Engineering Mathematics

PreviousYear

Questions

GATE(1994

- 2021)

Devendra Poonia

Q.1. The inverse of a matrix

(A) (B)

(C) (D)

GATE 1994

Q.2. The limit of as x → 0 is,

(A) 0

(B) 1

(C) 2

(D) ∞

GATE 1994

Q.3. Integrating factor for the differential equation,

(A) exp [∫ pdx]

(B) exp [∫ pdx]

(C) ∫ Pdx

(D) dP/dx

GATE 1994

Q.4. If i, j, k, are the unit vectors in rectangular coordinates, then the

curl of the vector:

i x + j y + k z

(A) k

(B) -k

(C) j + k

(D) i + k GATE 1994

Q.5. The solution for the differential equation

(A) C1 e-2t + C2 e3t

(B) C1sin 2t + C2 cos 2t

(C) C1 e2t + C2 e-3t

(D) C1 e-2t + C2 e-3t

GATE 1994

Q.6. Taylor’s series expansion of f(x) around x = a is ___________

GATE 1994

Q.7. For a differential function f(x) to have a maximum, df/dx should

be ______ and d2f/dx2 should be___________ GATE 1994

Q.8. M dx + N dy is an exact differential when____________ GATE 1994

Q.9. The integral of x sin x in ___________ GATE 1994

Q.10. The Green’s theorem relates __________ integrals to surface integrals

GATE 1994

Q.11. If ‘a’ is a scalar and ‘b’ is vector, the _______

GATE 1994

Q.12. The differential equation

with the conditions y(0)= 0 and y(1)= 1 is called a_____________ value

problem. GATE 1994

Q.13. (I) cosh(at) (A) a/(s2 + a2)

(II) sin(at) (B) a/(s2 – a2)

(C) s /(s2 – a2)

(D) s /(s2 + a2) GATE 1994

Q.14. (A) Linear first order O.D.E with

(i) constant coefficient

(B) Linear O.D.E. with variable

coefficient

(ii) (C) First order non liner O.D.E

(D) Linear second order O.D.E

GATE 1994

Q.15. Find the eigen value of the matrix

GATE 1994

Q.16. The rank of matrix

(A) 0 (B) 1

(C) 2 (D) 3 GATE 1995

Q.17. The angle between two vectors 2i – j + k and i + j + 2k is

(A) 00

(B) 300

(C) 450

(D) 600GATE 1995

Q.18. A function f (x) = 12x – x3 has maximum value at x = √12

(A) - 2 (B) 0

(C) 2 (D) 12 GATE 1995

Q.19.

(A) ∞ (B) 1

(C) 0 (D) -1 GATE 1995

Q.20. The second order Taylor series expansion for a function f (x) = x2 at x

= 1 is

(A) x2

(B) 1 +x2

(C) 1 + x + x2

(D) 1 –x + x2GATE 1995

Q.21. The average value of function f(x) = x3 in the interval 0 ≤ x ≤ 2 is

(A) 1 (B) 2

(C) 4 (D) 8 GATE 1995

Q.22. (I) y = x2 (A) linear O.D.E.

(II) dy/dx = 2x (B) nonlinear O.D.E.

(C) linear algebraic equation

(D) nonlinear algebraic equation

GATE 1995

Q.23.

(I) dy/dx + 5y = 0, y(0) = y0 (A) y = y0 + 5x

(II) dy/dx + 5 = 0, y(0) = y0 (B) y = y0 – 5x

(C) y = y0 e-5x

(D) y = y0 e5x

GATE 1995

Q.24. Find eigen values and eigen vectors of matrix.

GATE 1995

Q.25. The ratio

where 1/y is a monotonically increasing function of x, is :

(A) Less than unity (B) Equal to unity

(C) Greater than unity (D) Less than zero GATE 1996

Q.26. Solve

subject to y = 1 at x = 0 and dy/dx = 0 at x =1. GATE 1996

Q.27. Given the matrix

(i). Write down the characteristic equations.

(ii). Computer [A]4 without direct multiplication. GATE 1996

Q.28. Solve using

the integrating factor method given y=1 at x=0 GATE 1996

Q.29. The sum of the infinite series is

(A) 9 (B) 9/2 (C) 15/2 (D) infinity GATE 1997

Q.30.

