GATE Mechanical Solved Paper (2000-2013)

GATE 2014

MECHANICAL ENGINEERING

Topicwise Solved Paper

2013 - 2000

GATE MCQ MECHANICAL Engineering (Vol -1, 2, 3 and 4)

by nodia and company

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NODIA & COMPANY

VOL 1 APPLIED MECHANICS & DESIGN

Engineering Mechanics Strength of Material

Theory of Machine

Machine Design

VOL 2 FLUID MECHANICS & THERMAL SCIENCES

Fluid Mechanics

Heat and Mass Transfer Thermodynamics

VOL 2 FLUID MECHANICS & THERMAL SCIENCES

Industrial Engineering

Manufacturing Engineering

Operation Research

VOL 2 ENGG MATHS, GA & SOLVED PAPER

Engineering Mathematics General Aptitude

CONTENTS

UNIT 1

ENGINEERING MATHEMATICS

UNIT 2

ENGINEERING MECHANICS

UNIT 3

STRENGTH OF MATERIALS

UNIT 4

THEORY OF MACHINES

UNIT 5

MACHINE DESIGN

UNIT 6

FLUID MECHANICS

UNIT 7

HEAT TRANSFER

UNIT 8

THERMODYNAMICS

UNIT 9

REFRIGERATION & AIR-CONDITIONING

UNIT 10

MANUFACTURING ENGINEERING

UNIT 11

INDUSTRIAL ENGINEERING

2

ENGINEERING MATHEMATICS

YEAR 2013 ONE MARK

MCQ 2.1 Match the correct pairs:

Numerical Integration Scheme Order of Fitting Polynomial

P. Simpson’s 3/8 Rule 1. First

Q. Trapezoidal Rule 2. Second

R. Simpson’s 1/3 Rule 3. Third

(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

MCQ 2.2 The eigenvalues of a symmetric matrix are all

(A) Complex with non-zero positive imaginary part. (B) Complex with non-zero negative imaginary part. (C) real

(D) pure imaginary

MCQ 2.3 The partial differential equation

t u _u x u xu2 2 2 2 2 2 2 2 + = is a

(A) linear equation of order 2 (B) non-linear equation of order 1

(C) linear equation of order 1 (D) non-linear equation of order 2

MCQ 2.4 Choose the Correct set of functions, which are linearly dependent. (A) sin sinx, 2x _{and cos x}2 _{(B) cos sinx, x and tan x} (C) cos x2 , sin x2 _{and cos x}2 _{(D) cos2x,sinx and cos x}

YEAR 2013 TWO MARKS

MCQ 2.5 The function f t^ h satisfies the differential equation d f_dt f 0

2 2

+ = and the auxillary

conditions, f 0_{^ h=}0,df_dt^0_h=4_{. The Laplace Transform of f t^ h is given by}

(A) _s+2₁ (B) _s+4₁

(C)

s 1

4

2₊ (D) _s4₊2 ₁

MCQ 2.6 The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F=xi+yj+zk defined with respect to a Cartesian coordinate system having i j, and k as unit base vectors.

dA F n 4 1 S : ^ h

##

where S is the sphere, x2₊y2₊z2₌1 _{and n is the outward unit normal vector} to the sphere. The value of the surface integral is

GATE MCQ Mechanical Engineering (4-volumes)

(C) 3 4p/ (D) 4p

MCQ 2.7 The value of the definite integral e x ln x dx

1 ^ h

#

is (A) ₉4 e3₊₉2 _{(B) e} 9 2 9 4 3 -(C) ₉2 e3₊₉4 _{(D) e} 9 4 9 2 3

-MCQ 2.8 The solution of the differential equation d u_dx k 0 dx du

2 2

- = where k is a constant,

subjected to the boundary conditions u 0^ h=0 and u L^ h=U, is

(A) u=_{U L}x (B) u U e e 11 kL kx = - -c m (C) u U e e 1 1 kL kx = -d n (D) u U e e 1 1 kL kx = ++ d n

MCQ 2.9 The probability that a student knows the correct answer to a multiple choice question is ₃2_{. If the student does not know the answer, then the student guesses} the answer. The probability of the guessed answer being correct is ₄1_{. Given that} the student has answered the question correctly, the conditional probability that the student knows the correct answer is

(A) 2₃ (B) ₄3

(C) ₆5 (D) ₉8

YEAR 2012 ONE MARK

MCQ 2.10 The area enclosed between the straight line y=x and the parabola y₌x2 _{in the}

x-y plane is

(A) 1 6/ (B) 1 4/

(C) 1 3/ (D) 1 2/

MCQ 2.11 Consider the function f x( ) = x in the interval -1#x #1. At the point x=0 , f x( ) is

(A) continuous and differentiable (B) non-continuous and differentiable (C) continuous and non-differentiable (D) neither continuous nor differentiable MCQ 2.12 lim cos x x 1 x 0 2 -" b l is (A) 1 4/ (B) 1 2/ (C) 1 (D) 2

MCQ 2.13 At x=0, the function f x( )₌x3₊1 _has

(A) a maximum value (B) a minimum value

(C) a singularity (D) a point of inflection

MCQ 2.14 For the spherical surface x2₊y2₊z2₌1, _{the unit outward normal vector at the}

point , , 2 1 2 1 0 c m is given by (A) i j 2 1 2 1 + (B) i j 2 1 2 1

-(C) k (D) i j k 3 1 3 1 3 1 + +

YEAR 2012 TWO MARKS

MCQ 2.15 The inverse Laplace transform of the function ( )

( )

F s

s s1 1

= ₊ is given by

(A) f t( )=sint (B) f t( )₌e-tsint

(C) f t( )₌e-t _{(D) f t( )= -1 e}-t

MCQ 2.16 For the matrix A=>5_{1 3}3H, ONE of the normalized eigen vectors given as

(A) 2 1 23 J L K K KK N P O O OO (B) 2 2 1 1 -J L K K K K N P O O O O (C) 10 3 10 1 -J L K K K K N P O O O O (D) 5 1 5 2 J L K K K K N P O O O O

MCQ 2.17 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 selected set contains one red ball and two black balls is

(A) 1 20/ (B) 1 12/

(C) 3 10/ (D) 1 2/

MCQ 2.18 Consider the differential equation x d y dx2( 2 / 2)₊x dy dx( / )_-4y₌0 _{with the} boundary conditions of y 0( )=0 and y 1( )=1. The complete solution of the differential equation is (A) x2 _{(B) sin} x 2 p a k (C) exsin p₂x a k (D) e-xsin p₂x a k MCQ 2.19 x+2y+z =4 x y z 2 + +2 =5 x- +y z =1

The system of algebraic equations given above has (A) a unique solution of x=1,y=1andz=1.

