Maths

ENGINEERING MATHEMATICS

8. The Laplace transform of the function

f(t) = k, 0 < t < c (GATE-99)
= 0, c < t < ∞ , is d) (k/s) (1 - e-sc)

a) (k/s)e-sc b) (k/s)esc c) k e-sc

9. Laplace tranform of (a + bt)2 where ‘a’and ‘b’are constants is given by : (GATE-99)

a) (a + bs)2 b) 1/(a + bs)2

c) (a2/s) + (2ab /s2) + (2b2 / s3) d) (a2/s) + (2ab /s2) + (b2 / s3)

10. Let F(s) = L [f(t)] denote the Laplace transform of the function f(t). Which of the following statements

is correct ? (GATE-2000)

a) L [df /dt] = 1/s F(s) ; b) L [df /dt] = s F(s) - F(0) ;

t t

∫L f(τ(dτ =sF(s) - f(0) ∫L f(τ(dτ = -dF/ds
0 0

c) L [df /dt] = s F(s) - F(0) ; d) L [df /dt] = s F(s) - F(0) ;

{{ t t
∫L f(τ(dτ = F(s - a)
0 ∫L f(τ(dτ = 1 F(s)
s
0

11. The inverse Laplace transform of 1 / (s2 + 2s) is (GATE - 01)
d) (1- e-2t) /2
a) (1- e-2t) b) (1+ e+2t) /2 c) (1- e+2t) /2

12. The laplace transform of the following function is (GATE-02)

f(t) = sin t for 0 < t < π
0 for t > π

a) 1 / (1 + s2) for all s > 0 b) 1 / (1 + s2) for all s < π
c) (1 + e-πs) /(1+s2) for all s > 0 d) e-πs /(1+s2) for all s > 0

13. Using Laplace tranforms, solve (GATE-02)
(d2y /dt2) + 4y = 12t given that y = 0 and dy/dt = 9 at t =0

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MECHANICAL ENGINEERING

14. A delayed unit step function is defined as (GATE-04)
d) e as/a
0, t < a
u(t -a) = 1, t > a

Its laplace transform is -----------------------

a) a e-as b) e-as/s c) eas/s

15. The laplace transform of a function f(t) is (GATE-05)

F(s) = 5s2 + 23s + 6 . As t → α, f(t) approaches
s(s2 + 2s + 2)

a) 3 b) 5 c) 17/2 d) α

16. Laplace transform of f(t) = cos(pt +q) is (GATE-05)

a) s cos q - p sin q b) s cos q + p sin q c) s sin q - p cos q d) s sin q + p cos q
s2 + q2 s2 + q2 s2 + q2 s2 + q2

17. If F(s) is the Laplace trnasform of the function f(t) then Laplace trnasform of (GATE-07[ME])

∫f(x)dx is

1 1 c) s F(s) - f(0) ∫d) f(x)ds
a) s F(s) b) s F(s) - f(0)

1 (GATE-09[ME])
18. The inverse Laplace transform of (s2 + s) is

a) 1 + et b) 1 - et c) 1 - e-t d) 1 + e-t

1 (GATE-10[ME])
19. The inverse Laplace transform of f(t) is s2 (s+1) d) 2t + et

a) t - 1 + et b) t + 1 + e-t c) -1 + e-t

20. Given two continuous time signals x(t) = e-t and y(t) = e-2t which exists for t > 0 then the convolution

z(t) = x(t) * y(t) is ------------ (GATE-11)

a) e-t - e-2t b) e-2t c) e-1 d) e-t + e-3t

21. If F(s) = L{f(t)} = s2 2(s+1) 7 then the initial and final values of f(t) are respectively (GATE-11)
+ 4s +

a) 0, 2 b) 2, 0 c) 0, 2 d) 2 , 0
7 7

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ENGINEERING MATHEMATICS

22. The inverse Laplace transform of the function F(s) 1 is given by (GATE-12[ME,PI]
s (s+1)

a) f(t) = sin t b) f(t) = e-t sin t c) f(t) = e-t d) f(t) = 1 - e-t

d2f
23. The function f(t) satisfies the differential euqation dt2 + f = 0 and the auxiliary conditions,

f(0) 0, df (0) = 4 . The laplace transform of f(t) is given by (GATE-13[ME])
dt

a) 2 4 4 2
s +1 b) s +1 c) d) s4 +1

s2 +1

24. Laplace transform of cos (ωt) is s
s2 + ω2

The Laplace transform of e-2t cos (4t) is (GATE-14-ME-Set 4)

a) s - 2 s+2 c) s - 2 d) s + 2
(s - 2)2 + 16 b) (s - 2)2 + 16 (s + 2)2 + 16 (s + 2)2 + 16

25. If L denotes the laplace transform of a function, L{sin at} will be equal to (GATE-03[CE])

a a s d) s
a) s2 - a2 b) s2 + a2 c) s2 + a2 s2 - a2

26. Laplace transform of f(x) = cos h (ax) is (GATE-09[CE])
d) s
a s a
a) s2 - a2 b) s2 - a2 c) s2 + a2 s2 + a2

27. Laplace transform of 8 t3 is (GATE-08[PI])

8 b) 16 c) 24 48
a) s4 s4 s4 d) s4

28. Laplace transform of sin ht is (GATE-08[PI])

a) 1 1 c) s d) s
s2 - 1 b) 1 - s2 s2 - 1 s2 + a2

******************

“The toughest thing about success is that you’ve got
to keep on being a success”

5. TRANSFORM THEORY (ANS.) :
1 - sol, 2 - sol, 3 - d, 4 - sol, 5 - sol, 6 - a, 7 - d, 8 - d, 9 - c, 10 - d, 11 - d, 12 - c, 13 - sol., 14 - b,
15 - a, 16 - a, 17 - a, 18 - c, 19 - a, 20 - a, 21 - b, 22 - d, 23 - c, 24 - d, 25 - b, 26 - b, 27 - d, 28 - a.

