CUSAT CAT 2016 Question Paper Mathematics

1

MATHEMATICS (FINAL)

 1 3 2   x   1 
    
1. The system of equations  3 0 5   y    2 
 2 5 0   z   3
    

A. has no solution
B. has one and only one solution
C. has infinite number of solutions
D. None of the above

2. If a and b are real numbers, then sup a, b 

ab ab
A.
2
a b a b
B.
2
a b a b
C.
2
a b a b
D.
2

3. If, for x  ¡ ,   x  denotes the integer closest to x (if there are two such integers take the
12
larger one), then    x  dx equals
10

A. 22
B. 11
C. 20
D. 12

2

2 1 0
 
4. The eigen values of the matrix  0 2 1  are
0 0 2
 
A. 1,1,1
B. 2,2,2
C. –2, –2, –2
D. –1, –1, –1

 1
 n 1
5. The value of  is
n 1 n

A. log 2
B. e2
C. 0
D. 1

0 0 0 0 2 
 
1 0 0 0 4 
6. The rank of the matrix  0 1 0 0 5  is
 
0 0 1 0 1
0 0 0 1 13 


A. 5
B. 4
C. 3
D. 2


2n
7. 
n 1  n  1 !
is

A. e
B. 2e
C. 2e2
D. 2e2

3

 2 
8. If A    , and A2  125 , then the value of  is
2 

A. 5
B. 3
C. 2
D. 1

n n n
9. The algebraic equation x  y  z has no integer solution when

A. n  2
B. n  2
C. n  0
D. n  2

10. Which of the function f : R  R is one-one and onto?

A. f  x   x3  2

B. f  x   sin x

C. f  x   cos x

D. f  x   x 4  x 2

11. Let P  x  be a non-constant polynomial such that P  n   P  n  for all n  N .
Then P '  0  is

A. –1
B. 1
C. 0
D. –2

4

 3 1
12. If A    , then A2  5 A  7 I is equal to
 1 2

0 0 
A.  
0 0 
1 0 
B.  
0 1 
1 0 
C. 2  
0 1 
D. None of the above

13. If A is any non-identity 3  3 matrix such that A2  A , then A is

A. Singular
B. Non singular
C. Regular
D. None of the above

14. If every cross-section of a bounded surface in three dimensions is a circle, then the
surface must be a

A. cylinder
B. sphere
C. cone
D. third-degree surface

15. For any complex number z, the minimum value of z  z  1 is
A. 1
B. 0
C. 1/2
D. 3/2

5

16. The number having a recurring decimal representation 1.414141… is

A. real but irrational
B. not real
C. rational
D. neither rational nor real

17. The derivative with respect to x of the product
1  x  1  x 1  x 1  x   ....  1  x  at x  0 is
2 4 8 56

A. 0
B. 1
C. 8
D. 56

18. If  ,  and  are roots of the equation x3  px 2  qx  r  0, then  2   2   2 is equal to

A. p 2  2q

B. p 2  2q

C. 2 p  q 2

D. 2 p  q 2

19. Given that 2  i 3 is one of x 3  5 x 2  11x  7  0 then the other roots are

A. 2  i 3,  1

B. 2  i 3,1

C. 2  i 3,1
D. None of the above

6

20. The quadratic function on a single variable attains its maximum value 5 at 3. The
function is

A. ax 2  6ax  9a  5, a  0

B. ax 2  6ax  9a  5, a  0

C. ax 2  6ax  9a  5, a  0
D. None of the above

21. Let n be a two digit number. P  n  is the product of the digits of n and S  n  is the sum
of the digits of n. If n  P  n   S  n  then the unit digit of n is

A. 1
B. 5
C. 7
D. 9

22. The value of 20  20  20  ... is

A. – 4
B. 5
C. 4
D. – 5

23. If A, B, C are three sets with cardinality m, p, q respectively such that B I C   , then the
cardinality of  A  B  U  A  C  equals

A. mpq
B. m  p  q 

C. m  p  q  pq 

D. m  pq

7

24. Which among the following is not a group under usual multiplication?

A. ¡  0

B. ¤  0

C. ¤ 
D. ¥

25. If log 27 x  log3 27, then x is

A. 27
B. 3
C. 327
D. 27 3

26. If  is a homomorphism of a group G onto a group G with kernel K , then G/K is
isomorphic to

A. G
B. G
C. G / K
D. None of the above

27. If G is a group and for a  G no positive integer m exists such that a m  e , then the order
of a is

A. finite
B. m
C. m + 1
D. infinite

8

28. The derivative of et with respect to t is

et
A.
2 t

2 t
B.
et
C. 2 tet

D. 2 tet

1 1 1
29.  dx dy , is equal to
0 0
1  x 2  1  y 2 
2
A.
2
2
B.
3
2
C.
4
D. None of the above