(A) 0 (B)½ (C) 1 (D) infinity GATE 1997

Q.31. The value of is

(A) > 0 (B) < 0

(C) 0 (D) undefined GATE 1997

Q.32. Given f (x, y) = x2 + y2 , is

(A) 4 (B) 2

(C) 0 (D) 4 (x + y)2GATE 1997

Q.33. A polynomial, f (x), is sketched below

Its order is

(A) 3 (B) 2 (C) 4 (D) > 5 GATE 1997

Q.34. The cubic equation x3 – x + 10 = 0 has a root in the interval

(A) (-1 ,0) (B) (0, 1)

(C) (-3, -1) (D) (3, 4) GATE 1997

Q.35. The Fourier series of the function –

extended periodically, f (x +2π)= f (x), is

(A) a sine series (B) a cosine series

(C) a mixed series (D) a power series GATE 1997

Q.36. Find the value of λ such that the question given below has a

non-zero solution.

GATE 1997

Q.37. For a matrix A given below:

(i) Calculate eigen values and

(ii) Determine the eigen vector corresponding to the lowest eigen

value. GATE 1997

Q.38. The Laplace transform of the function e-at has the form :

(A) 1/ (s + a) (B) 1/ [s (s + a)]

(C) a / s (D) (s + a) GATE 1998

Q.39. The unit normal to the plane 2x + y + 2z = 6 can be expressed

in the vector form

(A)3i + 2j + 2k

(B) (2/3)i + (1/3)j + (2/3)k

(C) (1/3)i + (1/2)j + (1/2)k

(D)(- 2/3)i + (1/3)j – (2/3)k GATE 1998

Q.40. has the value

(A)1

(B) - ¼

(C) 0

(D) ∞ GATE 1998

Q.41. The function z = (x – 1)2 – 2y2 has

(A) A stationary point which is a minimum

(B) A stationary point which is a maximum

(C) A stationary point at which no extremum exists

(D) no stationary point GATE 1998

Q.42. The differenal equaon will have a solution of the form

(A) C1e3t + C2e

2t

(B) C1e-2t + C2e

-t

(C) C1e-3t + C2e

-2t

(D) C1e –5t

Where C1 and C2 are constants GATE 1998

Q.43. The integral is convergent for

(A) no value of p (B) p > 1

(C) p < 1 (D) all values of p GATE 1998

Q.44. The mass balance equations of a blender are

0.2 F1 + 0.8 F2 = 0.5 F

and

F1+F2=F

Express the equation set given above in a matrix form AX= B where

Find the inverse of the matrix A. Use matrix inverse A-1 to calculate F1/F and

F2/F. Find the dimensions of a hollow cylinder with both ends closed, which can

hold of water and has minimum outer surface area. GATE 1998

Q.45. The system of equations:

2x + 4y=10

and

5x +10y=25

(A) has no unique solution

(B) has only one solution

(C) has only two solutions

(D) has infinite solutions GATE 1999

Q.46. Four fair coins are tossed simultaneously. The probability

that at least one head turns up is

(A)1/16 (B) 15/16

(C) 7/8 (D) 1/8 GATE 1999

Q.47. The rank of the matrix is

(A) 0 (B) 1 (C) 2 (D) 3 GATE 1999

Q.48. The harmonic series

(A) converges for p > 1 (B) diverges for p > 1

(C) converges for p < 1 (D) diverges for p < 1 GATE 1999

Q.49. A box contains 8 balls, 2 of which are defective. The probability that

none of the balls drawn is defective when two are drawn at random without

replacement is

(A) 15/28 (B) 9/16 (C) 7/16 (D) 1/8 GATE 1999

Q.50. The gradient of xy2+ yz3 at the point (-1, 2,1) is

(A) 3i-3j+3k (B) 3i-3j+6k

(C) 4i-j+3k (D) 4i-3j+6k GATE 1999

Q.51. Evaluate by trapezoidal rule. Use a step size of 0.2.

Obtain the error bounds for this solution. Compute the absolute error of the

numerical solution by evaluating the integral analytically. GATE 1999

Q.52. Solve by following methods:

(i) Variations of parameters

(ii) Separation of variables GATE 1999

Q.53.