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MECHANICAL ENGINEERING

6 COMPLEX VARIABLES

1. i i, where i = √-1 is given by (GATE-1996[ME])

a) 0 b) e-π/2 π d) 1
c)
2

2. If φ (x, y) and ψ (x, y) are functions with continuous 2nd derivatives then φ (x, y) + i ψ (x, y) can be

expressed as a analytic function of x + iy (i = √-1) when (GATE-2007[ME])

a) ∂φ = - ∂ψ , ∂φ = ∂ψ b) ∂φ = - ∂ψ , ∂φ = ∂ψ
∂x ∂x ∂y ∂y ∂y ∂x ∂x ∂y

∂2φ + ∂2φ ∂2ψ + ∂2ψ =1 d) ∂φ + ∂φ ∂ψ + ∂ψ =0
∂ y2 = ∂ x2 ∂ y2 ∂x ∂y = ∂x ∂y
c) ∂ x2

∫3. The ingtegral f(z) dz evaluated around the unit circle on the complex plane for
(GATE-2008[ME])
Cos z
f(z) = z

a) 2 π i b) 4 π i c) - 2 π i d) 0

4. An analytic function of a complex variable z = x + i y is expressed as f(z) = u(x, y) + i v (x, y) where

i = √-1. If u = xy then the expression for v should be (GATE-2009[ME])

(x +y)2 x - y2 c) y2 - x2 (x - y)2
a) 2 + k b) 2 + k 2 +k d) 2 + k

5. The product of two complex numbers

1 + i & 2 - 5 i is (GATE-2011[ME])
d) 7 + 3i
a) 7 - 3i b) 3 - 4i c) - 3 - 4i

6. The argument of the complex number1 + i , where i = √ -1 is (GATE-14-ME-Set 1)
1-i

a) -π b) - π π d) π
2 c) 2

7. An analytic function of a complex variable z = x + i y is expressed as (GATE-14-ME-Set 2)
f(z) = u(x, y) + i v(x, y), where i = √-1.

If u(x, y) = 2xy , then v (x, y) must be

a) x2 + y2 constant b) x2 - y2 constant c) - x2 + y2 constant d) - x2 - y2 constant

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ENGINEERING MATHEMATICS

8. An analytic function of a complex variable z = x + iy is expressed as
f(z) = u(x, y) + iv(x, y), where i = √-1.

If u(x, y) = x2 - y2, then expression for v(x, y) in terms of x, y and a general constant c would be

(GATE-14-ME-Set 3)

a) xy + c x2 + y2 c) 2xy + c x2 - y2
b) 2 + c d) 2 + c

3i (GATE-14-ME-Set 4)
d) 0.511 + 1.57 i
∫9. If z is a complex variable, the value of dz is
z
5

a) - 0.511 - 1.75i b) -0.511 + 1.57 i c) 0.511 - 1.57 i

10. Given two complex numbers z1 = 5 + (5√3) i and (GATE-15-ME-Set 1)
d) 90
z2 = 2 + 2i , the argument of z1 in degrees i
√3 z2

a) 0 b) 30 c) 60

11. ez is a periodic with a period of (GATE-1997[CE])
c) π d) iπ
a) 2π b) 2πi

12. Which one of the following is Not true for the complex numbers z1 and z2 ? (GATE-2005[CE])

a) z1 z1 z2 b) |z1 + z2|<|z1| + |z2|
z2 = |z2 |2

c) |z1 + z2|<|z1| - |z2| d) |z1 + z2|2 + |z1 - z2|2 = 2|z1|2 + 2|z2|2

13. Consider likely applicability of Cauchy’s Integral theorem to evaluate the following interal counter

∫clock wise around the unit circle C I = sec z dz, z being a complete variable.
c (GATE-2005[CE])

The value of I will be

a) I = 0 ; Singularities set = φ b) I = 0 ; Singularities

p set = + (2n + 1) π / n = 0,1,2,- - -
c) I = 2 ; Singularities 2

set = {+ n π ; n = 0, 1, 2, - - -} d) None of the above.