(B) only the two solutions of (x=1,y=1,z=1)and(x =2,y=1,z=0) (C) infinite number of solutions

(D) no feasible solution

YEAR 2011 ONE MARK

MCQ 2.20 A series expansion for the function sin q is

(A) 1-₂q_!2+q_4!4-... (B) q-q_3!3+₅q_!5-... (C) 1+ +q q_2!2+q₃3_!+... (D) q+q_3!3+₅q_!5+...

GATE MCQ Mechanical Engineering (4-volumes)

MCQ 2.21 What is lim sin

0 q q " q equal to ? (A) q (B) sin q (C) 0 (D) 1

MCQ 2.22 Eigen values of a real symmetric matrix are always

(A) positive (B) negative

(C) real (D) complex

MCQ 2.23 The product of two complex numbers 1 +i and 2-5i is

(A) 7-3i (B) 3-4i

(C) - -3 4i (D) 7+3i

MCQ 2.24 If f x( ) is an even function and a is a positive real number, then f x dx( ) a a

-#

equals (A) 0 (B) a (C) 2a (D) 2 af x dx( ) 0

#

YEAR 2011 TWO MARKS

MCQ 2.25 The integral _{x dx}1 1

3

#

, when evaluated by using Simpson’s 1/3 rule on two

equal sub-intervals each of length 1, equals

(A) 1.000 (B) 1.098

(C) 1.111 (D) 1.120

MCQ 2.26 Consider the differential equation _dxdy ₌(1₊y x2) _{. The general solution with} constant c is

(A) y=tanx₂2+tanc (B) y₌tan2_ax₂₊c_k

(C) y₌tan2 x_{2 +}c

a k (D) y=tanbx₂2+cl

MCQ 2.27 An unbiased coin is tossed five times. The outcome of each toss is either a head or a tail. The probability of getting at least one head is

(A) ₃₂1 (B) ₃₂13

(C) ₃₂16 (D) ₃₂31

MCQ 2.28 Consider the following system of equations

x x x

2 1+ 2+ 3 =0

x2-x3 =0

x1+x2 =0 This system has

(A) a unique solution (B) no solution

(C) infinite number of solutions (D) five solutions

YEAR 2010 ONE MARK MCQ 2.29 The parabolic arc y= x, 1#x#2 is revolved around the x-axis. The volume

of the solid of revolution is (A) p/4

(B) p/2 (C) 3 4p/ (D) 3 2p/

MCQ 2.30 The Blasius equation,

d d f f d d f 2 0 3 3 2 2 h + h = , is a

(A) second order nonlinear ordinary differential equation (B) third order nonlinear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order nonlinear ordinary differential equation MCQ 2.31 The value of the integral

x dx 1₊ 2 3 3

-#

is (A) -p (B) -p/2 (C) p/2 (D) p

MCQ 2.32 The modulus of the complex number b₁3_-+₂4_iil is

(A) 5 (B) 5

(C) 1/ 5 (D) 1/5

MCQ 2.33 The function y= 2-3x

(A) is continuous 6 !x R and differentiable 6 !x R

(B) is continuous 6 !x R and differentiable 6 !x R except at x =3 2/ (C) is continuous 6 !x R and differentiable 6 !x R except at x =2 3/ (D) is continuous 6 !x R except x=3 and differentiable 6 !x R

YEAR 2010 TWO MARKS

MCQ 2.34 One of the eigen vectors of the matrix A=>2₁ 2₃H is

(A) > H_-2₁ (B) > H₁2

(C) > H4₁ (D) > H_-1₁

MCQ 2.35 The Laplace transform of a function f t( ) is

( )

s s2 1₊1 . The function f t( ) is

(A) t_{- +}1 e-t _{(B) t+ +1 e}-t

(C) _{- +}1 e-t _{(D) 2t+e}t

MCQ 2.36 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 MCQ Mechanical Engineering (4-volumes)

(C) 1/1260 (D) 1/2520

MCQ 2.37 Torque exerted on a flywheel over a cycle is listed in the table. Flywheel energy (in J per unit cycle) using Simpson’s rule is

Angle (Degree) 0 60c 120c 180c 240c 300c 360c

Torque (N-m) 0 1066 -323 0 323 -355 0

(A) 542 (B) 993

(C) 1444 (D) 1986

YEAR 2009 ONE MARK

MCQ 2.38 For a matrix 6M@=>3 5 4 5/_{x 3 5}/_/ H, the transpose of the matrix is equal to the

inverse of the matrix, 6M@T =6M@-1. The value of x is given by

(A) -₅4 (B) -₅3

(C) ₅3 (D) ₅4

MCQ 2.39 The divergence of the vector field 3xzi₊2xyj_-yz2k _{at a point ( , , )1 1 1 is equal to}

(A) 7 (B) 4

(C) 3 (D) 0

MCQ 2.40 The inverse Laplace transform of 1/(s2₊s) _is

(A) 1₊et _{(B) 1-e}t

(C) 1_-e-t _{(D) 1+e}-t

MCQ 2.41 If three coins are tossed simultaneously, the probability of getting at least one head is

(A) 1/8 (B) 3/8

(C) 1/2 (D) 7/8

YEAR 2009 TWO MARKS

MCQ 2.42 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=xy, the expression for v should be