(i) At what points are the Cauchy-Reimman equations satisfied for

the function, F(z) = xy2 + i x2y ?

Where is F(z) analytic ?

(ii) Compute the distinct cube roots of (1 + i) GATE 1999

Q.54. A pair of fair dice is rolled simultaneously. The probability that

the sum of the numbers from the dice equals six is

(A) 1/6 (B) 7/36 (C) 5/36 (D) 1/12 GATE 2000

Q.55. For an even function f (x)

(A) (B)

(C) (D)

GATE 2000

Q.56. The integrating factor for the differential equation is

(A)e tan x (B) cos 2x

(C) e –tan x (D) sin 2x GATE 2000

Q.57. The line integral of

where C is the unit circle around the origin traversed once in the

counter-clockwise direction, is

(A) - 2 π (B) 0 (C) 2 π (D) π GATE 2000

Q.58. The inverse of the matrix

(A)does not exist (B)

(C) (D)

GATE 2000

Q.59. The complex conjugate of 1/(1 + i) is

(A) 1/ (1 + i)

(B) (1 – i)

(C) 0.5 (1 – i)

(D) in the first quadrant of the complex plane GATE 2000

Q.60. The general solution of

(A) (B)

(C) (D)

GATE 2000

Q.61. Find the directional derivative of u = xyz at the point (1,2,3) in

the direction from (1,2,3) to (1,-1,-3). GATE 2000

Q.62. Find whether or not the vectors (1,1,2), (1,2,1) and (0,3,-3) are

linearly independent. GATE 2000

Q.63. The value of the following determinant is –

(A) 24 (B) 32 (C) -112 (D) 0 GATE 2001

Q.64. The value of (1 + i)8, where is

(A) 8 + 4i (B) 8 – 4i (C) 16 (D) 8 GATE 2001

Q.65. The function f (x, y) = x2 + y2 – xy – x – y + 5 has the

(A) Maximum at (1, 1)

(B) Saddle point at (1, 1)

(C) Minimum at (1,1)

(D) None of the above at (1 1) GATE 2001

Q.66. A fair die is rolled four times. Find the probability that six

shows up twice

(A)1/2 (B) 16/325

(C) 1/36 (D) 25/216 GATE 2001

Q.67. The parametric equation of a curve is

where t is the parameter.

(i) What type of conic (parabola, circle, ellipse, hyperbola) does

the curve represent ?

(ii) Find the unit tangent to the curve at t =1.

(iii) Find the unit normal to the curve at t = 1. GATE 2001

Q.68. Laplace Transform:

(i) Show that the Laplace transform of

(ii) Show from (i) that :

(iii) Show from (i) that :

GATE 2001

Q.69. In the complex plane, the angle between lines 1 + i and –1 + i

(where i = √-1) is

(A) π / 4

(B) π / 2

(C) 3π / 4

(D) π GATE 2002

Q.70. The inverse of the matrix

(A) (B)

(C) (D)

GATE 2002

Q.71. Which of the following holds for any non-zero vector a

(A) (B)

(C) (D) GATE 2002

Q.72. The coefficient of x2 in the Taylor series of cos2x about 0 is

(A) 2 (B) 0 (C) 1 (D) -1 GATE 2002

Q.73. Three grades of paint (A, B & C, production rates: 12, 24 & 18 batches

per day, respectively) are produced in independent batch production lines and

stored in separate areas. The number of off-specification batches in a day are

1, 3 and 2 for grades A, B & C, respectively. The probability of picking an

off-specification batch from a randomly chosen storage area is

(A) 23/216 (B) 24/216

(C) 19/216 (D) 18/216 GATE 2002

Q.74. (i) Reduce the following different equation to linear form

(ii) Find a general solution to the linearized equation

(iii) Determine the integration constants if z (0) = 0 GATE 2002

Q.75. Matrix

has the property that it satisfies Ax = x, for any vector x. Write the characteristic

equation to be solved for eigenvalues of A

(i) Based on visual observation, find one of the eigenvalues of A

(ii) Find the other two eigen values of A. GATE 2002

Q.76. A box contains 6 red balls and 4 green balls, one ball is randomly picked

and then a second ball is picked without replacement of the first ball. The

probability that both the balls are green is –

(A)1/15 (B) 2/25

(C) 2/15 (D) 4/25 GATE 2003

Q.77. The directional derivative of f (x, y, z) = x2+ y2+ z2 at the point (1, 1, 1)

in the direction i – k is

(A)0 (B) 1

(C) √2 (D) 2√2 GATE 2003

Q.78. The Taylor series expansion of the function : f (x) = x / (1+ x) around

x = 0 is

(A)x + x2+ x3+ x4…….