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MECHANICAL ENGINEERING

14. Using Cauhy’s integral theorem, the value of the integral (integration being taken in contour clock wise

direction) (GATE-2006[CE])

∫ z3 - 6 is where C is |z| = 1
dz
c 3z - i

a) 2π - 4πi π c) 2π - 6πi d) 1
81 b) 8 - 6πi 81

15. Potential function φ is given as φ = x2 - y2. What will be the stream function ψ with the condition

ψ = 0 at x = 0 , y = 0 ? (GATE-2007[CE])

a) 2xy b) x2 + y2 c) x2 - y2 d) 2x2 y2

16. The analytical function has singularities at, (GATE-2009[CE])
d) i and - i
z-1
where f(z) = z2 + 1

a) 1 and -1 b) 1 and i c) 1 and -i

∫ cos (2π z) (GATE-2009[CE])

17. The value of the interal (2z - 1) (z - 3) c) π i
(GATE-10[CE])
c d) 1

dz where C is a closed curve given by |z| = 1 is 5

a) - π i b) π i c) 2 π i
5 5

18. The modulus of the complex number 3 + 4i is
1 - 2i

a) 5 b) √5 c) 1
√5

19. For an analytic function

f(x + i y) = u(x, y) + i v(x, y), u is given by u = 3x2 - 3y2. The expression for v, considering k is to be

constant is (GATE-2011[CE])

a) 3y2 - 3x2 + k b) 6x - 6y + k c) 6y - 6x + k d) 6xy + k

2 - 3i (GATE-14-CE-Set 2)
20. z = - 5 + i can be expressed as d) 0.5 + 0.5i

a) - 0.5 - 0.5i b) - 0.5 + 0.5i c) 0.5 - 0.5i

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21. Consider the following complex function f(z) = 9 (GATE-15-CE-Set 1)

(z - 1)(z +2)2

Which of the following is ONE of the residues of the above function ?

a) -1 b) 9 c) 2 d) 9
16

22. The function w = u + iv = 1 log (x2 + y2) + i tan-1 y
2 x

is not analytic at the point (GATE-2005[PI])
d) (2, α)
a) (0, 0) b) (0, 1) c) (1, 0)

23. If a complex number z =√3 + i 1 then z4 is (GATE-2007[PI])
22

a) 2 √2 + 2 i b) - 1 + i √3 c) √3 - i 1 d) √3 - i 1
22 2 2 8 8

24. The value of the expression -5 + i10 (GATE-2008[PI]
3 + 4i

a) 1 - 2i b) 1 + 2i c) 2 - i d) 2 + i

25. The product of complex numbers (3 - 2i) & 3 + i 4) result in (GATE-2009[PI]
d) 17 + 6i
a) 1 + 6i b) 9 - 8i c) 9 + 8 i

26. If f(x+ iy) = x3 + -3xy2 + i φ(x, y) where i = √-1 and f(x + i y) is an analytic function then φ (x,y) is

(GATE-2010[PI])

a) y3 - 3x2y b) 3x2 y - y3 c) x4 - 4x3 y d) xy - y2

27. If a complex number ω satisfies the equation ω3 = 1then the value of 1 + ω + 1
ω

is ______________ (GATE-2010[PI])
a) 0 b) 1 c) 2 d) 4

∫ z2

28. The value of z4 - 1 dz, using cauchy’s integral around the circle |z + 1| = 1 where

c

z = x + i y is (GATE-2011[PI])

a) 2 π I b) - πi c) - 3πi d) π2 i
2 2

“To give happiness to others is a great act of charity”

6. COMPLEX VARIALES (ANS.) :
1 - b, 2 - b, 3 - a, 4 - c, 5 - a, 6 - c, 7 - c, 8 - c, 9 - b, 10 - a, 11 - b, 12 - c, 13 - a, 14 - a, 15 - a,
16 - d, 17 - c, 18 - b, 19 - d, 20 - b, 21 - a, 22 -a, 23 - b, 24 - b, 25 - d, 26 - b, 27 - a, 28 - b.

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MECHANICAL ENGINEERING

7 NUMERICAL METHODS

1. Simpson’s rule for integration gives exact result when f(x) is a polynomial function of degree less than

or equal to ____ (GATE-1933[ME])

a) 1 b) 2 c) 3 d) 4

2. Given the differential equation y1 = x - y with initial condition y(0) = 0. The value of y(0.1) calculated

numerically upto the third place of decimal by the 2nd order Runge-Kutta method with step size

h = 0.1 is (GATE-1993[ME])

3. The formula used to compute an approximation for the second derivative of a function f at a point
x0 is (GATE-1996[CS])

a) f(x0 + h) + f(x0 - h) b) f(x0 + h) - f(x0 - h)
2 2h

c) f(x + h) + 2f(x ) + f(x - h) d) f(x0 + h) - 2f(x0) + f(x0 - h)
0 0 0 h2

h2

4. The order of error in the simpson’s rule for numerical integration with a step size h is

(GATE-1997[ME])

a) h b) h2 c) h3 d)h4

5. Starting from x0 = 1, one step of Newton - Raphson method in solving the equation x3 + 3x - 7 = 0

gives the next value x1 as (GATE-2005[ME])

a) x = 0.5 b) x1 = 1.406 c) x = 1.5 d) x1 = 2
1 1

6. The area under the curve shown between x = 1 and x = 5 is to be evaluated using the trapezoidal rule.

The following points on the curve are given (GATE-2009)

Point x - coordinate y - coordinate
(m) (m)
1 1 1
2 2 4
3 3 9

The evaluated area (in m2) will be c) 9 d) 18
a) 7 b) 8.67 © Copyright : Ascent Gate Academy 58

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ENGINEERING MATHEMATICS

3 1
∫7. x
The integral dx when evaluated by using simpson’s 1/3rd rule on two equal sub interals each of