2 2 2
30.  sin x cos xdx is equal to
0


A.
8

B.
16

C.
32
D. 1

9

 1
31.  1 x dx is equal to
2 4
0
 
5
A.
32
5
B.
16
5
C.
32
5
D.
16


2 
32.  cos 2 is equal to
6
0

 1
A. 
12 4
 1
B. 
12 4
1 
C. 
4 2
1 
D. 
4 2

1
33. The series  p
is divergent if
n  log n 

A. p  1
B. p  1
C. p  1
D. p  1

10

34. A non-decreasing sequence which is bounded above is

A. divergent
B. convergent
C. oscillating
D. unbounded

35. Let f : ¡  ¡ be defined by


 1  cos 4 x if x  0
 x2

f  x    a, if x  0

 x
if x  0
 16  x  4


If f is continuous for all x, then the value of a is

A. 0
B. 4
C. 8
D. 5

36. The cardinal number of the empty set is

A. 1
B. 0
C. 
D. 1

37. Every closed and bounded set in R n is

A. empty
B. open
C. compact
D. convex

11

1 
38. In which subspace the sequence   is Cauchy but not convergent
n

A.  0,1

B.  0,1

C.  0,1

D.  0,1

1 2
39. The collection of open intervals  ,  , n  1, 2,3,.... is an open covering of the interval
n n

A.  0,1

B. 1, 2 

C.  0,  

D. 1,  

40. The set of all rational numbers is a

A. empty set
B. finite set
C. countable set
D. uncountable set

2 dx
41.   2  cos x  is equal to
0

2
A.
3

B.
3
2
C.
3

12

D. None of the above

2 2 2
42. The whole length of the asteroid x 3  y 3  a 3 is

A. 6a
B. 8a
C. 4a
D. None of the above

43. The condition for the point  x, y  to lie on the straight line joining the points
 0, b  and  a, 0 is
x y
A.  1
a b
x y
B.  1
a b
x y
C. 2
 2 1
a b
D. None of the above

44. The centroid of the triangle whose vertices are  2, 4,  3 ,  3,3,  5 and  5, 2,  1 is

A.  2,  3,  3

B.  3,3,  2 

C.  3,  2,  3

D.  2,3,  3

45. The triangle whose vertices are  1,1, 0  ,  3, 2,1 and 1,3, 2  is

A. isosceles triangle
B. right-angle triangle
C. equilateral triangle
D. None of the above

13

46. The coordinates of the point at which the line joining the points  4,3,1 and 1,  2, 6 
meets the plane 3 x  2 y  z  3  0 is

A.  2,  7,11

B.  2.7,11

C.  2,  7,  11

D.  2,7,11

47. The center of the sphere x 2  y 2  z 2  6 x  8 y  10 z  1  0 is

A.  5,  4,3

B.  3, 4,  5

C.  5,  4,  3

D.  3,  4,5

48. The equation of the right circular cone with its vertex at the origin, axis along z-axis and
semi-vertical angle  is

A. x 2  y 2  z 2 tan 2 

B. x 2  y 2  z 2 tan 2 

C. x2  y 2  z tan 2 

D. x2  y 2  z tan 2 

49.    A is equal to

A. 0
B.  2 A    . A

C. 2 A    . A

D.     A

14

15

r r r r
50. The curl of F  x 2i  y 2 j  z 2 k at 1, 2,  3 is

r r r
A. i  2 j  3k
r r r
B. i  2 j  3k
r r r
C. 2i  4 j  6k
r r r
D. 2i  4 j  6k

 yi  xj
51. Let F  . Then  F is
x2  y2

A. 0
B. 1
C. – 1
D. None of the above

52. The probability mass function or probability density function for which the mean in units
and the variance in square units are same is

A. Binomial
B. Poisson
C. Standard Normal
D. Geometric

53. The chance of getting at least 9, in a single throw with two dice is

4
A.
36
3
B.
36
5
C.
18
1
D.
18

16

C
54. A random variable X has a probability density function f  x   ,    x   . Then
1  x2
the value of C is

A. 
B. 1
1
C.

2
D.


55. If f  z  and f  z  are analytic in a region D, then

A. f  z  is constant in D

B. f  z  is continuous in D

C. f  z  is not differentiable everywhere in D

D. f  z   z

z2 1
56. The value of  dz , where C is a circle of unit radius with center at z  1 is
C z2 1

