Download - MATHS Matices
1 MATHEMATICS
MATRICES
1. A square matrix (aij) in which aij = 0 for ji and aij =k (constant) for i = j is1) Unit matrix 2) Scalar matrix3) Null matrix 4) Diagonal matrix
Ans:
2. If A is a symmetric or skew-symmetric matrix thenA2 is1) symmetric 2) skew-symmetric3) Diagonal 4) scalar
Ans:
3. Let A be a square matrix. consider1) A + AT 2) AAT 3) AT A 4) AT+A5) A - AT 6) AT - A , Then1) all are symmetric matrices2) (2),(4),(6) are symmetric matrices3) (1),(2),(3),(4) are symmetric matrices & (5),(6) are skew symmetric matrices4) 5,6 are symmetric
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4. If A is skew-symmetric matrix and n is odd positiveinteger, then An is1) a symmetric matrix2) skew-symmetric matrix3) diagonal matrix4) triangular matrix
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5. If 3 3ijA a
is a square matrix so that
2 2 ,ija i j then A is a1) unit matrix 2) symmetric marix3) skew symmetric matrix 4) orthogonal matrix
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6. isAthenCBACABIf ,1) non-singular 2) singular3) symmetric 4) Skew symmetric
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Model Questions
2 MATHEMATICS
7. If A is a 3 x 3 singular matrix then A(Adj A) =1) Det A 2) I 3) O 4) 1
Ans:
8. If A and B are two non-singular matrices then
)AB(Adj =
1) AAdj)B(Adj 2) BAdjAAdj
3) Both (1) and (2) 4) A BAns:
9. If A is an n x n non-singular matrix, then AAdj is
=
1) nA 2)
1nA 3)
1nA 4)
2nA
Ans:
10. A 3x3 is a non - singular matrix AA2 (Adj A) =
1) AA 2) I 3) IA 4) IA 2
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11. If ‘d’ is the determinant of a square matrix A of ordern, then the determinant of its adjoint is1) dn 2) dn-1 3) dn-2 4) d
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12. The system of equations which can be solved bycramer’s rule have1) unique solution 2) no solution3) infinitely many solutions 4) two solutions
Ans:
13.If A is invertible matrix and B is another matrix suchthat (AB) exists then1) rank (AB) = rank (A) 2) rank (AB) = rank (B)3) rank (AB) > rank (A) 4) rank (AB) > rank (B)
3 MATHEMATICS
EQUALITY OF MATRICES
14.. If ,23
704123
azazxyx
(x+y+z+a) =
1) -1 2) 0 3) 1 4) 8Ans:
15. If Tr (A) = 6 Tr (4A) =1) 3/2 2) 2 3) 12 4) 24
Ans:
16. If ,017654321
A ...)(,
540030001
BATrB
1) 40 2) 45 3) 39 4) 5Ans:
17. If Tr (A) = 8 , Tr(B) = 6, Tr (A - 2B) =1) -4 2) 4 3) 2 4) 11
Ans:
18. If A-2B =1 23 0
and 2A-3B = 3 3
1 1
then
B =
1)-5 7
5 1
2)
-5 7
-5 -1
3) -5 7
5 -1
4)
-5 -7
-5 -1
Ans:
SCALAR MULTIPE OF A MATRIX ANDMULTIPLICATION OF MATRICES
19.4,
0110
AthenAIf
=
1) I 2) 0 3) A 4) 4 IAns:
4 MATHEMATICS
20. If A2 = A, B2 = B, AB = BA = O then(A+B)2 =1) A - B 2) A + B 3) A2 - B2 4) 0
Ans:
21. If A = diagonal (3,3,3) then 4A 1)12A 2)81A 3)684A 4)27A
Ans:
22. Let P and Q be 2×2 matrices. Consider thestatements.I) PQ=0P=0 or Q=0 or bothII) PQ=I2P=Q-1
III) (P+Q)2=P2+2PQ+Q2
1) I and II are false but III is true2) I and III false and II is true3) All are false4) All are true
Ans:
23. Matrix A is such that A2 = 2A - I where I is the unitmatrix . Then for n 2, AAn =
1) InnA )1( 2) InA3) 12 ( 1)n A n I 4) 12 1 An
Ans:
24. If 3A + 4BT =7 -10 170 6 31
and 2B-3AT =
1 184 65 7
then B =
1)
1 31 02 4
2)
1 31 02 4
3)
1 31 0
2 4
4)
1 31 02 4
Ans:
25. If the order of A is 4 × 3, the order of B is 4 × 5 andthe order of C is 7 × 3, then the order of (A'B)'C' is1) 4 × 5 2) 3 × 7 3) 4 × 3 4) 5 ×7
Ans:
5 MATHEMATICS
SPECIAL TYPES OF MATRICES,SYMMETRIC & SKEW SYMMETRIC MATRICES
26.