1

length 1, equals to (GATE -2011[ME])

a)1.000 b) 1.008 c) 1.1111 d) 1.120

8. Match the CORRECT pairs (GATE-2013[ME])

Numerical Intergration Order of Fitting
Scheme Polynomial

P. Simpon’s 3/8 Rule P. Simpon’s 3/8 Rule
Q. Trapezoidal Rule Q. Trapezoidal Rule
R. Simpson’s 1/3 Rule R. Simpson’s 1/3 Rule

a) P -2, Q - 1, R - 3 b) P -3, Q - 2, R - 1 c) P -1, Q - 2, R - 3 d) P -3, Q - 1, R - 2

9. Using the trapezoidal rule, and dividing the inteval of intergration into three equal subintervals, the

+1 is (GATE-14-ME-Set 1)

∫difinite integral |x|dx
-1

4

∫10. The value of ln(x) calculated using the Trapezoidal rule with five subintervals is.
2.5 (GATE-14-ME-Set 2)

3

∫1

11. The definite integral x dx is evaluated using Trapezoidal rule with a step size of 1. The correct

1

answer is. (GATE-14-ME-Set 3)

12. The real root of the equation 5x - 2cosx = 0 (up to two decimal accurracy) is.
(GATE-14-ME-Set 3)

dx
13. Consider an ordinary differential equation dt = 4t + 4. If x = x0 at t= 0,

The increment in x calculated using Runge-Kutta foruth order multi-step method with a step size of

∆t = 0.2 is (GATE-14-ME-Set 4)

a) 0.22 b) 0.44 c) 0.66 d) 0.88

14. Simpson’s 1 rule is used to integrate the function f(x) =3 x2 + 9 between x = 0 and x = 1
3 55

using the least number of equal sub - intervals. The value of the intergral is (GATE-15-ME-Set 1)

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MECHANICAL ENGINEERING

15. The values of function (x) at 5 discrete points are given below : (GATE-15-ME-Set 2)

x 0 0.1 0.2 0.3

0.4

f(x) 0 10 40 90 0.4

∫Us1i6n0g Trapezodial rule with step size of 0.1, the value of f(x) dx is _____
0

2

∫16. Using a unit step, the value of intergral x ln xdx by trapezoidal rule is ____
1
(GATE-15-ME-Set 3)

17. Newton-Raphson method is used to find the root of the equation, x3 + 2x2 + 3x - 1 = 0. If theintitial
guess is x0 = 1, then the value of x after 2nd iteration is __________ (GATE-15-ME-Set 3)

18. Give a > 0, we wish to calculate its reciprocal value 1 by using Newton - Raphson method for
a

f (x)= 0. For a = 7 and starting with x0 = 0.2 the first two interation will be (GATE-2005[CE])

a) 0.11, 0.1299 b) 0.12, 0.1392 c) 0.12, 0.1416 d) 0.13, 0.1428

19. Given a > 0, we wish to calculate it reciprocal value 1 by using Newton - Raphson method for
a

f(x) = 0. The Newton Raphson algorithm for the function will be (GATE-2005[CE])

1 a a
a) xk+1 = 2 xk + xk b) xk+1 = xk + 2 x2k

c) xk+1 = 2xk - ax2k a
d) xk+1 = xk - 2 x2k

20. Given that one root of the equation x3 - 10x2 + 31x - 30 = 0 is 5 then other roots are

(GATE-2007[CE])

a) 2 and 3 b) 2 and 4 c) 3 and 4 d) -2 and -3

21. The following equation needs to be numerically solved using the Newton - Raphson method
x3 + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)
(GATE-2007[CE])

a) xk+1 = 2x3k + 9 b) x= 2x3k + 9
2x2k + 4 k+1 2x2k + 9

c) xk+1 = xk - 32k + 4 d) xk+1 = 4x2k + 3
9x2k + 2

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ENGINEERING MATHEMATICS

22. The table below gives values of a function F(x) obtained for values of x at intervals of 0.25
(GATE-2010[CE])

x 0 0.25 0.50 0.75 1
F(x) 1 0.9412 0.8 0.64 0.5

The value of the integral of the function between the limits 0 to 1, using Simpson’s rule is

a) 0.7854 b) 2.3562 c) 3.1416 d) 7.5000

23. The squre root of a number N is to be obtained by applying the Newton - Raphson iteration to the
equation x2 - N = 0. If i denotes the iteration index, the correct iterative scheme will be
(GATE-2011[CE])

a) xi+1 = 1 N 1N 1N 1N
2 xi + xi b) xi+1 = 2 x2i + x2i c) xi+1 = 2 xi + xi d) xi+1 = 2 xi - xi

1.5

∫24. The estimate of dx obtained using Simpson’s rule with three-point function evaluation exceeds
x
0.5

the exact value by (GATE-2012[CE])

a) 0.235 b) 0.0668 c) 0.024 d) 0.012

25. There is no value of x that can simultaneously satisfy both the given equations. Therefore, find the ‘least

squares error’solution to the two equations, i.e. find the value of x that minimizes the sum of squares

of the errors in the tow equations. (GATE-2013[CE])