(A) (x+₂y)2+k (B) x2-₂y2+k

(C) y2-₂ x2 +k (D)(x-₂y)2+k

MCQ 2.43 The solution of _xdxdy _{+ =}y x4 _{with the condition y 1( )} 5 6 = is

(A) y= x₅4+_x1 (B) y= 4₅x4+₅4_x

(C) y= x₅4+1 (D) y=x₅5+1

MCQ 2.44 A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x₊y)2 _{on path AB traversed in a counter-clockwise sense}

is

(A) p -₂ 1 (B) p +₂ 1

(C) p₂ (D) 1

MCQ 2.45 The distance between the origin and the point nearest to it on the surface

z2_{= +}1 xy _is

(A) 1 (B) ₂3

(C) 3 (D) 2

MCQ 2.46 The area enclosed between the curves y2₌4x _{and x}2_{=4y is}

(A) 16₃ (B) 8

(C) 32₃ (D) 16

MCQ 2.47 The standard deviation of a uniformly distributed random variable between 0 and 1 is (A) 12 1 _(B) 3 1 (C) 12 5 _(D) 12 7

YEAR 2008 ONE MARK

MCQ 2.48 In the Taylor series expansion of ex _{about x = , the coefficient of (2 x-2)}4 _is

(A) 1/4 ! (B) 2 /4!4

(C) /4!e2 _{(D) e}4_/4!

MCQ 2.49 Given that xp+3x=0, and (0)x =1, xo(0)=0, what is (1)x ?

(A) 0.99- (B) -0.16 (C) 0.16 (D) 0.99 MCQ 2.50 The value of ( ) lim x x 82 / x 8 1 3 - -" (A) 161 (B) 121 (C) 81 (D) 41

MCQ 2.51 A coin is tossed 4 times. What is the probability of getting heads exactly 3 times ?

(A) 41 (B) 83

GATE MCQ Mechanical Engineering (4-volumes)

MCQ 2.52 The matrix 4 p 1 3 1 2 0 1 6 R T S S SS V X W W

WW has one eigen value equal to 3. The sum of the other two eigen value is

(A) p (B) p -1

(C) p - (D) 2 p -3

MCQ 2.53 The divergence of the vector field (x-y)i+(y-x)j+(x+ +y z)k is

(A) 0 (B) 1

(C) 2 (D) 3

YEAR 2008 TWO MARKS

MCQ 2.54 Consider the shaded triangular region P shown in the figure. What is xydxdy

P

##

?

(A) 61 (B) 92

(C) 167 (D) 1

MCQ 2.55 The directional derivative of the scalar function ( , , )f x y z ₌x2₊2y2_{+ at the} z

point (1, 1, 2)

P = in the direction of the vector a=3i-4j is

(A) 4- (B) -2

(C) 1- (D) 1

MCQ 2.56 For what value of a, if any will the following system of equation in , andx y z have

a solution ?

2x+3y=4 4

x+ + =y z

3x+2y- =z a

(A) Any real number (B) 0

(C) 1 (D) There is no such value

MCQ 2.57 Which of the following integrals is unbounded ? (A) tan xdx/ 0 4 p

#

(B) x21 1dx 0 + 3

#

(C) xe dxx 0 3

-#

(D) ₁ 1_{x dx} 0 1

-#

MCQ 2.58 The integral

#

f z dz( ) evaluated around the unit circle on the complex plane for

( ) cos

(A) 2 ip (B) 4 ip

(C) 2 i- p (D) 0

MCQ 2.59 The length of the curve y_{= 3}2x3 2/ _{between x = and 0 x = is1}

(A) 0.27 (B) 0.67

(C) 1 (D) 1.22

MCQ 2.60 The eigen vector of the matrix >1_{0 2}2H are written in the form >_a1Hand>1_bH. What is a+ ?b (A) 0 _{(B) 2}1 (C) 1 (D) 2 MCQ 2.61 Let f₌yx_{. What is} x yf 2 2 2 2 _{at x=2,y= ?1} (A) 0 (B) ln2 (C) 1 _{(D) ln2}1

MCQ 2.62 It is given that ym+2yl+ =y 0, (0)y =0, (1)y =0. What is (0.5)y ?

(A) 0 (B) 0.37

(C) 0.62 (D) 1.13

YEAR 2007 ONE MARK

MCQ 2.63 The minimum value of function y_{= in the interval [1, 5] is}x2

(A) 0 (B) 1

(C) 25 (D) undefined

MCQ 2.64 If a square matrix A is real and symmetric, then the eigen values

(A) are always real (B) are always real and positive

(C) are always real and non-negative (D) occur in complex conjugate pairs MCQ 2.65 If ( , )j x y and ( , )y x y are functions with continuous second derivatives, then

( , )x y i ( , )x y

j + y can be expressed as an analytic function of x+iy(i= - , 1)

when (A) , x x y y 2 2 2 2 2 2 2 2 j _=- y j ₌ y _{(B) ,} y x x y 2 2 2 2 2 2 2 2 j _=- y j ₌ y (C) 1 x2 y x y 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 j₊ j ₌ y₊ y _{= (D) 0} x y x y 2 2 2 2 2 2 2 2 j₊ j ₌ y₊ y ₌

MCQ 2.66 The partial differential equation 0

x2 y x y 2 2 2 2 2 2 2 2 2 2 2 j₊ j₊ j₊ j _{= has}

(A) degree 1 order 2 (B) degree 1 order 1

(C) degree 2 order 1 (D) degree 2 order 2

YEAR 2007 TWO MARKS

MCQ 2.67 If y= +x x+ x+ x+...3, then y 2 =^ h

GATE MCQ Mechanical Engineering (4-volumes)

(C) 1 only (D) undefined

MCQ 2.68 The area of a triangle formed by the tips of vectors , anda b c is

(A) (₂1 a-b):(a- (B) c) (a b) (a c) 2 1 # - -(C) 1₂ a#b#c (D) ₂1(a#b):c

MCQ 2.69 _{The solution of dx}dy _{= with initial value (0)}y2 _{y = bounded in the interval1}

(A) -3#x#3 (B) -_{3 # #}x 1

(C) x<1,x> (D) 1 -2#x #2

MCQ 2.70 If F s( ) is the Laplace transform of function ( )f t , then Laplace transform of

( ) f d t 0 t t

#

is (A) _{s F s}1 ( ) (B) _{s F s}1 ( )-f(0) (C) sF s( )-f(0) (D)

#

F s d s( )

MCQ 2.71 A calculator has accuracy up to 8 digits after decimal place. The value of sin xdx

0 2p

#

when evaluated using the calculator by trapezoidal method with 8 equal inter-vals, to 5 significant digits is

(A) 0.00000 (B) 1.0000

(C) 0.00500 (D) 0.00025

MCQ 2.72 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 ?