(B) 1 + x + x2 + x3 + x4…….

(C) 2x + 4x2+ 8x3+ 16x4…….

(D) x – x2 + x3– x4……. GATE 2003

Q.79. The range of values for a constant ‘K’ to yield a stable system in the

following set of time dependent differential equations is

(A) 0 < K < 7 (B) 6.25 < K < 10

(C) -6 < K ≤ 6.25 (D) 0 ≤ K ≤ 7 GATE 2003

Q.80. The value of y as t → ∞ for the following differential equation

for an initial value of y (1) = 0 is

(A)1 (B) ½

(C) ¼ (D) 1/8 GATE 2003

Q.81. The equilibrium data of component A in the two phases B and C

are given below :

The estimate of Y for X= 4 by fitting a quadratic expression of a form

Y = m X2 for the above data is

(A)15.5 (B) 16

(C) 16.5 (D) 17 GATE 2003

X (moles of A/moles of B) Y (moles of A/moles of C)

1 0.5

2 4.125

Q.82. The fluid element has a velocity V = – y2x i + 2 yx2 j.

The motion at (x, y) = is

(A)rotational and incompressible

(B) rotational and compressible

(C) irrotational and compressible

(D) irrotational and incompressible GATE 2003

Q.83. The most general complex analytical function

f (z) = u (x, y) + iv (x, y) for u = x2 – y2 is

(A) Z (B) Z2 (C) Z3 (D) 1/Z2GATE 2003

Q.84. The differential equation , will have a

solution of the form

(A) (C1 + C2t) e-5t

(B) C1 e-2t

(C) C1 e-5t + C2 e5t

(D) C1 e5t + C2 e2t

where C1 and C2 are constants. GATE 2003

Q.85. The inverse Laplace transform of the function

(A) 1 + et

(B) 1 – et

(C) 1 + e-t

(D) 1 – etGATE 2004

Q.86. The function f (x) = 3x (x – 2) has a

(A) minimum at x = 1 (B) maximum at x = 1

(C) minimum at x = 2 (D) maximum at x = 2 GATE 2004

Q.87. The complex number 2 (1 + i) can be represented in polar

form as

(A) (B)

(C) (D)

GATE 2004

Q.88. The differential equation –

(A)first order and linear

(B) first order and non-linear

(C) second order and linear

(D) second order and non-linear GATE 2004

Q.89. The sum of the eigenvalues of the matrix .

For real and negative values of x is

(A)greater than zero

(B) less than zero

(C) zero

(D) dependent on the value of x GATE 2004

Q.90. The system of equations

(A) no solution

(B) only one solution

(C) two solutions

(D) infinite number of solutions GATE 2004

Q.91. A box contains three blue balls and four red balls. Another identical

box contains two blue balls and five red balls. One ball is picked at random,

from one of the two boxes and it is red. The probability that it came from

the first box is

(A) 2/3 (B) 4/9 (C) 4/7 (D) 2/7 GATE 2004

Q.92. The series converges for

(A) all Z (B) Z > 2

(C) (D)

GATE 2004

Q.93. The differential equation for the variation of the amount of salt X in a

tank with time t is given by

X is in kg and t is in minutes. Assuming that there is no salt in the tank

initially, the time (in min) at which the amount of salt increases to 100 kg is

(A) 10 ln 2 (B) 20 ln2

(C) 50 ln 2 (D) 100 ln 2 GATE 2004

Q.94. Value of the integral is

(A) 0 (B) 0.25 (C) 1 (D) ∞ GATE 2004

Q.95. The differential equation

can be reduced to (where α is a constant)

(A) (B)

(C) (D)

GATE 2004

Q.96. The value of

(A) 0 (B) 1/27 (C) 1/108 (D) ∞ GATE 2004

Q.97. Match the following, where x is the spatial coordinate and l is time,

Group I Group II

P. Wave equation 1.

Q. Heat equation 2.

3.

