5 marks) 14.
(91II01) Chords with slope equal to 1 are drawn in the ellipse
2 x 2 2 xy y 2 1 . Prove that the mid-points of these chords lie on a straight line, and find the equation of the line. (4 marks)
P. 246
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
SECTION B Set 1 15.
(90II09) Consider the hyperbola (H) : xy = c2 , c > 0 .
c t1
Let P(ct1 , ) and Q(ct2 ,
c ) be points on (H) where t12 t2 2 , t1 0 t2
and t2 0 . (a) Find the equation of the straight line joining the points P and Q , and hence, or otherwise, obtain the equations of the tangents to (H) at P and Q respectively. (3 marks) (b) Suppose R is the point of intersection of the tangents at P and Q . (i)
Find the coordinates of R .
(ii)
Show that if P and Q are moving in such a way that t1t2 is constant, then R lies on a straight line passing through the mid-point of PQ .
(iii)
If P and Q are moving in such a way that PQ always touches the ellipse 4x2 + y2 = c2 , show that R lies on an ellipse with centre at the origin. Also find the equation of this ellipse. (12 marks)
P. 247
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
16.
(94II09) Given an ellipse (E):
x2 y 2 1 a 2 b2
and a point P(h, k) outside (E) . (a) If y = mx + c is a tangent from P to (E) , show that
(h2 a 2 )m2 2km k 2 b2 0 . (4 marks) (b) Suppose the two tangents from P to (E) touch (E) at A and B . (i)
Find the equation of the line passing through A and B .
(ii)
Find the coordinates of the mid-point of AB . (6 marks)
(c) Show that the two tangents from P to (E) are perpendicular if and only if P lies on the circle x y a b 2
2
2
2
. (5 marks)
17.
(02II11) 2 2 Consider the parabolas C1 : y 4( x 1) and C2 : y 4 x .
2 Let P( p 1, 2 p) be a point on C1 . The two tangents drawn from P to 2 2 C2 touch C2 at the points S ( s , 2s) and T (t , 2t ) .
(a) Find the equations of PS and PT and hence show that s t 2 p ,
st p 2 1 . (4 marks) (b) Q(q , 2q) is a point on the arc ST of C2 . Prove that the area of 2
SQT is a maximum if and only if q = p . (6 marks) (c) Let Q be the point in (b) where the area of SQT is a maximum. If the straight line PQ cuts the chord ST at M , find the equation of the locus of M as P moves along C1 . (5 marks)
P. 248
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
Set 2 18.
(93II09) The equation of the hyperbola H is
x2 y 2 1 , where a , b > 0 . a 2 b2 1 1 1 1 Let P a(t ), b(t ) , where t 0 . t 2 t 2 (a) (i) (ii)
Show that P lies on H . Find the equation of the tangent to H at P . (4 marks)
(b) Let the tangents to H at P meet the asymptotes of H at the points S and T . Let O be the origin. (i)
Show that as t varies the locus of the centre of the circle passing through O , S and T is a hyperbola.
(ii)
Prove that OS OT a 2 b2 . Hence show that S , T and the two foci of H are concyclic. (11 marks)
P. 249
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
19.
(03II11) Consider the parabola P : y 4 x . 2
Let A(a , 2a) be a point on P , L1 be the tangent to P at A , and F 2
be the point (1, 0) . (a) (i) (ii)
Find the equation of L1 . Let F' be a point such that L1 is the perpendicular bisector of FF' . Prove that the x-coordinate of F' is 1 . Also find the y-coordinate of F' . (7 marks)
(b) Suppose a = 2 . The straight line x = 1 intersects L1 at B . Let L2 be the straight line passing through B and perpendicular to L1 . (i)
Prove that L2 is tangent to P and find the point of contact.
(ii)
Let L3 be the tangent to P at the point (1, 2) . L3 intersects L1 and L2 at C and D respectively. Prove that B , C , F and D are concyclic. (8 marks)
P. 250
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
Type 4: Others 20.
(99II12) (a) The equation of the ellipse E is
( x h) 2 y 2 2 1 , where a2 b
a, b, h R and a, b > 0 . (i)
By integration, find the area enclosed by E .
(ii)
If the straight line y = mx is tangent to E , show that
b2 . m 2 h a2 2
(6 marks) (b) For n = 1, 2, 3, … , let En be the ellipse given by
( x hn )2 y2 2 2 1 , an 2 p an where p > 0 , hn > hn + 1 and hn > an > 0 . Suppose for all n ,
En and En 1 touch each other externally and the straight line y = mx is a common tangent to all En as shown in the figure.
(i)
Express hn hn + 1 in terms of an and an + 1 .