2761
= P + Q, where P is a symmetric & Q is a
skew-symmetric then P =
1)
22
132
131
2)
22
132
131
3)
221
211
4)
0213
2130
Ans:
27. If AT BT = CT then C =1) AB 2)BA 3)BC 4)ABC
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28. If A =
25443543243321
iiiOiii
then A is
1) Hermitian2) Skew-Hermitian3) Symmetric 4) Skew-Symmetric
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DETERMINANTS AND ITS PROPERTIES
29. If
a c bA b a c
c b a
then the cofactor of a 32 in A + AAT
is
1) 22 ( )a b c b c 2) 2bac
3) bca 2 4) 2)(2 cacaa Ans:
30. The value of iiii
1122
is
1) A complex quantity 2) real quantity3) 0 4)cannotbe determined
Ans:
6 MATHEMATICS
31. det ....440198891988940894018
1) -8 2) -6 3) -1 4) 0Ans:
32.
11
1
2
2
2
= .....
1) 0 2) 1 3) 2 4) -1Ans:
33.333
222
111
zyxzyx =
1) 02) (x-y)(y-z)(z-x)(xy+yz+zx)3) (x+yx+zx) (x+y)(y+z) (z+x)4) (x+y+z)
Ans:
34.baccc
bacbbaacba
222222
=
1) 2(a+b+c)3 2) (a-b-c)3
3) 2(a-b-c)3 4) (a+b+c)3
Ans:
35. If A,B,C are the angles of triangle ABC, then
CABABCBCA
2sinsinsinsin2sinsinsinsin2sin
1) 1 2) 0 3) -1 4) 8
33
Ans:
36. The value of 14107853421
xxxxxxxxx
is
1) -2 2) 2 3) 0 4) 1Ans:
7 MATHEMATICS
37. If a,b,c are in A.P., then
cxxxbxxxaxxx
433221
1) 2ba
2) ab 3) 0 4) abc
Ans:
38.abc
c
cabb
bcaa
2
2
2
1
1
1
=
1) abc 2) a+b+c 3) 0 4) 4abcAns:
39. If x,y z are all different and if 32
32
32
111
zzzyyyxxx
= 0
then xyz =1) -1 2) 0 3) 1 4) 1
Ans:
40. If x<1 and ....08333
38333383
xthenx
xx
1) 2/3 2) -2/3 3) 0 4) 1/3Ans:
41. det
1990 1991 19921991 1992 19931992 1993 1994
=
1)1992 2)1993 3) 1994 4) 0Ans:
42. If 6a , b, c satisfy ba4cb3c2b2a
= 0, then abc=
1) a + b + c 2) 03) b3 4) ab + b - c
Ans:
8 MATHEMATICS
43.yx1zxz1yzy1x
=
1) 1 + x + y + z 2) x + y + z3) 0 4) 1
Ans:
44. If a + b + c = 0, then one root of xcab
axbcbcxa
= 0 iS1) a + b 2) 0 3) b + c 4) a + c
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45. If A is a 3 x 3 matrix and det (3A) = k. det A, then k =1) 9 2) 6 3) 1 4) 27
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46. det
CsinBsinAsinCsinBsinAsin
111
222 =
1) )ac)(cb)(ba(R81
3
2) )ac)(cb)(ba(R8 3
3) )ac)(cb)(ba(
4) )ac)(cb)(ba(R81
Ans:
47.0qrpr
rq0pqrpqp0
=
1) 0 2) (p - q) (q - r) (r - p)3) pqr 4) 3 pqr
Ans:
9 MATHEMATICS
48. The roots of the determinantal equation bxbmmmxaa
= 0 are1) a 2) b 3) a or b 4) 0, 0
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49. A is a 3 x 3 matrix and B is its adjoint matrix. If thedeterminent of B is 64, then the det A is1) 4 2) 4 3) 8 4) 8
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50.If the entries in a 3 3 determinant are either 0 or 1, thenthe greatest value of their determinats is1) 1 2) 2 3) 3 4) 9
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51.If
2 2 2111 1 1 1
a bc a b cb ca a b cc ab
then
1) 1 2) -1 3) 2 4) -3
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Adjoint of a Matrix :
52. If A =
222
222
222
543432321
then AAdj
1) 8 2) 16 3) 64 4) 128Ans:
53. If A3x3 and det A = 5 then det (adj A) =1) 5 2) 25 3) 125 4) 1/5
Ans:
54. If A3x3 and det A = 6, then det (2 adj A)1) 8 2) 48 3) 288 4) 1/12
Ans:
10 MATHEMATICS
55. If A is a square matrix such that A (AdjA)
4 0 00 4 00 0 4
then det (AdjA) =
1) 4 2) 16 3) 64 4) 256Ans:
56. Adj
1 0 21 1 2
0 2 1
=
5 21 1 02 2
a
b
a b =
1) 4 1 2) 4 1