(A) E XY( )=E X E Y( ) ( ) (B) C v X Yo ( , )=0

(C) Var(X+Y)=Var( )X +Var( )Y (D) (E X Y2 2)₌( ( )) ( ( ))E X 2 E Y 2

MCQ 2.73 lim x e 1 x x₂ x x 0 3 2 - + + = " b l (A) 0 (B) 1/6 (C) 1/3 (D) 1

MCQ 2.74 The number of linearly independent eigen vectors of >2₀ 1₂H is

(A) 0 (B) 1

(C) 2 (D) infinite

YEAR 2006 ONE MARK

MCQ 2.75 Match the items in column I and II.

Column I Column II

P. Gauss-Seidel method 1. Interpolation

Q. Forward Newton-Gauss method 2. Non-linear differential equations R. Runge-Kutta method 3. Numerical integration

S. Trapezoidal Rule 4. Linear algebraic equations

(A) P-1, Q-4, R-3, S-2 (B) P-1, Q-4, R-2, S-3

(C) P-1. Q-3, R-2, S-4 (D) P-4, Q-1, R-2, S-3

MCQ 2.76 The solution of the differential equation _dxdy 2xy e x2

+ = - _{with (0)y = is1} (A) (1 x e) x2 + + _{(B) (1 x e)} x2 + -(C) (1 x e) x2 - + _{(D) (1 x e)} x2 -

-MCQ 2.77 Let x denote a real number. Find out the INCORRECT statement. (A) S={ :x x >3} represents the set of all real numbers greater than 3 (B) S₌{ :x x <2 0} _{represents the empty set.}

(C) S={ :x x!Aandx!B} represents the union of set A and set B .

(D) S={ :x a< <x b} represents the set of all real numbers between a and b, where a and b are real numbers.

MCQ 2.78 A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective ?

(A) 51 (B) 251

(C) 9920 (D) 49519

YEAR 2006 TWO MARKS

MCQ 2.79 Eigen values of a matrix S=>3₂ 2₃H are 5 and 1. What are the eigen values of the matrix S2₌SS _?

(A) 1 and 25 (B) 6 and 4

(C) 5 and 1 (D) 2 and 10

MCQ 2.80 Equation of the line normal to function ( )f x ₌(x_-8)2 3/ _{+ at (0,5)}1 _{P is}

(A) y=3x- (B) 5 y=3x+5

(C) 3y= +x 15 (D) 3y= -x 15

MCQ 2.81 Assuming i= - and t is a real number, 1 / e dtit

0 3 p

#

is (A) ₂3 +i1₂ (B) ₂3 -i₂1 (C) ₂1+i ₂3 (D) 1₂+ic1- ₂3m MCQ 2.82 If ( )f x xx xx 522 127 39 2 =

-- + , then - lim f xx"3 ( ) will be

(A) 1/3- (B) 5/18

(C) 0 (D) 2/5

MCQ 2.83 Match the items in column I and II.

GATE MCQ Mechanical Engineering (4-volumes)

P. Singular matrix 1. Determinant is not defined

Q. Non-square matrix 2. Determinant is always one

R. Real symmetric 3. Determinant is zero

S. Orthogonal matrix 4. Eigenvalues are always real

5. Eigenvalues are not defined (A) P-3, Q-1, R-4, S-2 (B) P-2, Q-3, R-4, S-1 (C) P-3, Q-2, R-5, S-4 (D) P-3, Q-4, R-2, S-1 MCQ 2.84 For 4 3 3 dx d y dx dy _{y e} x 2 2 2

+ + = , the particular integral is

(A) ₁₅1 e2x _{(B) e}

5 1 2x

(C) 3e2x _{(D) C e} x _{C e} x

1 - + 2 -3

MCQ 2.85 Multiplication of matrices E and F is G. matrices E and G are cos sin sin cos E 0 0 0 0 1 q q q q = -R T S S SS V X W W WW and G 1 0 0 0 1 0 0 0 1 = R T S S SS V X W W WW What is the matrix F ?

(A) cos sin sin cos 0 0 0 0 1 q q q q -R T S S SS V X W W WW (B) cos cos cos sin 0 0 0 0 1 q q q q -R T S S SS V X W W WW (C) cos sin sin cos 0 0 0 0 1 q q q q -R T S S SS V X W W WW (D) sin cos cos sin 0 0 0 0 1 q q q q -R T S S SS V X W W WW

MCQ 2.86 Consider the continuous random variable with probability density function ( )

f t 1= +t for-1#t#0 1= -t for0#t#1

The standard deviation of the random variable is (A) 3 1 (B) 6 1 (C) 31 (D) 61

YEAR 2005 ONE MARK

MCQ 2.87 Stokes theorem connects

(A) a line integral and a surface integral (B) a surface integral and a volume integral (C) a line integral and a volume integral

(D) gradient of a function and its surface integral

MCQ 2.88 A 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

(A) 0.0036 (B) 0.1937 (C) 0.2234 (D) 0.3874 MCQ 2.89 (sin x sin x dx) a a _{6 7} +

-#

is equal to (A) 2 asin x dx6 0

#

(B) 2 asin x dx7 0

#

(C) 2 a(sin6x sin7x dx) 0 +

#

(D) zero

MCQ 2.90 A is a 3# real matrix and Ax4 = is an inconsistent system of equations. The b highest possible rank of A is

(A) 1 (B) 2

(C) 3 (D) 4

MCQ 2.91 Changing the order of the integration in the double integralI _x f x y dydx( , )

4 2 0 8 =

#

#

leads to I f x y dxdy( , ) p q r s =

#

#

What is q ? (A) 4y (B) 16 y2 (C) x (D) 8

YEAR 2005 TWO MARKS

MCQ 2.92 Which one of the following is an eigen vector of the matrix 5 0 0 0 0 5 0 0 0 0 2 3 0 0 1 1 R T S S S S SS V X W W W W WW (A) 1 2 0 0 -R T S S S S SS V X W W W W WW (B) 0 0 1 0 R T S S S S SS V X W W W W WW (C) 1 0 0 2 -R T S S S S SS V X W W W W WW (D) 1 1 2 1 -R T S S S S SS V X W W W W WW

MCQ 2.93 With a 1 unit change in b, what is the change in x in the solution of the system of equations x+ =y 2, 1.01x+0.99y=b?