A. 0
B. 2 i
C. 2 i
D. 1

17

1 x y
57. If  x  iy  3  a  ib , then the value of  is
a b

A. 4  a 2  b 2 

B. 4ab
C. 4  a 2  b 2 

D. 5ab

58. The value of 1  i  1  i 2 1  i 3 1  i 4  ...... 1  i 50  is

A. 50
B. 1
C. 0
D. i
z
59. If   and   1, then z lies on
z i

A. a circle
B. an ellipse
C. a parabola
D. a straight line

z2
60. The residue of at z  1 is
 z  1 z  2  z  3
A. –8
B. 1/2
C. –6
D. 0

18

61. If u  x, y   xy is a harmonic function, a harmonic conjugate of u is

x2
A.
2
B. x 2  y 2

x2 y2
C. 
2 2
y 2 x2
D. 
2 2

62. The solution of a homogeneous initial value problem with constant coefficient is
y  3xe2 x  6 cos 4 x . Then the least possible order of the differential equation is

A. 4
B. 5
C. 6
D. 3

x 2 x3
63. The differential equation of the curve y  1  x    ... is
2! 3!

dy
A. x
dx
dy
B.  x
dx
dy
C. y
dx
dx
D. y
dy

19

64. The differential equation of all circles which pass through the origin and whose centers
are on the x-axis is

dy
A. y 2  x 2  2 xy 0
dx
dy
B. x 2  y 2  2 x 0
dx
dy
C. y 2  x 2  2 xy 0
dx
dy
D. y 2  2 x 0
dx

65. The correlation coefficient lies between

A. 0 and 1
B. –1 and 0
C. –1 and 1
D. None of the above

66. The average monthly production of a factory for the first 8 months is 2500 units, the next
4 months is 1200 units. The average monthly production of the year will be

A. 2066.55
B. 5031.10
C. 4021.10
D. 3012.11

67. The equation of the curve that passes through the point 1,1 and has at every point the
y
slope is
x

A. xy  1
B. xy  1
C. xy  2
D. xy  2

20

d2y dy
68. The solution of the differential equation 2
 3  2 y  0, subject to initial
dx dx
conditions y  0   0, y  0   1 is

A. e x  e 2 x
B. e x  e2 x
C.  e x  e 2 x
D. None of the above

d2y
69. The general solution of the differential equation  y  x is
dx 2

A. sin x  cos x  x
B. sin x  cos x  1
C. sin x  cos x  x
D. sin x  cos x  1

70. The partial differential equation ut  kuxx is

A. one dimension wave equation
B. elliptic
C. one dimension heat equation
D. not parabolic

2
71. The partial differential equation  u xx   u yy  a  x, y  u x  b  x, y  u  4e x  0 is

A. linear
B. quasilinear
C. nonlinear
D. homogeneous

21

dy
72. The solution of the IVP  x 2 y  3x 2 , y  0   1 is y =
dx
3
A. 3  ce x / 3 , c is a constant
3
B. 3  2e x / 3
3
C. 3  3e x / 3
3
D. 3  2e x

73. If q  x   0, then any nontrivial solution of y  q  x  y  0

A. can have more than one zero
B. can have at most one zero
C. cannot have any zero
D. has exactly one zero

74. A solution of the partial differential equation ut  cux  0 is u  x, t  

A. sin  x  t 

B. cos  x  ct 

C. cos  cx  t 

D. cos xt

x x
75. The general solution of the partial differential equation  xy is
x y

x2 y 2
A. z  a  b
2 2a
x2 y2
B. z  a  b
2 2a
x2 y2
C. z  a  b
2 2a

22

x2 y 2
D. z  a  b
2 2a

x
76. The function f :  1,1  ¡ defined by f  x   is
1 x
A. one-one but not onto
B. not onto
C. one-one and onto
D. neither one-one nor onto

k 1
77. For each n  ¥ , let an  
n  1 . Then the sequence  an  is
k 1 k

A. not a Cauchy sequence
B. a convergent sequence
C. not a bounded sequence
D. convergent to 0

78. If a function f is continuous in  a, b and differentiable  a, b  , then there exists at least
one number    a, b  such that f  a  h   f  a   hf   a   h  . It is the statement of the

