Mathametics-120 nos ready.doc

7/27/2019 Mathametics-120 nos ready.doc

1/12

SolutionSThe innovators

MATHEMATICS

1. Adjoint of the matrix

211

223

121

is

a.

412

113

142

b.

411

114

232

c.

01

12d.

131

221

122

2. Rank of the matrix

432

220

101

is

a. 2 b. 0 c. 3 d. 1

3. Inverse of

1312

43

is

a.93

1

35

46b. 93

65

43c.

313

1213d. None of

these

4. Given A is a singular matrix where A =

29

51then a = ?

a.2

5b.

5

2

c. 5

2

d. 5

3

5. If A is a square matrix the A+ AT isa. Symmetric b. Skew symmetric c. 0 d. None of these

6. The rank of a null matrix is

a. 1 b. 0 c. 2 d. None of these

7. If A & B are square matrices of the same order then adj (AB) = ?a. (adj A).(adj B) b. (adj B).(adj A) c. (adj BA) d. none of these

8. adj (AT) = ?

a. adj A b. (adj A)T c. 1 d. None of these9. A square matrix A is orthogonal if A.AT = ?

a. 1 b. 0 c. A d. None of these

10.

+

sincos

sincosis

a. Singular b. Non singular c. Orthogonal d. None of these

1


2/12


11. Characteristic equation of

31

23is

a. 0672

=+ b. 0762

=+ c. 066

2=++ d. None of these

12. If A is an orthogonal matrix |A| = ?

a. 1 b. -1 c. 0 d. Both (a) and (b)

13. If A =

a3

25and Sum of eigen values is 15 then a = ?

a. 3 b. -3 c. 10 d. None of these

14. Given A =

300020

001

then eigen values are

a. 1, 2, 3 b. 0, 0, 0 c. 0, 2, 0 d. None of these

15. Every square matrix satisfies itsa. Characteristic equation b. Eigen value

c. Both (a) and (b) d. None of these

16. Write the quadratic form corresponding to the matrix

42

21

a. x2 + 4xy 4y2 b.x2 + 4xy + 4y2 c. x2 + 2xy 4y2 d. None of these

17. (A + B)2 = ? [If A and B are two square matrixes of the same order]a. A2 + 2AB + B2 b. A2 + AB + BA + B2 c. A2 + B2d. None of these

18. A =

64

32is

a. Singular b. Non singular c. Invertible d. None of these

19. The equation AX = B is consistent if rank of the coeiffient matrix and augmented matrix area. Equal b. Not equal c. 1 d. None of these

20. The maximum value of the rank of 4 x 5 matrix is

a. 4 b. 5 c. 20 d. 9

21. The matrix associated with quadratic form 3x12

+ 8x1x2-3x 22

is

a.

34

43b.

34

43c.

88

33d. None of these

22. Cofactor of 3 in

41

32

a. -1 b. 1 c. 4 d. None of these

23. If A =

41

23 then A (adj A) equals

2


3/12


a.

100

010b.

010

100c.

101

110d. None of these

24. If 5 , 5 are the eigen values of A =

12

21then eigen values of AT is

a. 5 , 5 b.5

1,

5

1 c. 5 , 5 d. None of these

25. Rank of a matrix can be found by converting the matrix toa. Normal form b. Characteristic form c. Eigen form d. None of these

26. Characteristic roots of a Hermitian matrix are

a. 0 b. Real c. Imaginary d. None of these

27. The sum of eigen values of a matrix

311

212

231

is

a. -3 b. 3 c. 0 d. None of these

28. Rank of the unit matrix of order 4 is


29. If A =

300

520

321

then eigen values of A-1 are

a. 1,3

1,

2

1b. 1, 2, 3 c. -1, -2, -3 d. None of these

30. A square matrix A is orthogonal ifa. A = A2 b. AT= A-1 c. A . A-1=1 d. None of these

31. Curvature of x2 + y2 = 25 is

a.5

1b. 5 c. 0 d. None of these

32. Curvature of a circle of radius r is

a.r

1b. r c.