(A) zero (B) 2 units

(C) 50 units (D) 100 units

MCQ 2.94 By a change of variable ( , )x u v =uv y u v, ( , )=v u/ is double integral, the integrand ( , )

f x y changes to ( , / ) ( , )f uv v u f u v . Then, ( , )f u v is

(A) 2 /v u (B) 2uv

(C) v2 _{(D) 1}

MCQ 2.95 The right circular cone of largest volume that can be enclosed by a sphere of 1 m radius has a height of

GATE MCQ Mechanical Engineering (4-volumes)

(A) 1/3 m (B) 2/3 m

(C) 32 2 m (D) 4/3 m

MCQ 2.96 If _{x dx}2dy ₊2xy₌ 2ln_x( )x _{and (1)y = , then what is ( )0 y e ?}

(A) e (B) 1

(C) 1/e (D) 1/e2

MCQ 2.97 The line integral

#

V:dr of the vector V r:( )=2xyzi+x z2 j+x y2 k from the origin to the point P (1, 1, 1)

(A) is 1 (B) is zero (C) is – 1

(D) cannot be determined without specifying the path

MCQ 2.98 Starting from x0= , one step of Newton-Raphson method in solving the equation 1

3 7 0

x3₊ x_{- = gives the next value ( )x}

1 as

(A) x1=0.5 (B) x1=1.406

(C) x1=1.5 (D) x1=2

MCQ 2.99 A single die is thrown twice. What is the probability that the sum is neither 8 nor 9 ?

(A) 1/9 (B) 5/36

(C) 1/4 (D) 3/4

Common Data For Q. 91 and 92

The complete solution of the ordinary differential equation 0 dx d y pdxdy qy 2 2 + + = is y c e x c e x 1 2 3 = - +

-MCQ 2.100 Then p and q are

(A) p=3,q= (B) 3 p=3,q=4

(C) p =4,q= (D) 3 p=4,q =4

MCQ 2.101 Which of the following is a solution of the differential equation

( 1) 0 dx d y pdxdy q y 2 2 + + + = (A) e-3x _{(B) xe}-x (C) xe-2x _{(D) x e}2 -2x

YEAR 2004 ONE MARK

MCQ 2.102 If x=a q( +sinq) and y=a(1-cosq)_{, then dx}dy will be equal to

(A) sin 2b lq (B) _{cos 2}b lq

(C) tan 2b lq (D) _{cot 2b l}q

(0.259, 0.966, 0)

Q will be

(A) 0c (B) 30c

(C) 45c (D) 60c

MCQ 2.104 The sum of the eigen values of the matrix given below is

3 1 1 3 2 5 1 1 1 R T S S SS V X W W WW (A) 5 (B) 7 (C) 9 (D) 18

YEAR 2004 TWO MARKS

MCQ 2.105 From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be Kings, if first card in NOT replaced ?

(A) 261 (B) 521

(C) 1691 (D) 2211

MCQ 2.106 A delayed unit step function is defined as ( ) , , for for U t a t a t a 0 1 < $ - =_* Its Laplace transform is (A) ae-as _(B) s e-as (C) seas (D) e_aas

MCQ 2.107 The values of a function ( )f x are tabulated below

x f x( )

0 1

1 2

2 1

3 10

Using Newton’s forward difference formula, the cubic polynomial that can be fitted to the above data, is

(A) 2x3₊7x2_-6x_{+ (B)} 2 _2x3_-7x2_+6x-2

(C) x3_-7x2_-6x2_{+ (D)} 1 _2x3_-7x2_+6x+1

MCQ 2.108 The volume of an object expressed in spherical co-ordinates is given by sin V / r2 drd d 0 1 0 3 0 2 f f q =

#

p

#

p

#

The value of the integral is

(A) 3p (B) p6

(C) 32p (D) p4

GATE MCQ Mechanical Engineering (4-volumes)

0 x 8 4 12 0 6 2 0 = R T S S SS V X W W WW (A) 4 (B) 6 (C) 8 (D) 12

YEAR 2003 ONE MARK

MCQ 2.110 _{lim sinx} x x 0 2 " is equal to (A) 0 (B) 3 (C) 1 (D) 1

-MCQ 2.111 The accuracy of Simpson’s rule quadrature for a step size h is

(A) ( )O h2 _{(B) O h( )}3

(C) ( )O h4 _{(D) O h( )}5

MCQ 2.112 For the matrix >4₁ 1₄H the eigen values are

(A) 3 and 3- (B) - and 53

-(C) 3 and 5 (D) 5 and 0

YEAR 2003 TWO MARKS

MCQ 2.113 Consider the system of simultaneous equations 2

x+ y+ z =6 2x+ +y 2z =6

x+ + y z =5 This system has

(A) unique solution

(B) infinite number of solutions (C) no solution

(D) exactly two solutions

MCQ 2.114 The area enclosed between the parabola y_{= and the straight line y}x2 _{= isx}

(A) 1/8 (B) 1/6

(C) 1/3 (D) 1/2

MCQ 2.115 The solution of the differential equation _dxdy ₊y2_{= is}0

(A) y_{= + (B) x} 1 _c y= - +₃x3 c

(C) cex _{(D) unsolvable as equation is}

non-line-ar

MCQ 2.116 The vector field is F=xi-yj (where i and j are unit vector) is

(A) divergence free, but not irrotational (B) irrotational, but not divergence free

(C) divergence free and irrotational (D) neither divergence free nor irrational MCQ 2.117 Laplace transform of the function sin tw is

(A)

s2₊s _w2 (B) _s2₊w_w2

(C)

s2_-s _w2 (D) _s2_-w_w2

MCQ 2.118 A box contains 5 black and 5 red balls. Two balls are randomly picked one after another form the box, without replacement. The probability for balls being red is

(A) 1/90 (B) 1/2

(C) 19/90 (D) 2/9

YEAR 2002 ONE MARK

MCQ 2.119 Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?