4.

(A) P – 4 Q – 2 (B) P – 2 Q – 4

(C) P – 3 Q – 1 (D) P – 1 Q – 3 GATE 2005

Q.98. Two bags contain ten coins each, and the coins in each bag are

numbered from 1 to 10, One coin is drawn at random from each bag. The

probability that one of the coins has value, 1,2,3 or 4, while the other has

value 7,8,9 or 10 is

(A) 2 / 5 (B) 4/25

(C) 8/25 (D) 16/25 GATE 2004

Q.99. Given i= √–1 the ratio is given by

(A) 1 (B) – 1

(C) i (D) – i GATE 2005

Q.100. How many solutions does the following system of equations have ?

4x + 2y + z = 7

x + 3y + z = 3

3x + 4y + 2z = 2

(A) 0 (B) 1

(C) 2 (D) ∞ GATE 2005

Q.100. The matrix A is given by

The eigen values of the matrix A are real and non-negative for the condition:

(A) (B)

(C) (D)

GATE 2005

Q.102. The divergence of a vector field A is always equal to zero, if the

vector field A can be expressed as

(A) The gradient of any scalar field ф

(B) The divergence of any scalar field ф

(C) The divergence of any vector field B

(D) The curl of any vector field B. GATE 2005

Q.103. In the limit x → 0, what is the liming value of the function F (x)

given below ?

(A) 0 (B) 1

(C) 2 (D) ∞ GATE 2005

Q.104. If z = x + iy is a complex number, where i=√-1, then which of the

following is an analytic function of z ?

(A) x2 + y2

(B) 2ixy

(C) x2 + y2 – 2 ixy

(D) x2 – y2 + 2 GATE 2005

Q.105. What condition is to be satisfied so that the solution of the

differential equation

is of the form y = (C1 + C2x) emx, where C1 and C2 are constants of

integration ?

(A) a2= b (B) b2= a

(C) a2 = 4b (D) b2 = 4a GATE 2005

Q.106. In the domain - ∞ < x < ∞, the function has

(A)no maximum and no minimum

(B) one maximum and no minimum

(C) no maximum and one minimum

(D) one maximum and one minimum GATE 2005

Q.107. If f (x) is the solution of the equation and g (x) is

the solution of the equation and the constant of integration

in f (x) is equal to that in g (x), then which of the following is true ?

(A) g (x) = f (x) + 2 (B) g (x) = f (x) + 1

(C) g (x) = f (x) – 1 (D) g (x) = f (x) – 2 GATE 2005

Q.108. If the constants, An are suitably chosen so as to satisfy the initial conditions,

and n is an integer, which of the following is a valid solution for the unsteady one

dimensional diffusion equation,

in the domain 0 ≤ x ≤ L with boundary conditions c = 0 at x = 0 and c = 0 at x = L ?

(A) (B)

(C) (D)

GATE 2005

Q.109. The function f (x) satisfies the equation f(x) = 0 at x = xe . The

Newton Raphson iterative method converges to the solution in one step,

regardless of the initial guess, if

(A)f(x) is a linear function of x

(B) f(x) is a quadratic function of x

(C) f(x) is a cubic function of x

(D) f(x) is an exponential function of x GATE 2005

Q.110. The ordinary differential equation dY/dt = f (Y) is solved using

the approximation Y (t + Δt) = Y(t) + f [Y(t)] Δt. The numerical error

introduced by the approximation at each step is

(A) proportional to Δt (B) proportional to (Δt)2

(C) independent of Δt (D) proportional to (1/ Δt) GATE 2006

Q.111. The trapezoidal rule of integration when applied to ∫ f(x)dx will

given the exact value of the integral

(A) if f (x) is a linear function of x (C) for any f (x)

(B) if f (x) is a quadratic function of x (D) for no f (x) GATE 2006

Q.112. The value of α for which the following three vectors are coplanar is

a = i + 2j + k

b = 3j + k

c = 2i + α j

(A)4 (B) zero

(C) – 2 (D) – 10 GATE 2006

Q.113. The derivative of |x| with respect to x when x ≠ 0 is

(A) |x| / x (B) - 1

(C) 1 (D) Undefined GATE 2006

Q.114. If the absolute error in the measurement of A is ΔA and the absolute error

in the measurement of B is ΔB, then the absolute error in the estimate of A – B is

GATE 2006

Q.115. If the following represents the equation of a line then the line passes

through the point.

