620188151058770 algebra and geometry

Algebra and solid Exam Q(13)A Tower guy wire is anchored by mean of a ring at A .The tension in the wire is 2600N determine the components I I Fz A ->(-3,4,0) , B (0,0,12) AB = B - A = (0,0,12)-(- 3,4,0) = (3,-4,12) F=

2600

(3-4,12) = ^ 2 0 (3 , _ 4 j +12K)

J 32 + 42 + 122

F = 600 i - 800 j + 2400 K

Q (14) Solve the following system of linear equations using the inverse m atrix where 4X+Y=0 , X+2Z=15 , Y-7Z=0 "-2 7 1" The cofactor m atrix 7 -28 -4

Adj (A) = 7 -28 -8 1 -4 -1 f Xl f 2 Y = -7 1 1

UJ

iH

<M

N

CVJ i

r

1 00 1

CO

4 1 0 |A| = 1 0 2 =- l 0 1 -7

f 2

" A =R

â&#x20AC;&#x153; -7 28 --1 4

-7 -2^ (-105) 28 8 15 = 480 loj I 60 J X a a

Q(15)Prove that

a a X a = (X + 2 a X X -a )2 a X

= (X + 2aXx - a f

9 + 0)

X + 2a a a X + 2a X a X + 2a a X

11

C1 +C2+c 3

|1 a a |1 a X a = (X + 2alO X - a |l a X |o 0

a 0 X -a

8 1

Algebra and solid Exam Q(16) Prove that: rx = j + t x(i + 2j - K ) , r2 = (i + j + K) + t 2(- 2i - 2j) Intersect at a point, then find their intersection point

r, = (0,1,0)+1,(1,2-1)->(1) , r2 = (1.1,1) + t 2(- 2 -2 ,0 ) -> (2) d, -â&#x2013;ş(1,2-1) , d2

(-2 -2 ,0 ) v | l = - L

* ^ =-^2

.-.The two straight lines are not parallel to get the point of intersection r, = r2 (0 + 1, , 1+ 2t, , 0 - 1, ) = (1 - 2t2 ,1 - 2t2 ,1) /. t2 = 1 substitute in (2) point = r2 = (1,1,1)+1(-2 -2,0) = (-1 ,-1,1)

Q(17) Find the volume of the parallelepiped which three of its adjacent sides are represented by the vectors - 1 2 i - 3 k , 3 j - K and2i + j-1 5 K

V=

12 0 - 3 0 3 - 1 = 546 2 1-15

Q(18 )lf the coefficients of the 4th, 5th and 6th term respectively in the expansion of (2X + Y)n form an arithmetic sequence find the value of n 2coeff.T5 = coeffT4 + coeff.T6 coeff.T, coeff.T. -------

coeff X;

n-4+1

t-

___ 6 = 2

coeff.T5

2 + n - 5 +1x l = 2 5 2

8 + ^ -= 2 n-3 10 n = 19 or n = 8

-r T5

Algebra and solid Exam Q(19) Put the number Z! =-1 + V3 i ,if Z1Z2 = 8e3TTI in the exponential form then find the square root of Z2 in the trigonometric form r = i/l2+(V3)f =2 Z1= 2 e 3'

,tan0 = ^ .\ - > e = - ^

(-,+)2n d 0 = 120° = ^

Z1= 2 [ c o s ^ + i s i n ^

1

8[ jf<cos^TT + isin^ Z, = A ---- ---------= 4(c o s 3 tt + isin3-n) 2 2 tt , ■ ■ 2 tt 21 cos-^+ isin 3 ----- 3 ) = 4 (co stt + isinTr) = Jz~ = 2

3tt

■ ■ 3tt

c o s y + isiny|2^ cos-—+isin-— 2 2

v° Q(20) In the expansion of ^aX -+ in descending powers of X if the bX term free of X is equal to the coefficient of the 7 term prove that 6ab = 5 Tr+1=10Cr(jlx-1J (aX)10_r= 10C r(b)_r(a)10_rX 10_2r1 0 - 2r = 0 /. r = 5 /. T6= 10C 5(b)“5(a)5 T7= ’»Cs(b) 6(a)4X 5 . 1 0 -6 + 1 1 -> --- t:--- x——=1„ 6 ab

» C«(b) 6(a)4= ,0C 5(b) 5( a f .