2x =3

4x = 1

26. Find the magnitude of the error (correct to two decimal places) in the estimation of following integral

1 4
using Simpson’s
(GATE-2013[CE])
∫3
rule. Take the step length as 1.(x4+10)dx

0

27. The quadratic equation x2 - 4x + 4 = 0 is to be solved numerically, staring with the initial guess x0 = 3.
The Newton-Raphson method is applied once to get a new estimate and then the Secant method is

applied once using in the initial guess and this new estimate. The estimated value of the root after the

application of the Secant method is ____________. (GATE-2015-CE-Set 1)

28. In Newton-Raphson iterative method, the initial guess value (xini) is considered as zero while finding
the roots of the equation : f(x) = -2+6x-4x2+0.5x3. The correction, ∆x to be added to xini in the first

iteration is ______ . (GATE-2015-CE-Set 2)

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MECHANICAL ENGINEERING

29. For step-size, ∆x = 0.4, the value of following integral using Simpson’s 1/3 rule is _________

0.8 (GATE-2015-CE-Set 2)

∫(0.2 + 25x - 200x2 +675x3 - 900x4 + 400x5)dx
0

30. For solving algebraic and transcendental equation which one fo the following is used ?

(GATE-2015[PI])

a) Coulomb’s theorem b) Newton - Raphson method

c) Euler’s method d) Stoke’s theorem

31. Newton - Raphson formula to find the roots of an euqation f(x) = 0 is given by

a) xn+1 = x0 - f(xn) b) xn+1 = x0 + f(xn) (GATE-2015[PI])
f1(xn) f1(xn)
xnf1(xn)
c) xn+1 = f(xn) d) x = f1(xn)
xnf1(xn) n+1

32. The real root of the equation xex = 2 is evaluated using Newton - Raphson’s method. If the first

approximation of the value x is 0.8679, the 2nd approximation of the value x corrected to three decimal

place is (GATE-2015[PI])

a) 0.865 b) 0.853 c) 0.849 d) 0.838

33. Matching exercise choose the correct one out of alternatives A, B, C, D (GATE-2007[PI])

Group - 1 Group - II

P. 2nd order differential equations (1) Runge - Kutta method

Q. Non-linear algebraic equations (2) Newton - Raphson method

R. Linear algebraic equations (3) Gauss Elimination

S. Numerical integration (4) Simpson’s Rule

a) P - 3, Q -2, R - 4, S - 1 b) P - 2, Q - 4, R - 3, S - 1

c) P - 1, Q - 2, R - 3, S - 4 d) P - 1, Q - 3, R - 2, S - 4

b

∫34. The following algorithm computes the integral J = f(x)dx from the given values
fj = f(xj) at equidistant points x0 = a, x1 = x0 + h, a
x2 = x0 + 2h, ... x2m = x0 + 2mh = b compute
S0 = f0 + f2m (GATE-2010[PI])
S1= f1 + f3 + ..... + f2m-1
S2= f2 + f4 + ..... + f2m-2

h
J = 3 [S0 + 3(S1) + 2(S2)]

The rule of numerical intergration, which uses of the above algorithm is

a) Rectangle rule b) Trapezoidal rule c) Four - point rule d) Simpson’s rule

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ENGINEERING MATHEMATICS

∫35. Using the Simpson’s 1/3rd rule, the value of the ydx computed, for the data given below, is
(GATE-2014-PI-Set 1)
x135

y246

36. If the equation sin(x) = x2 is solved by Newton Raphson’s method with the initial guess of x= 1, then

the value of x after 2 interations would be ________ (GATE-2014-PI-Set 1)

37. In numerical integration suing Simpson’s rule, the approximating function in the interval is a

(GATE-2015[PI])

a) constant b) straight line c) cubic B-Spline d) Parabola

**********************

“Success demands singleness of purpose”

7. NUMERICAL METHODS (ANS.) :
1 - c, 2 - 1/200. 3 - sol, 4 - d, 5 - c, 6 - c, 7 - c, 8 - d, 9 - 1.11, 10 - 1.7532, 11 - 1.16, 12 - 0.54, 13
- d, 14 - 2, 15 - 22, 16 - 0.6931, 17 - 0.3043, 18 - b, 19 - c, 20 - a, 21 - a, 22 - a, 23 - a, 24 - d, 25 -
0.3, 26 - 0.533, 27 - 2.33, 28 - 0.33, 29 - 1.367, 30 - b, 31 - a, 32 - b, 33 - c, 34 - d, 35 - 20, 36 -
0.73, 37 - d.

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8 PROBABILITY & STATISTICS

1. A lot had 10% defective items. Ten items are chosen randomly from this lot. The probability that

exactly 2 of the chosen items are defective is (GATE-05[ME])

a) 0.0036 b) 0.1937 c) 0.2234 d) 0.3874

2. A single die is thrown two times. What is the probability that the sum is neither 8 nor 9 ?
(GATE-05[ME])

a) 1 b) 5 c) 1 3
9 36 4 d)