(A) (0, 0) (B) (3, 4) (C) (4, 3) (D) (4, 4) GATE 2006

Q.116. If A = , then the eigenvalues of A3 are

(A) 27 and 8 (B) 64 and 1 (C) 12 and 3 (D) 4 and 1 GATE 2006

Q.117. With y = eax , if the sum

approaches 2y as n → ∞, then the value of a is

(A) 1/3 (B) 1/2 (C) 2/3 (D) 2 GATE 2006

Q.118. Determine the following integral where r is the position vector field

(r = ix + jy + kz) and S is the surface of a sphere of radius R.

(A) 4 π R2

(B) 3/4 π R3

(C) πR3

(D) 4 π R3GATE 2006

Q.119. The solution to the following equation is given by

(A) y = C1x + C2x–2 + C3

(B) y = C1x2 + C2x

–2 + C3

(C) y = C1x2 + C2x

–1 + C3

(D) y =C1x + C2x–1 + C3 GATE 2006

Q.120. The value of the contour integral where C is the circle |z| = 2 is

(A) (B)

(C) Zero (D)

GATE 2006

Q.121. The Newton-Raphson method is used to solve the equation,

(x – 1)2 + x – 3 = 0. The method will fail in the very first iteration if

the initial guess is

(A)zero (B) 0.5 (C) 1 (D) 3 GATE 2006

Q.122. A pair of fair dice is rolled three times. What is the probability that

10 (sum of the numbers on the two faces) will show up exactly once ?

(A)121/1728 (B)363/1728

(C)121/576 (D)363/576 GATE 2006

Q.123. A company purchased components from three firms P, Q, and R as shown in

the table below,

The components are stored together. One of the components is selected at random,

and found to be defective. What is the probability that it was supplied by Firm R ?

(A) 1/250 (B) 1/12 (C) 1/8 (D) 1/6 GATE 2006

Firm Total number of components purchased Number of components likely to be defective

P 1000 5

Q 2500 5

R 500 2

Q.124. A weighing machine is calibrated at 250C and he output reading R

(in mm) is related to the weight W (in kg) by the equation R = sW where

the sensitivity s = 20 mm/kg. At a temperature of 300C, the weighing

machine undergoes a zero drift (change in instrument output reading at

zero value of weight) of +2 mm and its sensitivity changes to 20.5 mm/kg.

The weighing machine when used at 300C shows a reading of 50 mm. The

true weight in kg) of the object is

(A) 2.34 (B) 2.40 (C) 2.44 (D) 2.50 GATE 2006

Q.125. Give i = √–1 , the ratio is given by

(A) i (B) – 2 (C) – i + 2 (D) i + 1 GATE 2007

Q.126. The value of “a” for which the following set of equations

y + 2z = 0

2x + y + z = 0

ax + 2y = 0

have non-trivial solution, is

(A) 0 (B) 8 (C) – 2 (D) 3 GATE 2007

Q.127. The initial condition for which the following equation has infinitely

many solutions, is

(A)y (x = 0) = 5

(B) y (x = 0) = 1

(C) y (x = 2) = 1

(D) y (x = –2) = 0 GATE 2007

Q.128. Given that the Laplace transform of the function shown below over a

single period 0 < t < 2 is , the Laplace transform of

the periodic function over 0 < t < ∞ is

(A) (B)

(C) (D)

GATE 2007

Q.129. If z = x + iy is a complex number, where i =√ –1 then the derivative

of z z ̅ at 2 + i is

(A) 0 (B) 2 (C) 4 (D) does not exist GATE 2007

Q.130. A and B are two 3 x 3 matrix such that

and AB = 0 Then the rank of matrix B is

(A) r = 2 (B) r < 3 (C) r ≤ 3 (D) r = 3 GATE 2007

Q.131. The solution of the following differential equation is

(A) 0 (B)

(C) c1x + c2 x2 (D)