A. Roll’s Theorem
B. Lagrange’s Mean Value Theorem
C. Cauchy’s Mean Value Theorem
D. Taylor’s Mean Value Theorem

79. The set of all polynomials with rational coefficients

A. is not countable
B. is finite
C. does not contain Q
D. countable

23

  1  if x  0
 x sin  
80. The function f : ¡  ¡ defined by f  x    x .
0 if x  0


Then at the point 0

A. f is continuous
B. f is not continuous
C. f is differentiable
D. None of the above

2 n1  3n 1
81. lim is equal to
n  2 n  3n

A. 3
B. 2
C. 1
D. 0

 x  1 if x  1
82. The function f : ¡  ¡ defined by f  x   
1  x if x  1
Then at the point 1

A. f is continuous
B. f is not continuous
C. f is differentiable
D. None of the above

83. The smallest number with 18 divisors is

A. 360
B. 175
C. 185
D. 180

24

sin z
84. The value of the  z 2  2
dz

z 2 
 

A. –1
B. 1
C. 0
D. 

4
 1 i 
85.   is equal to
 2

A. 1
B. 0
C. 2
D. –1

86. The set  z  £ : z  3i  3 geometrically represents a

A. Parabola
B. Circle
C. Ellipse
D. Hyperbola

25

87. The bilinear transformation which maps the points z1  0, z2  i and z3  1 into
w1  i, w2  1 and w3  0 respectively is

i  z  1
A.
z 1
i  z  1
B.
z 1
1 z  i 
C.
z i

D.
 z  i
z i

z
88. The first two terms of the Laurent’s series of f  z   , valid for z  2, is
 z  1 2  z 
1 3
A.  2
z z
1 3
B.   2
z z
1 3
C.  2
z z
D. None of the above

89. Z6 is

A. a ring
B. an integral domain
C. a field
D. None of the above

26

90. The set of all square matrices of order 2 over the set of real number is

A. Ring
B. Integral Domain
C. Field
D. None of the above

91. Let S be the set of functions f : ¡  ¡ which are solutions to the differential equation
f   f   2 f  0. Then S is

A. not a vector space
B. a vector space of dimension greater than 3
C. a vector space of dimension 3
D. a vector space of dimension less than 3

92. The dimension of the subspace W   x1 , x2 , x3 , x4   ¡ 4 : x1  x2  x3  x4  0

A. 1
B. 2
C. 3
D. 4

93. Let W be the subspace spanned by
S  1, 0, 0, 0  ,  0,1, 0, 0  , 1,1, 0, 0  , 1,1,1, 0  ,  2, 0,3, 0  .
Then the dimension of W is

A. 4
B. 5
C. 2
D. 3

27

94. The number of subsets (including the empty subset and the whole set) for a set of n
elements is

A. n
B. n2
C. nn
D. 2n

95. Let G be the complete graph on n vertices. Then the number of edges in G is

A. n
B. n2
C. 2n
n  n  1
D.
2

96. For n  4, let G be a graph with n vertices and n edges. Then

A. G is a star
B. G should contain a cycle
C. G is acyclic
D. G is a complete graph

97. If Z is the optimal solution of a LPP and Z  is the optimal solution of its dual, then

A. Z  Z 
B. Z  Z 
C. Z  Z 
D. Z  Z 

28

98. The differential equation obtained by eliminating f from z  f  x 2  y 2  when
f f
p and q  is
x y

A. py  qx
B. pq  xy
C. px  qy
D. x  y

99. The differential equation of the family of curves y  e2 x  A cos x  B sin x 
where A and B are constants is

d2y dy
A. 2
 4  4y  0
dx dx
d2y dy
B. 2
 4  5y  0
dx dx
d2y dy
C. 2
 4y  5y  0
dx dx
d2y dy
D. 2
 4  4x  0
dx dx

2
2  dy   dy 
100. Let  y  c   cx be the primitive of the differential equation 4 x    2 x    y  0.
 dx   dx 
Then number of integral curve(s) which will pass through 1, 2  is

A. one
B. two
C. three
D. four

29

101. If y1 and y2 are two independent solutions of the differential
2
d y dy
equation 2
 P  x   Q  x  y  0 , where P  x  and Q  x  are continuous functions
dx dx
of x , then which of the following is true?