4

3. Let X and Y be two independent random variables. Which one of the relations between expectation

(E), variance (Var) and covariance (Cov) given below is FALSE ? (GATE-07[ME])

a) E(XY) = E(X) E(Y) b) cov (X,Y) = 0

c) Var (X+Y) = Var(X) + Var(Y) d) E(X2Y2) = (E(X)2 (E(Y))2

4. A coin is tossed 4 times. What is the probability of getting heads exactly 3 times ?

(GATE-08[ME])

a) 1/4 b) 3/8 c) 1/2 d) 3/4

5. If three coins are tossed simultaneously, the probability of getting at least one head is

(GATE-09[ME])

a) 1/8 b) 3/8 c) 1/2 d) 7/8

6. The standard deviation of a uniformly distributed random variagble between o and 1 is
(GATE-09[ME])

a) 1 b) 1 c)√512 d) 7
√12 √3 √12

7. A box contains 2 washers, 3 nuts and 4 bolts. Items are drawn from the box at random one at a time

without replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently

the 4 bolts is (GATE-10[ME])

a) 2/315 b) 1/630 c) 1/1260 d) 1/2520

8. An unbiassed coin is tossed five times. The outcome of each loss is either a head or a tail. Probability

of getting at least one head is ___ (GATE-11[ME])

a) 1 b) 13 c) 16 d) 31
32 32 32 32

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9. Let X be a normal random variable with mean 1 and variance 4. The probability P{X<0} is

a) 0.5 b) greater than zero and less than 0.5

c) greater than 0.5 and less than 1.0 d) 1.0 (GATE-2013[ME])

10. The probability that a student knows the correct answer to a multiple choice question is 2
3

If the student does not know the answer, then the student guesses the answer. The correct is 1
4

Given that the student has answered the question correctly, the conditional probability that the student

knows the correct answer is (GATE-13[ME])

a) 2 b) 3 c) 5 d) 8
3 4 6 9

11. In the following table, x is a discrete random variable and p(x) is the probability density. The standard
deviation of x is
(GATE-14-ME-Set 1)
x 1 23
p(x) 0.3 0.6 0.1

a) 0.18 b) 0.36 c) 0.54 d) 0.6

12. A box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a
random manner from the box. The probability of both the parts being good is
(GATE-14-ME-Set 2)

a) 7 42 c) 25 d) 5
20 b) 125 29 9

13. Consider an unbiased cubic die with opposite faces coloured identically and each face coloured red,
blue or green such that each colour appears only two times on the die. If the die is thrown thrice, the
probability of obtaining red colour on top face of the die at leat twice is _____ .
(GATE-14-ME-Set 2)

14. A group consists of equal number of men and women. Of this group 20% of the men and 50% of the

women are unemployed. If a person is selected at random from this group, the probability of the

selected person being employed is ________. (GATE-14-ME-Set 3)

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15. A machine produces 0, 1 or 2 defective pieces in a day with associated probability of 1 , 2 and 1
63 6

respectively. Then mean value and the variance of the number of defective pieces produced by

(GATE-14-ME-Set 3)

a) 1 and 1/3 b) 1/3 and 1 c) 1 and 4/3 d) 1/3 and 4/3

16. A nationalized bank has found that the daily balance available in its savings accounts follows a normal
distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings
account holders, who maintain an average daily balance more than Rs. 500 is ______
(GATE-14-ME-Set 4)

17. The number of accidents occuring in a plant in a month follws Poisson distribution with mean as 5.2.

The probability of occurence of less than 2 acceidnts in the plant during a randomly selelcted month

is ______ (GATE-14-ME-Set 4)

a) 0.029 b) 0.034 c) 0.039 d) 0.044

18. Among the four normal distributions with probability density functions as shown below, which one has

the lowest variance ? (GATE-15-ME-Set 1)

a) I
b) II
c) III
d) IV

19. The probability of obtaining at leat two ‘SIX’ in throwing a fair dice 4 times is

(GATE-15-ME-Set 1)

a) 425/432 b) 19/144 c) 13/144 d) 125/432

20. The chance of a student passing an exam is 20% The chance of a student passing the exam and getting

above 90% mark in it is 5%. Given that a student passes the examination, the probability that the

student get above 90% marks is (GATE-15-ME-Set 2)

a) 1 b) 1 c) 2 d) 5
18 4 9 18

21. If P(X) = 1/4, P(Y) = 1/3, and P(X ∩Y) = 1/2, the value of P(Y/X) is (GATE-15-ME-Set 3)

a) 1 b) 4 c) 1 d) 29
4 25 3 50

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22. The probability that two friends share the same brith-month is (GATE-98)
d) 1/24
a) 1/6 b) 1/12 c) 1/144

11 and P(E1 ∩E2) = 1
23. Consider two events E1 an E2 such that P(E1) = 2 P(E2) = 3 5

Which of the following statemenet is true ? (GATE-99)

a) (E1 ∩E2) = b) E1 and E2 are independent
d) P(E /E ) = 4/5
c) E1 and E are note independent
2 12

24. E1 and E2 are events in a probability sapce satisfying the following constraints P(E1) =P(E2) ;

P(E1 U E2) = 1 ; E1 and E2 are independent then P(E1) = (GATE-2000)

a) 0 b) 1 c) 1 d) 1
4 2

25. In a manufacturing plant, the probability of making a defective bolt is 0.1. The mean and standdard

deviation of defective bolts in a total of 900 bolts are respectively. (GATE-2000)

a) 90 and 9 b) 9 and 90 c) 81 and 9 d) 9 and 81

26. Seven car accidents occured in a week, what is the probability that they all occured on the same

day ? (GATE-01)

a) 1 1 1 d) 7
77 b) 76 c) 27 27

27. Four fair coins are tossed simultaneously. The probability that atleast one heads and atleast one tails

trun up is (GATE-02)