3) 4 1 4) 4 1Ans:
INVERSE OF A MATRIX
57. If
1tan
tan1
1
1tantan1
=
1
cossinsincos
then =
1) 0 2) 2
3) 4
4) 6
Ans:
58. If A=2 23 2
B = 0 11 0
then
(B-1 A-1)-1 =
1)2 23 2
2) 2 22 3
3) 2 22 3
4) 2 2
2 3
Ans:
59. The matrix A is such that A
1321
=
7714
then A =
1)
3211
2)
32
11
3)
3211
4)
3211
Ans:
11 MATHEMATICS
60. The inverse of the matrix
100210321
is
1)
100210
721
2)
100210721
3)
127012001
4)
100210721
21
Ans:
61. i) If the inverse of matrix A exist. Thenmatrix A is non singular.ii) If a square, non-singular matrix A statisfies A2-A+2I=0
then A-1=12 (I+A).
Which of the following statement is correct.
1. Only i 2. Only ii3. Both i and ii 4. Neither i nor ii
Ans:
62. If A is a n n non singular matrix
List 1 List 2A.(AB)-1 1.adj A
B. 1A A II.1nA
C. adjA III.B-1A-1
D. nk A IV. n nk AWhere k is a scalarA B C D
1. I II III IV2. III I II IV3. I III II IV4. II III IV IAns:
RANK OF A MATRIX
63. The rank of the matrix A =
642321
is
1) 3 2) 2 3) 1 4) 0Ans:
64. If I is a (9 × 9) unit matrix, then rank ( I ) =1) 0 2) 3 3) 6 4) 9
Ans:
12 MATHEMATICS
SOLUTION OF SIMULTANEOUS EQUATION65. The number of solutions of the equations
2x - 3y = 5, x + 2y=7 is ....1) 1 2) 2 3) 4 4) 0
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66. The system of equations2 6 11 0x y , 6 18 1 0y z
6 20 6 3 0x y z 1) is consistent 2) has unique solution3) is inconsistent 4) cannot be determined
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67. The equations 4 2 3x y z
3 5 7x y z
2 3 5x y z have
1) a unique solution2) infinite number of solutions3) no solution4) two solutions
Ans:
68. If the system of equations x + y + z = 6, x + 2y + z =
0, x + 2y + 3z = 10 has no solution, then =
1) 2 2) 3 3) 4 4) 5Ans:
69. The system of linear equations x + y + z = 2, 2x + y -z = 3, 3x + 2y + kz = 4 has a unique solution, if
1) 0k 2) -1 < k < 1
3) -2 < k < 2 4) k = 0Ans:
70. x,y,z not all zeros and the equations
, ,x cy bz y az cx z bx ay
are consistent then a relation among a,b,c is1) a2+b2+c2 = 0 2) a2+b2+c2-2abc=03) a2+b2+c2+2abc=1 4) a2+b2+c2+2abc=0
Ans:
13 MATHEMATICS
71. 2x - 3y + 4z = 0, 5x - 2y -z = 0 ,21x - 8y + az = 0 has infinity solutions then a =1) -5 2) -4 3) 2 4) 4
Ans:
72. The number of nontrivial solutions of the system:
0, 2 0, 2 3 0x y z x y z x y z is
1) 0 2) 1 3) 2 4) 3Ans:
*73. Let 1 2 1 2, ,and be the roots of
2 0ax bx c and 2 0px qx r
respectively. If the system of equaions
1 2 0y z and 1 2 0y z has a
non-trivial solution, then
1) 2
2
b acq pr
2) 2
2
c abr pq
3) 2
2
a bcp qr
4) abc=pqr
Ans:
KEY1) 2 2) 1 3) 3 4) 25) 3 6) 1 7) 3 8) 39) 3 10) 1 11) 2 12) 113) 2 14) 3 15) 4 16) 117) 1 18) 2 19) 1 20) 221) 4 22) 2 23)1 24)325)4 26)1 27)2 28)129)1 30)1 31)3 32)133)2 34)4 35)2 36)137)3 38)3 39)1 40)141)4 42)1 43)3 44)245)4 46)1 47)1 48)349)3 50)2 51)1 52)353)2 54)3 55)2 56)357)1 58)2 59)3 60)161)1 62)2 63)2 64)465)1 66)3 67)3 68)269)1 70)3 71)1 72)173)1
14 MATHEMATICS
PRACTICE QUESTIONSEQUALITY OF MATRICES
1. If A=0 23 4
, KA=0 32 24
ab
then arrange the
values of k,a,b, in ascending order
1) k, a, b 2) b, a, k 3) a, k, b 4) b, k, aAns:
2. If the traces of the matrices A and B are 20 and 8, thentrace of A + B =1) 28 2) 20 3) - 8 4) 12
Ans:
3. i. Trace of the matrix is called sum of the elements ina principle diagonal of the square matrix.