A. y1 y1'  y2 y2'  Ce 
 Pdx

B. y1 y2'  y2 y1'  Ce 
 Pdx

C. y1 y1'  y2 y2'  Ce 
Pdx

D. y1 y2'  y2 y1'  Ce 
Pdx

1 1 2
102. Let the chances of solving a problem given to three students are , , . The probability
2 3 5
that the problem will be solved is

1
A.
15
1
B.
5
3
C.
5
4
D.
5

103. To measure flatness or peakedness of a distribution we use

A. skewness
B. kurtosis
C. correlation
D. standard deviation

30

104. A method for solving linear programming problem without using artificial variables is

A. Two-phase method
B. Big-M method
C. Dual simplex method
D. Revised simplex method

105. In an assignment problem of order n , in the reduced cost matrix, the minimum number
of lines needed to cover all zeros will be

A. n
B. n  1
C. n  1
D.  n  1 n  1

106. The angle between two forces of equal magnitude P when their resultant also has the
same magnitude P is

A. 120o
B. 60o
C. 30o
D. 45o

107. The horizontal range of the projectile is maximum when the particle is projected at an
angle of

A. 90o to the horizontal
B. 30o to the horizontal
C. 60o to the vertical
D. 45o to the vertical

31

108. The period of oscillation of a simple pendulum is

g
A. 2
l

2 l
B.
g

2 l
C.
g

l
D. 2
g

109. The series x  x3 / 3!  x5 / 5!  ... represents the function

A. cos x
B. cosh x
C. sin x
D. sinh x

1
110. The value of
2
 log 1  x   log 1  x   equals

A. tan x
B. tanh x
C. tan 1 x
D. tanh 1 x

111. The value of p  4  , where p is the number of possible partitions of 4 is

A. 10
B. 5
C. 4
D. 1

32

112. Let f  x   R  x   a0  a1 x  ...  an x n : ai  R a ring , where n is a non-negative
integer. If f  a   f '  a   0 , then  x  a  divides
2

A. f  x 

B. f '  x 

C. f  x   a

D. f '  x   a

113. Let G be a graph with n  2 vertices. If all the n vertices in G form a cycle, then the
degree of any vertex in G is

A. 1
B. 2
C. n  1
D. n

114. Consider the graph G with 6 vertices given by v1 , v2 , v3 , v4 , v5 , v6 and edges
a, b, c, d , e, f , g , h . Then which among the following is a possibility for a path?

A. v1 a v2 b v3 c v3 d v4 e v2 f v5

B. v2 b v3 d v4 e v2 a v1

C. v1 a v2 b v3 d v4

D. v2 b v3 d v4 h v5 f v2

115. The maximum possible number of edges in a simple graph with n vertices and 2
components is

A.  n  1 n  2  / 2

B. n  n  1 / 2

C. n  n  1 / 2

D. n2 / 2

33

116. A Hamiltonian circuit is possessed by every graph with three or more vertices if it is

A. connected
B. a tree
C. simple
D. complete

117. The rank of incidence matrix A  G  of a disconnected graph G with n vertices and k
components is

A. k
B. k  1
C. n  k
D. n  k  1

118. Let T be a tree with four vertices v1 , v2 , v3 , v4 . Then the possible number of paths
between the vertices v1 and v4 is

A. atleast two
B. atleast one
C. exactly one
D. exactly two

119. For a graph G , both the incidence matrix A  G  and adjacency matrix X  G  contain the
entire information about G if,

A. G has no self-loops
B. G has no parallel edges
C. G has self-loops but no parallel edges
D. G is simple

34

120. Which of the integrals does not have a definite value?


A.  sin x dx, a  0
a

 1
B.  dx
1 x2
0
C.  e x dx


1 1
D.  dx
0
x

121. If p is a prime number, then  p  1 ! 1 is

A. an odd number
B. an even number
C. a prime number
D. divisible by p

122. The curvature of a circle or radius r is

1
A.
r2
1
B.  2
r
C. r 2
1
D.
r

123. The area enclosed by the curve x  y  1 is

A. 1
B. 2
C. 2
D. 4

35

n
124. If 1  x   C0  C1 x  C2 x 2  ...  Cn x n , then the value of C0  2C1  3C2  ...   n  1 Cn is

A.  n  2  2 n 1

B.  n  2  2 n

C.  n  1 2 n 1

D.  n  1 n  2  2n

125. The value of   4  is

A.  2 /12
B.  4 / 20
C.  4 / 90
D. 

126. The area under one arc of the cycloid x  a   sin   , y  a 1  cos   is

 a2
A.
8
3 a 2
B.
16
C. 3 a 2
3 a 2
D.
32

36

127. The area between the parabola y 2  4ax and the line y  x is

3a 2
A.
8
8a 2
B.
3
a2
C.
8
5a 2
D.
8

 y u u
128. If u  tan 1   , then x  y is equal to
x x y

2xy
A.
x  y2
2

B. 1
C. 0
x2
D.
x2  y 2

129. The velocity of a particle moving in a straight line is given by v 2  se8 , where s is the
displacement in time t . The acceleration of the particle is given by