(A) ₃₆5 (B) ₁₈5

(C) ₄1 (D) ₃1

MCQ 2.120 Which of the following functions is not differentiable in the domain [-1 1, ] ?

(A) f x( )₌x2 _{(B) f x( )= -x 1}

(C) f x( )=2 (D) f x =( ) maximum (x,-x)

MCQ 2.121 A regression model is used to express a variable Y as a function of another variable X.This implies that

(A) there is a causal relationship between Y and X

(B) a value of X may be used to estimate a value of Y

(C) values of X exactly determine values of Y

(D) there is no causal relationship between Y and X

YEAR 2002 TWO MARKS

MCQ 2.122 The following set of equations has

x y z 3 +2 + =4 x- +y z =2 x z 2 2 - + =5

(A) no solution (B) a unique solution

(C) multiple solutions (D) an inconsistency

MCQ 2.123 The function f x y( , )₌2x2₊2xy_-y3 _has (A) only one stationary point at ( , )0 0

(B) two stationary points at ( , )0 0 and _b61,-₃1_l (C) two stationary points at ( , )0 0 and ( ,1 -1) (D) no stationary point

GATE MCQ Mechanical Engineering (4-volumes)

maximum waiting time at either stop can be 8 min each but any time of waiting up to 8 min is equally, likely at both places. He can afford up to 13 min of total waiting time if he is to arrive at D on time. What is the probability that Manish will arrive late at D ?

(A) ₁₃8 (B) ₆₄13

(C) ₁₂₈119 (D) ₁₂₈9

YEAR 2001 ONE MARK

MCQ 2.125 The divergence of vector i=xi+yj+zk is

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

(C) 0 (D) 1

MCQ 2.126 Consider the system of equations given below

x+y =2

x y

2 +2 =5 This system has

(A) one solution (B) no solution

(C) infinite solutions (D) four solutions

MCQ 2.127 What is the derivative of f x( ) = x at x =0 ?

(A) 1 (B) -1

(C) 0 (D) Does not exist

MCQ 2.128 The Gauss divergence theorem relates certain (A) surface integrals to volume integrals (B) surface integrals to line integrals

(C) vector quantities to other vector quantities (D) line integrals to volume integrals

YEAR 2001 TWO MARKS

MCQ 2.129 The minimum point of the function f x( )=b lx₃3 -x is at

(A) x=1 (B) x =-1

(C) x=0 (D) x

3 1 =

MCQ 2.130 The rank of a 3#3 matrix C(=AB), found by multiplying a non-zero column matrix A of size 3#1 and a non-zero row matrix B of size 1#3, is

(A) 0 (B) 1

(C) 2 (D) 3

MCQ 2.131 An unbiased coin is tossed three times. The probability that the head turns up in exactly two cases is

(A) ₉1 (B) ₈1

GATE MCQ Mechanical Engineering (4-volumes)

SOLUTION

SOL 2.1 Option (D) is correct.

Numerical Integration Scheme Order of Fitting Polynomial

P. Simpson’s 3/8 Rule 3. Third order

Q. Trapezoidal Rule 1. First order

R. Simpson’s 1/3 Rule 2. Second order

SOL 2.2 Option (C) is correct. Let a square matrix

A = >x_y y_xH

The characteristic equation for the eigen value is given by

A-lI =0 x y y x l l -- =0 x_-l 2_-y2 ^ h =0 or _^x_-lh2 _=y2 or x-l =!y or l =x!y it is a real value.

So, eigen values are real if matrix is real and symmetric. SOL 2.3 Option (D) is correct.

We have xu2 u ux ut 2 2 2 2 2 2 2 - - =0

Order is determined by the orders of the highest derivative present in it. So, it is a second order partial differential equation.

It is also a non-linear equation because in linear equation, the product of u with x

u 2

2 _{is not allow. Therefore, it is a second order, non-linear partial differential} equation.

SOL 2.4 Option (C) is correct. We know

cos x2 ₌2cos x2 _-1 cos x2 _{= -}1 2sin x2

cos x2 _{= -1 sin x}2 The linear equation is given by

y =mx+c

This equation satisfy the above three equations, so that cos x2 , sin x2 _{, cos x}2 _are linearly dependent.

SOL 2.5 Option (C) is correct. We have dt d f _f 2 2 + =0 D2₊1 f ^ h =0 The auxiliary equation is

m =!1

Thus the solution of this equation becomes

f t^ h =C1cosx+C2sinx

and df_dt =-C1sinx+C2cosx

From given conditions f 0^ h=0

C1 =0

and df_dt^ h0 =4

4 =C2+0&C2=4

So that f t^ h =4sin x

Hence, the laplace transform is

L f t^ h =4L6sinx@

s24 1 =

+ SOL 2.6 Option (A) is correct.

We have F =xi+yj+zk

and x2₊y2₊z2 ₌₁

We know, the Gauss divergence theorem is

dA F n S $ ] g

##

F dV V $ d =

###

^ h

Thus the gauss theorem transformed surface integral to volume integral.

F $ d _xi _yj _yk $ xi yj zk 2 2 2 2 2 2 =c + + m ^ + + h 1 1 1 3 = + + = So that F dV V $ d ^ h

###

3dV V =

###

3 # = volume of sphere 3 ₃4_p 1 3 4_p # # = ^ h =

Hence the given integral becomes

dA F n 4 1 S $ ^ h

##

= ₄1#4p=p

SOL 2.7 Option (C) is correct.

Let I e x ln x dx

1

=

#

^ h

From ILATE, consider ln x^ h as first and x as second function.