GATE 2007

Q.131. The directional derivative of at (1, 1) in the direction of b

= i – j is

(A) 0 (B) 1/√2 (C) √2 (D) 2 GATE 2007

Q.133. Evaluate the following integral (n ≠ 0) within the area

of a triangle with vertices (0,0), (1,0) and (1,1) (counterclockwise)

(A) 0 (B) 1/(n + 1) (C)1/2 (D) n/2

GATE 2007

Q.134. The family of curves that is orthogonal to xy = c is

(A) y = c1x

(B) y = c1/x

(C) y2 + x2 =c1

(D) y2 – x2 = c1 GATE 2007

Q.135. The Laplace transform of is

(A) (B)

(C) (D) Does not exist

GATE 2007

Que.136. The thickness of a conductive coating in micrometers has a probability

density function of 600 x-2 for 100 μm < x < 120 μm. The mean and the variance

of the coating thickness is

(A)1 μm, 108.39 μm2 (B) 33 83 μm, 1 μm2

(C) 105 μm, 11 μm2 (D) 109.39 μm, 33.83 μm2GATE 2007

Que.137. Which ONE of the following is NOT an integrating factor for

the differential equation xdy – ydx = 0 ?

GATE 2008

Que.138. Which ONE of the following is NOT a solution of the differential equation ?

(A) y = 1 (B) y = 1 + cos x (C) y = 1 + sinx (D) y = 2 + sin x + cos x GATE 2008

Que.139. The limit of is

(A) -1 (B) 0 (C) 1 (D) ∞ GATE 2008

Que.140. The unit normal vector to the surface of the sphere

x2 + y2 + z2 = 1 at the point is (i, j, k are unit normal

vectors in the Cartesian coordinate system)

GATE 2008

Que.141. A nonlinear function f(x) is defined in the interval –1.2 < x < 4 as

illustrated in the figure below. The equation f(x) = 0 is solved for x within this

interval by using the Newton – Raphson iterative scheme. Among the initial

guesses (I1, I2, I3 and I4), the guess that is likely to lead to he root most rapidly is

(A) I1 (B) I2 (C) I3 (D) I4 GATE 2008

Que.142. Which ONE of the following transformations {u= f (y)} reduces

to a linear differential equation? (A and B are positive

constants)

(A) u = y-3

(B) u = y-2

(C) u = y-1

(D) u = y2GATE 2008

Que.143. The Laplace transform of the function f(t) = t sint is

GATE 2008

Que.144. The value of the surface integral

evaluated over the

surface of a cube having sides of length a is (n is unit normal vector).

(A) 0

(B) a3

(C) 2 a3

(D) 3 a3GATE 2008

Que.145. The first four terms of the Taylor series expansion of cos x

about the point x = 0 are

GATE 2008

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GATE 2009

Que.162. The inverse of the matrix is

GATE 2010

Que.163. . The Laplace transform of the function shown in the figure below

is

(A) (B)

(C) (D)GATE 2010

Que.164. The Maxwell – Boltzmann velocity distribution for the x component

of the velocity, at temperature T, is . The standard deviation

of the distribution is

GATE 2010

Que.165. Given that i = √ –1 , ii is equal to

GATE 2010

Que.166. A root of the equation x4 – 3x + 1 = 0 needs to be found using the

Newton-Raphson method. If the initial guess, x0, is taken as 0, then the new

estimate, x1, after the first iteration is

GATE 2010

Que.167. The solution of the differential equation

with the initial conditions = - 1 is

(A) – t sin t (B) – e –1 (1 – cos t) (C) –(t + sin t)/2 (D) – e –1 sin t

GATE 2010

Que.168. If and , then curl is

GATE 2010

Que.169. X and Y are independent random variables. X follows a binomial

distribution, with N = 5 and p = 1/2. Y takes integer values 1 and 2, with

equal probability. They the probability that X = Y is

GATE 2010

Que.170. A box contains three red and two black balls. Four balls are

removed from the box one by one, without replacement. The probability of

the ball remaining in the box being red, is

GATE 2010

Que.171. For a function g (x) if g (0) = 0 and gʹ (0) =2. then

is equal to

(A) ∞ (B) 2 (C) 0 (D) – ∞ GATE 2010

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GATE 2021