2

1

rd. None of these

33. Radius of curvature at any point of the curve 3cosax = , 3sinay = is

a. 0 b. cossin3a c. 3a d. None of these

34. Radius curvature of the parabola y2 = 4ax at (at2, 2at) is

a. ( ) 23212 ta + b. ( ) 231 ta + c. ( )t

t232

12 +d.

( )2

2321

2t

ta

+

35. Locus of centres of curvature of a curve is calleda. Circle of curvature b. Evolute c. Envelope d. None of these

36. Centre of curvature of y = x2 at

=4

1,

2

12atxy is

3


4/12


a.

4

3,

2

1b.

4

3,

2

1c. [10, 0] d. None of these

37. Centre of curvature of xy = c2 at (c, c)

a. [2c, 2c] b. [c, c] c.

cc

1,

1d. None of these

38. Envelope of the family of linesn

anxy += , where a is a constant

a. Parabola y2 = 4ax b. Circle c. Ellipse d. None of these

39. u = x3 + y3 3a has a maximum or minimum at

a. [a, a] b. [0, 0] c. [1, 1] d. None of these

40. Shortest distance from the point (1, 0) to the parabola y2 = 4xa. 1 b. 0 c. -1 d. None of these

41. Radius of curvature of sinar=a.

3

ab. 1 c. 0 d. None of these

42. ( ) 53322 2, xyxyxyxyxu +++= has.at (0, 0)a. Maximum of minimum b. Neither maximum nor minimum

c. Maximum only d. Minimum only

43. Sin X . Sin Y . Sin Z has maximum at

a. === zyx b.3

=== zyx

c.2

=== zyx d. None of these

44. If ?sin1 =

=

x

uthen

y

xu

a. 22

1

xy b. 22

xy

xy

c.

y

x1cos d. None of these

45. If sin,cos ryrx == then ?= y

a. 22yx

x

+b. 22

xy

y

+

c. 22

1

yx +d.

22

2

yx

x

+

46.3223

53 yxyyxx +++ is a homogeneous function of ordera. 3 b. 2 c. 0 d. None of these

47. ( )33 yxxU = is homogeneous of degreea. 2 b. 1 c. 0 d. None of these

48. nth derivative of eax

a. aneax b. eax c. an d. None of these

49. nth derivative of cos (ax + b)

a. )2cos(n

baxan

++ b. )2sin(n

bax ++ c. Sin [ax + b] d. None of these

4


5/12


50. nth derivative of log (ax + 1)

a. ( ) ( )( )n

nn

x

an

+

12!11

1b.

na

1c.

( )nn

bax

a

+d. None of these

51. If ( )yxU tantanlog += then ?=

x

u

a.yx

x

tantan

sec2

+b.

yx

y

tantan

sec2

+c.

yx tantan

1

+d. None of these

52. If cotsq = then sq is equal to

a. cot b. 2

cosec c. 2coscot ecs d. cot2

1

53. If vuyvux =+= , then ?=

x

u

a. 2

1

b. 2

1

c. 0 d. None of these

54. If ( )22log yxz += then ?=

x

z

a. 22 yx

x

+b. 22 yx

y

+c. 22

1

yx +d. None of these

55. If teyttxt

cos,sinlog +=+= then ?=dx

dy

a. tt

tet

cos1

sin

+

b. t

tet

cos1

sin

+

c. te

t

sin d. None of these

56. If nnn ayx =+ then ?=

dx

dy

a.1

1

n

n

y

xb.