A. 1  s  v / 2

B. v 2 / 2 1  s 

C. v / 2  s  1

D. v / 2 1  1/ s 

37

130. The directional derivative of f  x, y   2 x 2  3 y 2  z 2 at point  2,1,3 in the
r r
direction i  2k is

A. 4 5

B. 4 5

C. 5 4

D.  5 4

131. The value of  xyz dx dy dz taken through the positive octant of the sphere
x 2  y 2  z 2  a 2 is

a3
A.
48
a6
B.
48
a6
C.
8
a5
D.
48

132. If  is the boundary surface of a three dimensional region V having  as the unit vector
along the exterior normal to the surface  and y is a vector function, then
 . ydV   y. d  is called
V 

A. Stokes theorem
B. Gauss theorem
C. Green’s theorem
D. Cauchy’s theorem

38

133. The vector equation of a sphere one of whose diameters has the extremities with the
position vectors a and b is

A.  r  a    r  b   0

B.  r  a    r  b   0

C.  r  a  .  r  b   0

D.  r  a  .  r  b   0

r
134. Consider a closed surface S surrounding a volume V . If r is the position vector of a
r
ˆ
point inside S with n̂ the unit normal on S , the value of the integral  5r .ndS is

A. 3 V
B. 5 V
C. 10 V
D. 15 V

135. The dimension of the vector space V of all polynomials in ¡  x  of degree  20 is given
by

A. 21
B. 20
C. 10
D. 8

136. If A   a i j  is an n  n matrix defined over a field F , then trace of A is

A. 0
B. I
n
C.  ai j
i 1
n
D.  ai j
i j 1

39

cos x  sin x 0 
137. If F  x    sin x cos x 0  , then F  x  F  y  
 
 0 0 1 

A. F  xy 

B. F  x   F  y 

C. F  x  y 

D. F  x  y 

 
1 0 0 
 
1  i 3
138. If A   i 0  , then the trace of A102 is
 2 
 
0 1  i 3 
 1  2i 
2

A. 0
B. 1
C. 2
D. 3

 3 1 1 
139. If the matrix A   15 6 5 is singular, then the value of x is
 
 x 2 2 

A. 6
B. 5
C. 3
D. 2

40

140. If V and W are vector spaces of dimensions m and n respectively over a field F , then
the dimension of the vector space of all homomorphisms of V into W is

A. m  n
B. m  n
C. mn
D. m / n

141. The eigen values of a real symmetric matrix are always

A. positive
B. imaginary
C. real
D. complex conjugate pairs

142. If A is an n  n matrix with diagonal entries a and other entries b , then one eigen value
of A is a  b . Another eigen value of A is

A. b  a
B. nb  a  b
C. nb  a  b
D. 0

143. What is the degree of the first order forward difference of a polynomial of degree n ?

A. n
B. n  1
C. n  2
D. n  1

144. The rate of convergence of bisection method is

A. linear
B. faster than linear but slower than quadratic
C. quadratic

41

D. cubic

145. The set ¥ of natural numbers where x  y  max  x, y is a

A. ring
B. complete lattice
C. semi group
D. field

146. In a three dimensional Euclidean space, the direction cosines of a line which is equally
inclined to the axes are

A. 1, 1, 1
B. 1/ 3, 1/ 3, 1/ 3

C. 1/ 3, 1/ 3, 1/ 3
D. 1/ 2, 1/ 2, 1/ 2

147. In two dimensions, the distance between the origin and the centroid of the triangle joining
the points  1, 0  ,  4, 0  and  0, 3  is

A. 0
B. 1
C. 2

D. 3

148. Let P be the point 1, 0  and Q a point on the locus y 2  4 x . Then the locus of mid
point of PQ is

A. y 2  2 x  1  0

B. y 2  2 x  1  0

C. x 2  2 y  1  0

D. x 2  2 y  1  0

42

sin at
149. The Laplace transform of is
at

a
A. tan  
s
a
B. tan 1  
s
s
C. tan 1  
a
s
D. tan  
a

s2
150. The inverse Laplace transform of 2
is
s  4s  13

A. e 2t cos3t
4 2t
B. e sin 3t
3
4
C. e2t cos 3t  e 2t sin 3t
3
4
D. e2t cos 3t  e2t sin 3t
3

***