81. The value of
Wronskian W (x, x

^{2}, x^{3}) is
(A) 2x

^{2}
(B) 2x

^{4}
(C) 2x

^{3}
(D) x

^{2}
Answer: C

82. The general
solution of (∂l

^{2}u)/( ∂x^{2}) + (∂^{2}u)/( ∂y^{2}) = 0 is of the form
(A) u= f(x + iy) - g (x - iy)

(B) u = f(x - iy) - g (x - iy)

(C) u = f(x + iy) + g (x - iy)

(D) u = f(x - iy) + g (x + iy)

Answer: C

83. The partial
differential equation formed by eliminating the arbitrary function from z=f((y)/(x))
is

(A) x(∂z)/( ∂x) +(∂z)/( ∂y) = 0

(B) (∂z)/( ∂x) +(∂z)/( ∂y) = 0

(C) (∂z)/( ∂x) + y (∂z)/( ∂y) = 0

(D) x(∂z)/( ∂x) + y (∂z)/( ∂y) = 0

Answer: D

84. The orthogonal
trajectory of the family of curves x

^{2}-y^{2}= k is given by
(A) x

^{2}+y^{2}=c
(B) xy=c

(C) y=c

(D) x=0

Answer: B

85. The general
solution of the wave equation (∂

^{2}y)/( ∂t^{2}) = c^{2}(∂^{2}y)/( ∂x^{2}) is
(A) y (x, t) = Phi (x+ct) + psi (x - ct)

(B) y (x, t) = f (x+ct)

(C) y (x, t) = f (x-ct)

(D) No general solution exists

Answer: A

86. Stirling's
formula is the …………… of Gauss' forward and backward formulae.

(A) Arithmetic mean

(B) Geometric mean

(C) Harmonic mean

(D) None of the above

Answer: A

87. The
interpolating polynomial of the highest degree which corresponds the functional
values f (-1) = 9, f(0)=5, f (2) = 3, f (5) = 15 is

(A) x

^{3}+x^{2}+2x+5
(B) x

^{2}-3x+5
(C) x

^{4}+4x^{3}+5x^{2}+5
(D) x+5

Answer: B

88. The solution
of the integral equation Phi (x) = x+ int_0^x (Xi -x) Phi (Xi) dXi is

(A) cos x

(B) tan x

(C) sin x

(D) sec x

Answer: C

89. The minimizing
curve must satisfy a differential equation called

(A) Lagrange's equation

(B) Euler-Lagrange equation

(C) Gauss equation

(D) None of the above

Answer: B

90. A solid figure
of revolution, for a given surface area, has maximum volume is in the case of

(A) a circle

(B) a sphere

(C) an ellipse

(D) a parabola

Answer: B

91. A rigid body
moving in space with one point fixed has degree of freedom

(A) 3

(B) 1

(C) 6

(D) 9

Answer: A

92. A particle of
unit mass is moving under gravitational field, along the cycloid x = phi - sin
phi, y =1 + cos phi.

Then the Lagrangian for motion is

(A) phi^2 (1+cos phi) - g (1- cos phi)

(B) phi^2 (1-cos phi) + g (1+ cos phi)

(C) phi^2 (1-cos phi) - g (1+ cos phi)

(D) 2phi^2 (1-cos phi) - g (1+ cos phi)

Answer: C

93. L^-1 [(1)/(s
(s

^{2}+a^{2}))] is
(A) (1)/(a

_{2}) (1- cos at)
(B) (2 sin h t)/(t)

(C) (1)/(a

^{2}) (e^{at} -1)
(D) (1)/(a

^{2}) sin h at
Answer: A

94. int_0^∞
e^{-x^2}dx is

(A) (1)/(2)

(B) (pi)/(2)

(C) (sqrt(pi))/(2)

(D) -sqrt(pi)

Answer: C

95. Using Fourier
series, representing x in the interval [-pi, pi], the sum of the series

1-(1)/(3)+(1)/(5)-(1)/(7)+... is

(A) 0

(B) 1

(C) (pi)/(2)

(D) (pi)/(4)

Answer: D

96. The only
idempotent t-conorm is

(A) algebraic sum

(B) drastic union

(C) standard fuzzy union

(D) bounded sum

Answer: C

97. Using fuzzy
arithmetic operations on intervals [4,10]/[1,2] is

(A) [4,5]

(B) [2,10]

(C) [2,8]

(D) [4,20]

Answer: B

98. The language
generated by the grammar G = ({S}, {a,b}, S, P) where P is given by is

S -> aSb, S->lambda is

(A) {a

^{n}b^{n}: n>=0}
(B) {a

^{n}b^{n+1}: n>=0}
(C) {a

^{n+1}b^{n}: n >= 0}
(D) {a

^{n+2}b^{n}: n >= 1}
Answer: A

99. Which of the
following is not true in the derivative of a smooth vector field X ?

(A) grad_v (X+Y) = grad_v X + grad_v Y

(B) grad_v (fX) = (grad_v f) X (p) + f(p) (grad_v X)

(C) grad_v (X * Y) = (grad_v X) * Y (p) + X (p) * (grad_v Y)

(D) grad_v (fX) = f(grad_vX)

Answer: D

100. Let X be a
non-empty compact Hausdorff space. If every point of X is a limit point of X,
then

(A) X is disjoint

(B) X is countable

(C) X is uncountable

(D) None of the above

Answer: C

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