ii. The trace of the matrix
8 7 25 8 27 2 8
is 24
Which of the following statement is correct.1. Only i 2. Only ii3. Both i and ii 4. Neither i nor ii
Ans:
Sum and difference of the matrices
4. If A =
11651
,3419
B &
3A + 5B + 2X = 0 then X =
1)
32211416
2)
32211416
3)
32211416
4)
32211416
Ans:
SCALAR MULTIPE OF A MATRIXAND MULTIPLICATION OF MATRICES
5. If P =
431
, Q = 512 , then PQ =
1)
20481536512
2) 2032
3)
20
32
4) 19
Ans:
15 MATHEMATICS
6.
dcba
yx0
0=
1)
dyycbxax
2)
dy
ax0
0
3)
dybxcyay
4)
0
0dy
ax
Ans:
7. If A =
3021
and B = 13 then BA = =
1)
3003
2) 03 3) 33 4) 30
Ans:
8.
1001
IIf and
0010
E then 332 EI
1) EI 188 2) EI 364 3) EI 368 4) EI 32
Ans:
9. A=1 0
0 2
A3-A2 =
1) 2A 2) 2I 3) A 4) IAns:
10. If A =
4320
, KA =
24230
ba
, then the values of
k, a, b are respectively1) -6, -12, -18 2) -6, 4, 93) -6, -4, -9 4) -6, 12, 18
Ans:
11. The order of [x y z]
zyx
cfgfbhgha
is
1) 3 x 1 2) 1 x 1 3) 1 x 3 4) 3 x 3Ans:
16 MATHEMATICS
12. If 1 tan
tan 1
1 tantan 1
= a bb a
1) a =1, b = -1 2) a = sec2 , b=0
3) a=0, b = sin2 4)a = sin2 , b=cos2Ans:
13. If 1 20 1
A
then nA
1) 10 1
n
2) 20 1
n
3) 10 2
n
4) 1 20 1
n
Ans:
Transpose and properties of transpose ofmatrix
14. If 2A+BT =2 34 7
, AAT-B =4 50 1
then A = ----
1) 6 311 83
2)2 31 8
3)2 311 82
4) 0
Ans:
Problems on Order of Matrices :
15. symmetricaisPQPL ,121214532
matrix, Q is a skew-symmetric matrix thenP =
1)
146417672
2)
146417672
3)
123212
732
72
4)
246427674
Ans:
17 MATHEMATICS
16. List - I List - II
A=
2
2
2
11
1
1) Symmetric Matrix
is complex cube root of 1
B =
2 2 41 3 4
1 2 3
2) Skew symmetric
C=
a h gh b fg f c
3) Nil potent Matrix
D=
O c bc O a
b a O
4) Singular Matrix
5) Idempotent MatrixMatch of list-I from list - I
A B C D1. 4 3 1 22. 4 3 2 53. 4 5 2 34. 4 5 1 2
Ans:
17. If A = [aij] is a square matrix such that List - I List - II
A. aij = 1 if i=j 1. symmetricmatrix = O if i j
B. aij = O if i j 2. skew symmetric matrix
C. aij = O if i>j 3. unit matrixD. aij = i2-j2 i, j 4. diagonal matrix
5. upper triangular matrix
Correct match of List-I from List-IIA B C D
1. 3 4 5 12. 3 4 5 23. 3 5 4 24. 1 5 3 2
Ans:
DETERMINANTS AND ITS PROPERTIES
18. If A =
273408651
, then the cofactor of -7 =
................1) 44 2) 43 3) 40 4) 39
Ans:
19.