—x——=51 6 ab

1„ 6ab = 5

M

M

C5lb)

=1

Algebra and solid Exam Answer the following questions 20 questions From Ito 12 choose the correct answer Q ( l) In the opposite figure, a right circular cone , the perimeter of its Base =12iTcm C is the midpoint of AM then OC*OA = ©

9

©

36

2 u r = 1 2 tt

CD = 4cm

© 18 © 54 .\ r = 6 OA = 6 /. MO = 8cm , CO = 5

A DO = 3cm

. OC• OA = ||OC||x||OA||cos(ZCOA) = 5 x 6 x 1 = 1 8

Q (2) In the given figure ||BC|j = Jt BA = (-1,0,1) then BA*BC=....

and ||AC|| = Jz

® 1

©

2

®

©

3

4

v ||BA|| = ^1 + 0 + 1 = J Z

m (ZA)= m (ZB)

cos(ZB) = cos(ZC) = 2 + 6 ~^. = £■ __________________________________ Z

B A • BC = J l x J 6 x

=3

J Z x J h ___ i ___________________________ i ______

Q (3) The coefficient of the middle term in the expansion [ 3X - -g-j

equals

63

8 63 8 coeff.T6 = 10a f - i j < 3 5 = - M

©

67 8

©f

Q (4) The distance between the two planes Y=4 and Y=-2 is

®

3units

® 6units

®

2units

© 8units

4 +2 = 6

Algebra and solid Exam Q(5) In the opposite figure: I f

and

and Z ^ a r e complex

numbers then Z, = ©

-2i

© ©

-i 2i

re 2 i _ „ ei c re Q(6) If ( X - 2 ) 2+(Y + 4)2+ ( Z - 2 ) 2 =1 , (X + 4)2+ ( Y - 4 ) 2+ ( Z - 2 ) 2 = 4 Are the equations of two spheres then the distance between their centers © 10

© J lO

© 20

© zJE

J(Z + 4 f + ( - 4 - 4 f + ( Z - Z f =10 Q(7) ' 3 + 5uj + 5 + 3u)2''8 3 + 5w 5 + 3w ©

81

®

27

9

©

3

81 Q(8) The equation of the line of intersection of the planes X + 2Y - 3Z = 6 and 2X - Y + Z = 7 x - 4 1

y -i 7

z 5

x+ i 1

y+ i 7

Z 5

x - 4

y - i

z 5

X +4 y -1 z 1 “ 7 “ 5

® x-i - 1 "

y-i

z

- 7

" 5

Algebra and solid Exam

Q(9)

-

^

-

-

lfA = 2 i + 3 j + m k , B = - 6 i - 4 j + 4 k and A _L B , then m

© 8

© 6

4

©

(2 x - 6 + 3 ) x ( - 4 + m x 4 ) = 0

-4

.-.4111 = 24

.’.111 = 6

Q (10) Number of solution(natural) such that a + b + c =7

©

© 2 4

© 3 6

©

" '"'Cr = ' _,C 7 = 36

35 210

(n) number of variables , ( r) their sum

Similar example Number of ways to distribute 4 identical balls among 3 boxes n+r 1 c _ 3+4 iq 3 number of boxes , ( r) number of balls

Q (ll) 2 1 + u,r = r =l

0 0 1+w

© 6 ©1

l4 + W+ U)2 !-U)3 !-U)4 +W5 +W6 =1^

0 (12) C„ +" c, +■ c, +■ C, + .....................+" c.

