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

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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

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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

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21. If A = diagonal (3,3,3) then 4A 1)12A 2)81A 3)684A 4)27A

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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

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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

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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

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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

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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

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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

Ans:

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

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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

Ans:

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

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