(ii)
Using (a)(ii) and the result of (b)(i), or otherwise, show that
an 1 h1 a1 . an h1 a1 (iii)
Let Sn be the area enclosed by the ellipse En . Evaluate
S n 1
n
in terms of a1 , h1 and p . (9 marks)
P. 251
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
21.
(97II12) Consider the curves :
xy = 1 (x > 0) ,
:
xy = 1
:
xy = 1 (x < 0) ,
:
xy = 1
(x < 0) , (x > 0) ,
And the region determined by xy 1 in the rectangular coordinate plane as shown in the figure. For 0 < a 1 , let P , Q , R and S be points on , , and respectively , where
1 1 P ( a, ) , Q ( , a ) , a a 1 1 R (a, ) and S ( , a) . a a (a) Show that PQRS is a square. (3 marks) (b) If the straight line PQ intersects
at another point Q , find the
coordinates of Q in terms of a . Hence find the range of values of a such that the line segment PQ lies inside the region
. (8 marks)
(c) If PQRS lies within the region
, determine a such that its area
is maximized. You may use the fact that PQRS lies within the region and only if PQ lies inside
if
. (4 marks)
P. 252
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
22.
(01II09) In the figure, C is the circle
x2 ( y 1)2 42 and F is the point (0, 1) . For any point P on C , let Lp be the perpendicular bisector of the line segment FP . It appears that as P moves on C , all the Lp ’s are tangents to an ellipse inside C . (a) Suppose for every P on C , the line Lp is tangent to the same ellipse. Write down the equations of the two horizontal tangents and the two vertical tangents to the ellipse. Hence guess the equation of the ellipse. (Note that C is symmetric about the y-axis and F lies on this line of symmetry.) (5 marks) (b) Let E be the ellipse found in (a). (i)
For any point P(p, q) on C , let M be the point (m, n) where m
3p 4(q 1) and n . Show that 7q 7q
(I)
M lies on E ,
(II)
the tangent at M to E is the perpendicular bisector of FP .
(ii)
For any point M on E , show that there is a point P on C such that the perpendicular bisector of FP is the tangent to E at M . (10 marks)
P. 253
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
23.
(96II11) Figure A shows a circle with radius b rolling externally without slipping on a fixed circle with radius a and centred at the origin. Let P be a point fixed on the rolling circle with initial position at A(a, 0) . (a) Referring to Figure B, show that the parametric equations of the locus of P are given by
ab x (a b) cos b cos b . y (a b) sin b sin a b b (6 marks) (b) Suppose a = 2b . (i)
Write down the parametric equations of the locus of P . 2
dx dy d d
2
in terms of .
(ii)
Express
(iii)
[Out of Syllabus: Arc Length] Find the distance travelled by P before it meets the fixed circle again. (9 marks)
Figure A
Figure B P. 254
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
24.
(98II12) Figure A shows a circle with radius 1 and centre C touching a line L with slope m > 0 at Q(a, b) . R(x0, y0) is another point on the circle and QCR .
2 sin (cos m sin ) x0 a 2 2 2 2 1 m (a) Show that . 2 y b sin (m cos sin ) 0 2 2 2 1 m2 (7 marks)
(i)
3 2 ( x 1) 2 where x 1 . 3 Find the slope of tangent of at x = a .
(ii)
[Out of Syllabus: Arc Length] Find the length of arc
(b) Consider the curve : y
from x 1 to x a .
2 32 Ans: (a 1) 3 (iii)
Figure B shows a circle with radius 1 rolling tangentially below the curve without slipping. Let P be a point fixed on the circle with initial position at (1, 0) . Find the x-coordinate of P when the circle touches at x = 4 . (8 marks)
Figure A
Figure B P. 255
HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry
25.
(93II10)
In the figure, O is the origin and is the curve whose equation is
x3 y3 3axy ( a > 0 ) . L is the asymptote of . (a) Evaluate lim
x
y , where (x, y) . x
(You may assume that lim
x
y exists.) x
Hence, or otherwise, show that the equation of L is x y a 0 . (3 marks) (b) [Out of Syllabus: Polar Coordinate] Find the polar equations of and L . (3 marks) (c) [Out of Syllabus: Area of region characterized by Polar Equation] Find the area of the region enclosed by (i.e. 1). (3 marks) (d) Suppose a straight line through O cuts at A and L at B in the second quadrant. Let be the angle between OB and the positive x-axis. Let A be the area of the region bounded by the x-axis , , L and AB (i.e. 2) . [Out of Syllabus: Area of region characterized by Polar Equation] Show that
a2 1 3 A 2 . 3 2 1 tan 1 tan
[Evaluation of limit is included in the syllabus] Hence evaluate lim A .
3 4
P. 256
(6 marks)