a) 1 b) 1 c) 7 d) 15
16 8 8 16

28. Let P(E) denote the probability of an event E. Given P(A) = 1, P(B) = 1 the value of
2

P(A/B/) and P(B/A) respectively are (GATE-03)

a) 1 , 1 b) 1 , 1 c) 1 , 1 1
42 24 2 d) 1 , 2

29. Abox contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement.

The probability taht none of the two screws is defective will be (GATE-03)

a) 100% b) 50% c) 49% d) none

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30. A hydraulic structure has four gates which operate independently. The probability of failure of each

gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is

(GATE-04)

a) 0.240 b) 0.200 c) 0.040 d) 0.008

31. From a pack of regular playing cards, two cards are drawn at random. What is the probability taht

both cards will be kings, if the card is NOT replaced (GATE-04)

a) 1/26 b) 1/52 c) 1/269 d) 1/221

32. The following data about the flow of liquid was observed in a continous chemical process plant

Flow rate 7.5 7.7 7.9 8.1 8.3 8.5
(liters/sec) to to to to to to
7.7 7.9 8.1 8.3 8.5 8.7
Frequency 1 5 35 17 12 10

Mean flow rate of the liquid is (GATE-04)
d) 8.26 litres/sec
a) 8.00 litres/sec b) 8.06 litres/sec c) 8.16 litres/sec

33. Lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly

2 of the chosen items are defective is (GATE-05[CE])

a) 0.0036 b) 0.1937 c) 0.2234 d) 0.3874

34. Using given data points tabulated below, a straight line passing through the origin is fitted using least

squares method. The slope of the line is (GATE-2005)

x 123
y 1.5 2.2 2.7

a) 0.9 b) 1 c) 1.1 d) 1.5

35. The standard normal probability function can be approximated as (GATE-09[CE])

F(XN) = 1 + exp 1 | XN|0.12)
(-1.7255XN

X = standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and
N

27 cm respectively, the probability that the annual precipitatin will be between 90 cm and 102 cm is

a) 66.7% b) 50.0% c) 33.3% d) 16.7%

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36. Two coins are simultaneously tossed. The probability of two heads simultaneously appearing is

(GATE-10[CE])

a) 1/8 b) 1/6 c) 1/4 d) 1/2

37. The annual precipitation data of a city is normally distributed with mean and standard deviation as

1000mm and 200 mm, respectively. The probability taht the annual precipitation will be more than

1200 mm is (GATE-12[CE])

a) <50% b) 50% c) 75% d) 100%

38. In an experiment, positive and negative values are equally liekly to occur. The probability of obtaining

at most one negative value in five trials is (GATE-12[CE])

a) 1 b) 2 c) 3 d) 6
32 32 32 32

39. Find the value of λ such that the function f(x) is a valid probability density function

{ f(x) = λ(x -1) (2 - x) for 1 < x < 2 = 0 otherwise (GATE-13[CE])

40. The probability density function of evaporation E on any day during a year in a watershed is given by

1 (GATE-14-CE-Set 1)
f(E) = 5 0 < E < 5 mm/day

0 otherwise

The probability taht E lies in between 2 and 4 mm/day in the watershed is (in decimal) _____

41. Atrafic office imposes on an average 5 number of penalties daily on traffic violators.Assume that the

number of penalties on different days is independent and follows a Poisson distribution. The probability

that there will be less than 4 penalties in a day is ___________. (GATE-14-CE-Set 1)

42. A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes ;

(i) Head, (ii) Head, (iii) Head, (iv) Head. The probability of obtaining a “Tail” when the coin is tossed

again is (GATE-14-CE-Set 1)

a) 0 b) 1 c) 4 d) 1
2 55

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43. If {x} is a coninuous, real valued random variable defined over the interval (-∞, +∞) and its occurrence

1 x-a 2 (GATE-14-Set 1)

is defined by the density function given as ; f(x) = 1 e2 b
√2π *
b

where ‘a’ and ‘b’ are the statistical attributes of the integral

a 1 x-a 2
b
∫-∞ 1 e 2
√2π * dx is
b
b) 0.5
a) 1 c) π π
d) 2

44. An observer counts 240veh/h at a specific highway location. Assume that the vehicle arrival at the

location is Poisson distributed, the probability of having one vehicle arriving over a 30-second time

inerval is ____ (GATE-14-CE-Set 2)

45. Consider the following probability mass function (p.m.f) of a random variable X.

p(x, q) = q if X = 0
1 - q if X = 1

0 otherwise

If q = 0.4, the variance of X is __________. (GATE-15-CE-Set 1)

46. The probability density function of a random vriable, x is (GATE-15-CE-Set 2)

f(x) = x (4 - x2) for 0 < x < 2
4

= 0 otherwise

The mean, µ of the random variable is ____________ .
x

47. Four cards are randomly selected from a pack of 52 cards. If the first two cards are kings, what is the

probability that the third card is a king ? (GATE-15-CE-Set 2)

a) 4/52 b) 2/50 c) (1/52)x(1/52) d) (1/52)x(1/51)x(1/50)

48. Two dice are thrown simultaneously. The probability that the sum of numbers on both exceeds 8 is
(GATE-05[PI])

4 7 9 10
a) 36 b) 36 c) 36 d) 36

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49. The life of a bulb (in hours) is a random variable with an exponential distribution f(t) = αe-αt, 0 < t < ∞.