I ln x e x dx e _dxd ln x e x dx dx 1 1 1 = ^ h

#

-

#

; ^ h

#

E ln x ₃2x3 2/ e e_x1 ₃2x dx/ 1 1 3 2 # # =: ^ h D -

#

e x 3 2 ₀ 3 2 3 2 / / e 3 2 3 2 1 # =; - E-; E 1 e e 3 2 9 4 / / 3 2 3 2 = - 6 - @ e e 3 2 9 4 9 4 / / 3 2 3 2 = - + e e 9 2 9 4 9 2 9 4 / 3 2 3 = + = +

SOL 2.8 Option (B) is correct. We have dx d u kdxdu 2 2 - =0

GATE MCQ Mechanical Engineering (4-volumes)

or _^D2_-kD u_{h =0}

The auxiliary equation is

m2_-km ₌₀ m m^ -kh =0

or m =0, k

Thus the complete solution is

u C ex C ekx 1 0 2 = + or u C C ekx 1 2 = +

From the given condition

u 0^ h=0: 0 =C1+C2 C1+C2 =0 ....(i) and u L^ h=U: U C C ekL 1 2 = + ....(ii)

Subtracting equation (i) from (ii), we get

U C ekL 1 2 = ^ - h or C2 ekLU 1 = -^ h

From equation (i), we have

C1 C e U 1 kL 2 =- = -^ h

Substitute these values in the expression for u, we get

u e U e U _e 1 1 kL kL kx = -- ₊ -^ h ^ h or u U e e 1 1 kL kx = -c m

SOL 2.9 Option (D) is correct.

Let A be the event when student knows the answer and B be the event when student guesses the answer. Therefore

P A^ h =P A^ +Bh= ₃2

and P B^ h = +₃2 ₃1#₄1 =₁₂9

where ₃2 _{is the probability of correct answer and} 3

1 _{is the probability that student} does not know the answer. So guesses the answer and probability of correct guess is ₄1_{. Therefore total probability of correct answer}

3 2 3 1 4 1 129 # = + =

Conditional probability that student knows the correct answer / P A B^ h P B P A+B = ^ _{^ h} h 3 2 129 98 ' = =

SOL 2.10 Option (A) is correct.

For y =x straight line and

y ₌x2 _{parabola, curve is as given. The shaded region} is the area, which is bounded by the both curves (common area).

We solve given equation as follows to gett the intersection points : In y₌x2 _{putting y=x we have x =x}2 _or

x2_-x _{=0 & x x( -1) =0 & x =0 1,}

Then from y=x, for x =0 &y=0 and x =1 &y=1 Curve y₌x2 _{and y=x intersects at point ( , )0 0 and ( , )1 1} So, the area bounded by both the curves is

A dydx y x y x x x 0 1 2 = = = = =

#

#

dx dy x x y x y x 0 1 2 = = = = =

#

#

dx y x x x x 0 1 2 = = = 6 @

#

(x x dx) x x 2 0 1 = -= =

#

x x 3 2 3 2 0 1

=: - D = - =-₃1 ₂1 ₆1 _{= 6}1unit2 _{Area is never negative} SOL 2.11 Option (C) is correct.

Given f x( )= x (in -1#x#1)

For this function the plot is as given below.

At x = 0, function is continuous but not differentiable because.

For x >0 and x<0 ( ) f xl =1 and f xl( )=-1 ( ) lim f x x"0+ l =1 and _xlim f x"₀- l( )=-1

R.H.S lim=1 and L.H.S lim=-1

Therefore it is not differentiable. SOL 2.12 Option (B) is correct.

Let y lim( cos )

x x 1 x 0 2 = -"

It forms : D₀0 condition. Hence by L-Hospital rule

y ( ) ( ) lim cos x x 1 x dxd dxd 0 2 = -" lim sinx x 2 x 0 = "

GATE MCQ Mechanical Engineering (4-volumes)

y 2 ( ) ( ) lim sin x x x _dxd dxd 0 _# = " =lim cos xx"0 2 cos 20 21 = =

SOL 2.13 Option (D) is correct.

We have f x( ) ₌x3₊1

( )

f xl =3x2+0 Putting f xl( ) equal to zero

( )

f xl =0

x

3 2₊0 _{=0 & x =0}

Now f xll( ) =6x

At x=0, fll(0) =6#0=0 Hence x=0 is the point of inflection. SOL 2.14 Option (A) is correct.

Given : x2₊y2₊z2 ₌₁

This is a equation of sphere with radius r=1

The unit normal vector at point , ,

2 1 2 1 0 c m is OA Hence OA i j (0 0)k 2 1 ₀ 2 1 ₀ =_c - _m +_c - _m + - i j 2 1 2 1 = +

SOL 2.15 Option (D) is correct.

First using the partial fraction : ( ) F s ( ) s s1 1 As sB1 = ₊ = _{+ +} ( ) ( ) s s A s Bs 1 1 = +₊+ ( ) s s1+1 ( ) ( ) ( ) s s A B s s sA 1 1 = +₊ + ₊

Comparing the coefficients both the sides,

(A+B) =0 and A =1, B =-1 So ( ) s s1+1 = - +s1 s 11 ( ) F t ₌L F s-1[ ( )] ( ) L 1 _{s s}1 ₁ L 1 _s1 _s 1₁ = - ₊ = - _{- +} ; E : D ₌L 1 _s1 _-L 1 _s 1 ₁ + - _{: D} - _{: D} 1 e t = - -SOL 2.16 Option (B) is correct.

Given A =>5_{1 3}3H

For finding eigen values, we write the characteristic equation as

A-lI =0 5 1 3 3 l l -- =0

& (5-l)(3-l)-3 =0 8 12 2

l - l+ =0 & l =2 6, Now from characteristic equation for eigen vector.

x A-lI 6 @" , =6 @0 For l = 2 X X 5 2 1 3 3 2 1 2 -> H> H => H0₀ & 1 X X 3 1 3 1 2 > H> H => H0₀ X1+X2 =0 & X1=-X2 So eigen vector _{= -}* 1₁4

Magnitude of eigen vector ₌ ( )1 2₊( )1 2 ₌ 2

Normalized eigen vector 2

1 2 1 = -R T S S S SS V X W W W WW SOL 2.17 Option (D) is correct.