111098
AIf , then cofactor of a12 is
1) 11 2) 10 3) -11 4) -10Ans:
20.bacacbcba
=
1) a3 + b3 + c3 2) a3 + b3 + c3 - 3abc3) 3abc - a3 - b3 - c3 4) 0
Ans:
18 MATHEMATICS
21. 222 234
234111
1) 2 2) -2 3) 1 4) 0Ans:
22.bacbbaacba
baccc
2222
22
=
1) (a+b+c)2 2) (a+b+c)3
3) (a+b+c) 4) (a+b+c)4
Ans:
23.c
ba
111111111
=
1) 0 2) 1 3) abc 4)
cbaabc 1111
Ans:
24.
yxzzyxzyxxyz
1) xyz 2) 4 xyz 3) 2 xyz 4) 3 xyzAns:
25.zyyxxzxzyxzy
=
1) (x+y+z) (x-z)2 2) xyz3) 2(x+y+z) 4) xyz(x-z)2
Ans:
26.
2
2
2
cbcacbcbabacaba
1) 4abc 2) 4a2b2c2
3) a2b2c2 4) a2+b2+c2-ab-bc-caAns:
19 MATHEMATICS
27.133121211222
aaaaa
=
1) (1-a)3 2) (a-1)2 3) (a-1)3 4) (a+1)2
Ans:
28. If A =
00
0
abacbc
then (a2+b2+c2) A =
1) abc 2) a + b + c3) (a3 + b3+c3) 4) 0
Ans:
29. If x65432001
= 45 then x =
1) 4 2) 7 3) -5 4) -7
Ans:
30. If
1 24 1 72 4 6
x
is a singular matrix,
then x =1) 0 2) 1 3) -3 4) 3
Ans:
31. If A =
1231
, then the determinant A2A2 is
1) 5 2) 25 3) -5 4) -25Ans:
32. The real part of determinant of
cosisincosisinsinicossinicos
is
1) 2 cos 2) 2 sin 3) 0 4) 1
Ans:
20 MATHEMATICS
33. If the matrix
1000cossin0sincos
is singular, then =
1) 2) 2/ 3) 3/ 4) 4/Ans:
34. If x 0 and 0431
53121
xxxx
, then x =
1) 1 2) -1 3) 2 4) -2Ans:
35. If a, b, c are distinct and 1ccc1bbb1aaa
32
32
32
= 0, then
1) a + b + c = 1 2) ab + bc + ca = 03) a + b + c = 0 4) abc = 1
Ans:
36.cbcb
ccaababa
=
1) 4 abc 2) abc 3) 2a2 b2 c2 4) 4a2 b2 c2
Ans:
37. If A is
42476
268
is a singular matrix then =
1) 3 2) 4 3) 2 4) 5Ans:
38.
212101xsinxcos1xcosxsin
22
22
1) 0
2) 2 212cos x 10sin x 2
3) 2 212sin x 10cos x 2 4) 10 sin 2x
Ans:
21 MATHEMATICS
39.Maximum value of a second order determinant whose ev-ery element is either 0,1and 2 only1) 0 2) 1 3) 2 4) 4
Ans:
40.If 10,2
2 1 0
a b a bb c b c
then a, b, c, are in
1) A.P 2) H.P 3) G.P 4) AGPAns:
41.abccabbca
cba111
2222
=
1) 0 2) 2 3) 4 4) - 4Ans:
Adjoint of a Matrix :
42. )(,300030003
AAdjthenA
1) 3A 2) 6A 3) 9AT 4) 2AT
Ans:
43. A: If Matrix P=
1 2 11 1 22 1 1
then
det[adj (adj P)]=124
R: A is square matrix of order n×n then
det Adj (Adj A) = 2( 1)nA
The correct answer is1) Both A and R are true R is correct explanation to A2) Both A and R are true but R is not correct explanationto A3) A is true R is false4) A is false R is true
Ans:
INVERSE OF A MATRIX
44. If
111111111
A then det (A-1) =
1) 1 2) 1/2 3) 1/3 4) 1/4Ans:
22 MATHEMATICS
45. 10 0
0 0 det0 0
aIf P b then P
c
1) abc 2) a2b2c2 3) abc1
4) 222
1cba
Ans:
46. The inverse of the matrix
7 3 31 1 01 0 1
=
1)
1 1 13 4 33 3 4
2)
1 3 14 3 83 4 1
3)
1 1 13 3 43 4 3
4)
1 3 31 4 31 3 4
Ans:
47. If A =
23
22, B =
0110
, then 111 )( AB =
1)
3222
2)
2223
3)
32
22101
4)
22
23101
Ans:
48. If
0281
02yx 3
, then
1
02yx
=
1)
1220
2)
1001
3)
1280
4)
41
21
210
Ans:
49. If A = )x(f1000xcosxsin0xsinxcos
then A-1 =1) f(- x) 2) f(x) 3) - f(x) 4) -f (-x)
Ans:
23 MATHEMATICS
50. A =
1232
then 8A-1 =
1)
2231
2)
22
31
3)
2231
4)
2231
Ans:
51.