The probability that its value lies between 100 and 200 hours is (GATE-05[PI])

a) e-100α - e-200α b) e-100α - e-200 c) e-100α + e-200α d) e-200α - e-100α

50. Two cards are drawn at random in succession with replacement from a deck of 52 well shuffled cards

Probability of getting both ‘Aces’ is (GATE-07[PI])

1 b) 2 c) 1 d) 2
a) 169 13 13

169

51. The random variabele X takes on the values 1, 2 (or) 3 with probability 2 + 5P , 1 + 3P and
55

1.5 + 2P respectively the values of P and E(X) are respectively (GATE-07[PI])
5

a) 0.05, 1.87 b) 1.90, 5.87 c) 0.05, 1.10 d) 0.25, 1.40

52. If X is a continuous random variabel whose probability density fucntion is given by

k(5x - 2x2), 0 < x < 2 (GATE-07[PI])
f(x) =
d) 17/28
{ 0, otherwise
Then P(x >1) is
a) 3/14 b) 4/5 c) 14/17

53. For a random variable x(-∞ < x < ∞) following normal distribution, the mean is µ = 100 If the
probability is P= α for x > 110. Then the probability of x lying between 90 and 110. i.e. , P(90<x<110)

and equal to b) 1 - α c) 1 - α/2 (GATE-08[PI])
a) 1 - 2α d) 2α

54. In a game, who two players X and Y toss a coin alternately. Whoever gets a ‘head’first, win the game

and the game is terminated.Assuming that player X starts the game the probability of player X winning

the game is (GATE008[PI])

a) 1/3 b) 1/4 c) 2/3 d) 3/4

55. If a random variable X satisfies the possion’s distribution with a mean value of 2, then the probability

that X > 2 is (GATE-10[PI])

a) 2e-2 b) 1- 2e-2 c) 3e-2 d) 1- 3e-2

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56. Two white and two black balls, kept in two bins, are arranged in four ways as shown below. In each

arrangement, a bin has to be chosen randomly and only one ball needs to be picked randomly from the

chosen bin. Which one of the following arrangements has the highest probability for getting a white ball

picked ? (GATE-10[PI])

a) b)

c) d)

57. It is estimated that the average number of events during a year is three. What is the probability of

occurence of non more that two events over a two-year duration ?Assume that the number of events

follow a poisson distribution. (GATE-11[PI])

a) 0.052 b) 0.062 c) 0.072 d) 0.082

58. A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after

another, without replacement. The probability that the selcted set contains one red ball and two black

ball is (GATE-12[ME, PI])

a) 1/20 b) 1/12 c) 3/10 d) 1/2

59. An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies 60%

and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test. The

ones that pass the quality test are considered reliable. Of X’ a shock absorbers, 96% are reliable. Of

Y’ shock abosrbers, 72% are reliable. The probability that a randomly choosen shock absorber,

which is found to reliabel is made by Y is (GATE-12[ME, PI])

a) 0.288 b) 0.334 c) 0.667 d) 0.720

60. A simple random sample of 100 observations was taken form a large population. The sample mean

and the standard deviation were determined to be 80 and 12, respectively. The standard error of

mean is ____ . (GATE-14-PI-Set 1)

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ENGINEERING MATHEMATICS

61. Marks obtained by 100 students in an examination are given in the table. (GATE-14-PI-Set 1)

S. No. Marked Number of
Obtained students
1
2 25 20
3 30 20
4 35 40
40 20

What would be the mean, median and mode of the marks obtained by the students ?

a) Mean 33; Median 35; Mode 40 b) Mean 35; Median 32.5; Mode 40

c) Mean 33; Median 35; Mode 35 d) Mean 33; Median 32.5; Mode 35

62. In a given day in the rainy season, it may rain 70% of the time. If it rains, chance taht a village fair will

make a loss on that day is 80% . However, if it does not rain, chance that the fair will make a loss on

that day is only 10%. If the fair has not made a loss on a given day in the rainy season, what is the

probability that it has not rained on that day. (GATE-14-PI-Set 1)

a) 3/10 b) 9/11 c) 14/17 d) 27/41

*****************

“If you go as far as you can see, you will then see enough
to go even farther”

8. PROBABILITY & STATISTICS (ANS.) :
1 - b, 2 - d, 3 - d, 4 - a, 5 - d, 6 - a, 7 - c, 8 -d, 9 - b, 10 - d, 11 - d, 12 - a, 13 - 0.26, 14 - 0.65,
15 - a, 16 - 50, 17 - b, 18 - d, 19 - b, 20 - b, 21 - c, 22 - b, 23 - c, 24 - d, 25 - a, 26 - b, 27 - c, 28 -
d, 29 - d, 30 - c, 31 - d, 32 - c, 33 - b, 34 - b, 35 - d, 36 - c, 37 - a, 38 - d, 39 - 6, 40 - 0.4, 41 -
0.625, 42 - b, 43 - b, 44 - 0.27, 45 - 0.24, 46 - 1.067, 47 - b, 48 - d, 49 - a, 50 - a, 51 - a, 52 - d, 53
- a, 54 - c, 55 - d, 56 - c, 57 - b, 58 - d, 59 - b, 60 - 1.2, 61 - c, 62 - d.

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