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