1 2

©

©

©2"

© 4"

("C0 +n C j = 2"

Algebra and solid Exam Q(13) Find the volume of the parallelepiped in which three adjacent sides are represented by the vectors A = (2, 1, 3), B = (-1 , 3 , 2) C = (1 ,1 , -2) 2

1

3

2 = |- 28| = 28 1 -2

The volume = |A â&#x20AC;˘ Bx C| = - 1 3 1

X+ y+ 2

X

Y

1

2X + Y +1

Y

1

X

X+2Y+1

2X + 2Y + 2

X

Y

2X + 2Y + 2

2X + Y+ 1

Y

2X + 2Y + 2 r2" r! >r3- ri

X

X+ 2Y+ 1

Q(14)prove that = 2(X + Y + 1)3

Ci + C2+ c 3

I X

H

+ >+

CsT II

x

X+Y+l

0

0

0

X+Y+l

= 10

1

X

Y X+ 2Y+ 1

= 2(X + Y + 1)2

2X find the value o 51 + X 7K-3I

2Y 2Z X y 5m + y 5n + Z = 2 51+ X 5m+ y 7F - 3m 7g - 3n 7 K - 31 7 F - 3 m

y

Y

= (2X + 2Y + 2) 1 2X+ Y+ 1

0

51 5m 7 K -3 I 7 F-3 m

X

Y

Q(15) X y Z If 1 m n = 2 K f 9 2X 51+ X 7 K -3 I

1

z

x

Z 5n + Z 7g - 3n

y

5n = 2(5) I m 7g - 3n 7 K -3 I 7 F-3 m

X y Z X y Z 1 m n = 70 1 m n = 70 x 2 = 140 7K 7F 7g K F 9

2Y 2Z 5m+ Y 5n + Z 7F - 3m 7g - 3n

z n 7g - 3n

P3+3f2

Algebra and solid Exam Q(16) Find the equation of the straight line passing through the point ( 2 , - 1 , 3) and intersects the straight line n = (1 , -1 , 2 ) + t ( 2 , 2 , -1 ) orthogonally. d1= AC = C - A = ( 2 t - 1 , 2t , - 1 - t ) . L., _L L 2 .*.d1«d2 =0 ( 2t —1 ,2t ,-1-t ) • (2,2,-l) = 0 4 t - 2 + 4t + 1+ t = 0

(2-1,3)

d2=(2,2-1)

►

/. 9t - 1 = 0 /. t = ^

(l +2t , -1+2t, 2-1)

•'•d' = H ’ f ’ l r ) = ( -7’ 2 ’ - 10) Equitation of the line is r = (2,-1,3)+t(-7,2,-10) Q(17) Find all the different forms of the equation of the straight line 3X+1 Y - 1 5 - Z

Let M

3X + 1 = t .. X = 2t —1 2 3 Y -1 =t 2t Y = +1 2 1 5 - Z = t .. Z = 3 t- 5 -1

±1 = Y ^ L = 5^ Z = t

3 3 1 +2t 5 -3 t

" r _ ( 3 ,1’5) + t( s ,2, 3)

Q(18) If n+1C r : n+1C M = 3 :5 [n = 720 = [6.\ n = 6

,j_n = 720 calculate the value ofn+1Pr.2 ,

Cr

3

----------- =

7C’ r-13 r = 40 - 5 r

r =5

5 n+1

7 - r+ 1 -----------------=

r

P,.2 =7P3 =210

3 —

5

Algebra and solid Exam Q (19) If Z = ^ cos-y+ isin-yj, find each of the two numbers Z 1 =Z-1 ,

, Z 2 =Z+1

Z1 then prove that----is a pure imaginary number Zo

z = i +4 i

Z i= - i +;f i

IZ- I = J H = 1

, tane = ^ |(-,+ ) 2nd -> 0 = 120° Z A= cos 120° + i sin 12(T____

z 2 = |+ :f i

N = 1/ R ^ , t a n e ^ . e = 30<

Z2 = */3(cos 30° + i sin 30°)

! i = J L [cos(l 20° - 30°)+ isin(l 20° - 30°)]

z2 V3

= -5L[c o s 90° + isin 9 0 °l= -^ i

Jz

1 Jz

Q (20) If 35, 21 , 7 are the coefficient of three consecutive terms in the expansion (1 + X)n find the value of n and the order of these terms Let the terms are Tr ,Tr+1 ,Trr+2 coofTr+1 coofT,

n - r + 1 . 1 21 3 x —= ——= — —>5n —8r + 5 = 0 r 1 35 5

coofTr+2 n - ( r + l)+1 1 7 1 0 a a rx r tL = ----1— t -— x —= —- = — —>3n —4r —1 = 0 coofTr+1 r +1 1 21 3 n = 7, r = 5