1nny c. 1nnx d. 0

57. If xyzeU= then ?=z

u

a.xyzxye b. xyze c. 0 d. None of these

58. If x= re cos , thenx is equal to

a. --y

y2sin, b. re c.

y

y2sind. None of these

59. nth derivative ofx

1

a.1

!)1(+

n

n

x

nb.

nx

1c. nx d. None of these

60. nth

derivative ofx

ex.a.

xex b.

xn ex c. )( nxex + d. None of these

5


6/12


61.0x

Lt ( ) xSinx tan

a. 0 b. 1 c. -1 d. None of these

62. 0xLt

x

xSin1

1

tan


63.0x

Lt )1log(

1sin

x

xex

++


64.1x

Lt ( ) 2tan2 xx

a. 0 b. 1 c. a d. None of these

65. If ( )xyzzyxU 3log 333 ++= then ?=

x

u

a.xyzzyx

yzx

3

33333

2

++

b.xyzzyx 3

1333 ++

c.xyzzyx

x

3

3333

2

++d. None of these

66.yx

yx

++ 44

is homogeneous of order

a. 3. b. 4 c. 2 d. None of these

67. If ( ) ( ) cos1,sin =+= ayax then ?2

2

=dx

yd

a.2

sec4

1 4

ab.

2sec

2 c. 0 d. None of these

68. The partial derivative of22

2 byhxyax ++ with respect to x isa. byax 22 + b. hyax 22 + c. hxax 22 + d. byhx 22 +

69. If ( )yxf , is a homogeneous function of degree n, possessing continuous partial derivatives of firstorder then

a. nf

x

fy

x

fX =

+

.. b.

22

2

2

2

2

.. fn

y

fy

x

fX =

+

c. fx

fy

x

fX =

+

.. d. nf

y

f

x

f=

+

2

2

2

2

70. The series ...........5

1

4

1

3

1

2

11

222

2

+++++ is

a. Absolutely convergent b. Not absolutely convergent

c. convergent d. Both (a) and (c)

71. The series ...............12 +++ rr diverges if

a. 1


7/12


72. The sum of the alternating series ...........4

1

2

11 + is

a. 0 b. Infinite c. log2 d. None of these

73.

=

+1

11log

n n

is

a. Convergent b. Divergent c. Oscillatory d. None of these

74. Let nu be a series of positive terms. Given that nu is convergent and also If 1tn

,1

ku

u

n

n =+

then nu coverages for K

a. < 1 b. 1 c. > 1 d. 1

75. The series 1 + 2 + 3 +.+ .................+n isa. Convergent b. Divergent c. Oscillatory d. None of these

76. The series ....................4

5

3

4

2

3

1

22222=+ is

a. Conditionally convergent b. Absolutely convergent

c. Divergent d. None of these

77. The series 1 1 + 1 1 +isa. convergent b. Divergent c. Oscillatory d. None of these

78. The series pn1

is convergent if P is

a. Greater than one b. Equal to one

c. Less than one d. None of these

79. ....................4

1

3

1

2

11

222=++ is

a.6

2

b.3

22

c.

6

d.

8

80. The series ...................2

11 ++ is

a. Divergent b. Convergent c. Oscillatory d. None of these

81. The series ..................

4

1

3

1

2

11

432+++ is


82. An absolutely converging series isa. Divergent b. Conditionally convergent c. Convergent d. Oscillatory

83. The series ..............1

...........2

1

1

1++++

kkk nis divergent if k

a. k > 1 b. k < 1 c. k 1 d. None of these

84. The sum of the alternating series ....................4

1

3

1

2

11 ++ is

a. 0 b. Infinite c. log 2 d. None of these

7


8/12


85. The series ...........21 ++uu is convergent ifLt ?=

n

nu


86. If ..................21 ++aa converges to s, then ..................32 ++aa converges toa. S b. S - a c. S + a d.None of these

87. Every absolutely convergent series is


88. The series

=1

1

nnn

nis..

a. Convergent b. Divergent c.Oscillatory d. None of these

89. ( ) ?=ateL

a.a

1b.

as1

c.as +

1d. None of these

90. ( )natteL .

a.1

1+ns

nb.