1
10331
=
1)
13310
2)
13310
3)
10331
4)
10331
Ans:
52. List I List II
A.
2 2 22 2 22 2 2
= I.
2 55 8
B. A=
0 1 01 0 00 0 1
thenA-1 = II. AT
C. A=1 23 4
thenA+AT= III.5 44 1
D. A=3 11 2
, IV. 32
A2=A+B then BMatch the followingA B C D
1. IV II I III2. II IV III IV3. IIII II IV I4. I II III IVAns:
RANK OF A MATRIX
53. The rank of the matrix A = is
111111111
1) 3 2) 2 3) 1 4) 0Ans:
24 MATHEMATICS
54. The Rank of 1 0 42 1 3
is
1) 1 2) 2 3) 0 4) 3Ans:
55. The ranks of the matrices in decending order
A.
1 4 12 3 00 1 2
B.
1 1 11 1 11 1 1
C.
1 2 32 3 40 1 2
1) A,B,C 2) A,C,B3) B,C,A 4) C,A,B
Ans:
SOLUTION OF SIMULTANEOUS EQUATION
56. If the lines 01332 yx
0104 yx
0215 yaxare concurrent then the value of a is1) 1 2) 2 3) 3 4) -2
Ans:
57. The equations
2763,522,042 zyxzyxzyx
have
1) unique solutions
2) no solution
3) infinite number of solutions
4) two solutions
Ans:
58. The number of solutions of the system equations 2x+y-
z=7, x-3y+3z=1,
x+4y-3z=5 is.....
1)3 2)2 3)1 4)0
Ans:
25 MATHEMATICS
59. A: The number of values of k for which the equations.
1 8 4k x y k and kx+(k+3)y=3 k-1 has
infinitely many solution is 1
R : Two linear equations, a1x+b1y+c1=0, a2x+b2y+c2=0
has infinite numbers solution if
det 1 1
2 2
a ba b
=0
The correct answer is
1) Both A and R are true R is correct explanation to A
2) Both A and R are true but R is not correct explanation
to A
3) A is true R is false
4) A is false R is true
Ans:
60. The set of equations 0432 zyx ,
0643 zyx , 0854 zyx
1) Infinite solutions 2) is inconsistent
3) has unique solution 4) has two solutions
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61. The real value of 'a' for which the equations ax+y+z
=0, -x+ay+z=0, -x-y+az = 0 have non-zero solution is
1) 1 2) 0
3) -1 4) all the above
Ans:
62. The equations x + y - 2z = 0, 2x - 3y - z = 0x - 5y + 4z = k are consistent if k =1) 1 2) -13) 0 4) any real value
Ans:
63. If x2 + y2 + z2 0, x = cy + bz, y = az + cx, and z = bx+ ay, then a2 + b2 + c2 + 2abc =1) 2 2) a + b + c3) 1 4) ab + bc + ca
Ans:
64. The system of equations 3x - 2y + z = 0,0z15y14x and x + 2y - 3z = 0 have non -
zero solution, then =1) 1 2) 3 3) 5 4) 0
Ans:
26 MATHEMATICS
KEY1) 4 2) 1 3) 3 4) 35) 1 6) 1 7) 3 8) 39) 1 10) 3 11) 2 12) 213) 4 14) 1 15) 3 16) 417) 2 18) 1 19) 4 20) 321) 2 22) 2 23)4 24)225) 1 26)2 27)3 28)429)2 30)3 31)2 32)333)4 34)2 35)4 36)137)1 38)1 39)4 40)341)1 42)1 43)1 44)445)3 46)4 47)1 48)449)1 50)4 51)2 52)153)3 54)2 55)2 56)357)2 58)3 59)3 60)161)2 62)4 63)3 64)3