( ) 11

+ nasn


91. ( )attL cos

a.22 as

s

+b.

22

22

as

as

+

c.22 as

s

d.

( )2221

as +

92.( )

t

eL

t1

a.

s

s 1log b.

1

11

ss

c.2

1

1

11

s

ss

d.None of these

93. ( )tetL 33.

a.( )43

6

+sb.

( )33

1

+sc.

( )43

1

+sd.

( ) 43

3

+s

94.

3

1 1

sL

a.2

2

1t b. t

3

2c. 24t d. None of these

95. ( )

+

2

1

2s

s

L

8


9/12


a. tt tee22

2 b. te t 22 c. tt tee 22 2 d. None of

these

96.( )

+

22

1 1

assL

a.2

cos1

a

atb.

2

cos1

a

at+c. at

asin

1

d. None of these

97. If ( ) ( ){ }sftfL = then ( ){ } ?=atfL

a. ( )sfa

1b. ( )asaf c.

a

sf d. None of these

98. ( )atteL

a.( ) 1

!+ nas

nb.

( )2!1

as c.

1

12 +s

d. None of these

99. ( )btL sin

a.22 sb

b

b.

22 bs

b

+c.

1

12 +s

d. None of these

100.

49

12

1

sL

a. t7sinh7

1b. t7cosh

7

1c.7

1d. None of these

101.

ns

L 11

a.( )!1

1

n

tnb.

!n

tnc. !n d. None of these

102.

+

2

11

sL

a. te 2 b. te2 c. t d. None of these

103. ( )tL 2cosh

a.42 s

s b.42 +s

s c.4

22 +s

d.4

22 s

104.

0

2 .sin dttte t

a.25

4b.

25

2c. 1 d. None of these

105.

0

4.. dtee

att

a.

a4

1b.

a+4

1c.

a4

1d. None of these

9


10/12


106. ( ) 22 +=xxf isa. Even b. Odd c. Neither d. Both

107. xx sin2 is

a. Even b. Odd c. Does not exist d. Neither

108. If )(xf is odd a

adxxf ).(

a. 0 b. 1 c. a

dxxf0

).(2 d. None of these

109. Value of 0a in the Fourier series expansion of2x in ( )ll,

a.3

2 2lb.

3

2l

c.3

4 2ld. None of these

110. Value of 0a in the Fourier series expansion of4

2xin ( ), is

a.12

2

b.6

2


111. If


11/12


120. Formulae for a0 in Fourier series is

a. ( ) dxxf .1

b. ( ) dxxf .2

c. dxxf ).( d. None of these.

LET MATHAMATICS KEY

TEST PAPER

1. a 2. c 3. d 4. b 5. a 6. b 7. b 8. b 9. a 10. a 11. b 12. d

13. c 14. a 15. a 16. a 17. b 18. a 19. a 20. a 21. a 22. a 23.a 24. a

25. a 26. b 27. b 28. b 29. a 30. b 31. a 32. a 33. b 34. a 35. b 36. b

37. a 38. a 39. a 40. a 41. a 42. b 43. b 44. a 45. a 46. a 47. b 48. a

49. a 50. a 51. a 52. a 53. a 54. a 55. a 56. a 57. a 58. a 59. a 60. c

61. a 62. b 63. a 64. c 65. a 66. a 67. a 68. b 69. a 70. a 71. b 72. c

11


12/12


73. a 74. c 75. b 76. a 77. c 78. a 79. a 80. b 81.ca 82. c 83. c 84. c

85. a 86. b 87. a 88. a 89. a 90. b 91. a 92. a 93. a 94. a 95. d 96. a

97. a 98. a 99. b 100. a 101. a 102. a 103. a 104. a 105. a 106. a 107. b 108. a

109. a 110. a 111. a 112. a 113. a 114. c 115. b 116. a 117. c 118. a 119. c 120. a

12

Documents

Mathametics-120 nos ready.doc