Given : No. of Red balls =4 No. of Black ball =6

3 balls are selected randomly one after another, without replacement. 1 red and 2 black balls are will be selected as following

Manners Probability for these sequence

R B B 104 #96#85 = 61 B R B 106 #94#85 =16 B B R 106 #95#84 =16 Hence Total probability of selecting 1 red and 2 black ball is

P = + + =₆1 ₆1 ₆1 ₆3= ₂1 SOL 2.18 Option (A) is correct.

We have x dx d y xdxdy 4y 2 2 2 + - =0 ...(1)

Let x₌ez _{then z =log x}

dx dz x1 = So, we get _dxdy =b_dzdylb_dxdzl=_{x dz}1dy xdxdy =Dy where _dzd =D Again dx d y 2 2 dxd dxdy dxd x dz1dy = b l= b l = -_{x dz}21dy +_{x dz}1 d b_dzdyl_dxdz

GATE MCQ Mechanical Engineering (4-volumes)

x dz dy x dzd ydxdz 1 1 2 2 2 = - + x dz d y dz dy 1 2 2 2 = _c - _m dx x d y 2 2 2 (D2 D y) D D( 1)y = - =

-Now substitute in equation (i)

[ (D D-1)+D-4]y =0

(D2_-4)y _{=0 & D =!2}

So the required solution is y =C x1 2+C x2 -2 ...(ii)

From the given limits y 0( ) =0, equation (ii) gives 0 =C1#0+C2

C2 =0 And from y 1( )=1, equation (ii) gives

1 =C1+C2

C1 =1

Substitute C1&C2 in equation (ii), the required solution be

y ₌x2 SOL 2.19 Option (C) is correct.

For given equation matrix form is as follows

A 1 2 1 2 1 1 1 2 1 = -R T S S SS V X W W WW, B 4 5 1 = R T S S SS V X W W WW The augmented matrix is

: A B 8 B : : : 1 2 1 2 1 1 1 2 1 4 5 1 = -R T S S SS V X W W WW R2 R2 2R1, " - R₃"R₃-R₁ : : : 1 0 0 2 3 3 1 0 0 4 3 3 + -R T S S SS V X W W WW R3 R3 R2 " -: : : 1 0 0 2 3 0 1 0 0 4 3 0 + - -R T S S SS V X W W WW R2 R2/ 3 " -: : : 1 0 0 2 1 0 1 0 0 4 1 0 + R T S S SS V X W W WW

This gives rank of A, r( )A =2 and Rank of 8A : BB=r8A : BB=2 Which is less than the number of unknowns (3)

A

r6 @ =r8A : BB=2<3 Hence, this gives infinite No. of solutions. SOL 2.20 Option (B) is correct.

sin q ...

3 5 7

3 5 7

q q q q

= - + - +

SOL 2.21 Option (D) is correct.

Let y lim sin

0 q

q

= " q

( ) ( ) lim sin dd dd 0 q q = " q _q q _{lim cos} 1 0 q = "

q Applying L-Hospital rule

cos 10

= =1

SOL 2.22 Option (C) is correct Let a square matrix

A =>x_y y_xH

We know that the characteristic equation for the eigen values is given by

A-lI =0 x y y x l l -- =0 (x_-l)2_-y2 ₌₀ (x_-l)2 _=y2 x-l =!y & l=x!y

So, eigen values are real if matrix is real and symmetric. SOL 2.23 Option (A) is correct.

Let, z1=(1+i), z2=(2-5i)

z =z1#z2 =(1+i)(2-5i)

i i i

2 5 2 5 2

= - + - = -2 3i+5 = -7 3i i2_=-1 SOL 2.24 Option (D) is correct.

For a function, whose limits bounded between -a to a and a is a positive real number. The solution is given by

( ) f x dx a a

-#

2 ( ) ; ( ) 0 ; ( ) is even is odd f x dx f x f x a 0 =

_*

#

SOL 2.25 Option (C) is correct.

Let, f x( ) _{x dx}1

1 3 =

#

From this function we get a=1, b=3 and n= - =3 1 2 So, h = - = - =b _na 3₂ 1 1

We make the table from the given function y =f x( )= _x1 as follows :

x f x( )= =y _x1

x=1 y1= =₁1 1

x=2 y2= =₂1 0 5.

x=3 y3=1₃=0.333

Applying the Simpson’s 1 3/ rd _formula

x dx1

1 3

GATE MCQ Mechanical Engineering (4-volumes)

[1.33 2] . 3 1 ₃ 3 3 333 = + = =1.111

SOL 2.26 Option (D) is correct.

Given : _dxdy ₌(1₊y x2)

( y)

dy

1₊ 2 =xdx

Integrating both the sides, we get

y dy 1₊ 2

#

=

#

xdx tan y-1 x _c 2 2 = + & y tan x c 2 2 = b + l

SOL 2.27 Option (D) is correct.

The probability of getting head p= ₂1

And the probability of getting tail q= - =1 ₂1 ₂1 The probability of getting at least one head is

( 1) P x $ 1 5C p( ) ( )q 0 5 0 = - = -1 1#b₂1l5b₂1l0 1 2 1 5 = - = ₃₂31 SOL 2.28 Option (C) is correct.

Given system of equations are,

x x x

2 1+ 2+ 3 =0 ...(i)

x2-x3 =0 ...(ii)

x1+x2 =0 ...(iii)

Adding the equation (i) and (ii) we have

x x

2 1+2 2 =0

x1+x2 =0 ...(iv)

We see that the equation (iii) and (iv) is same and they will meet at infinite points. Hence this system of equations have infinite number of solutions.

SOL 2.29 Option (D) is correct.

The volume of a solid generated by revolution about x-axis bounded by the function f x( ) and limits between a to b is given by

V y dx a b ₂ p =

#

Given y = x and a=1, b =2 Therefore, V ( x dx)2 1 2 p =

#

xdx 1 2 p =

#

x₂2 1 2 p = : D =p:₂4-₂1D= 3₂p SOL 2.30 Option (B) is correct.

Given: d d f f d d f 2 3 3 2 2 h + h =0

Order is determined by the order of the highest derivation present in it. So, It is third order equation but it is a nonlinear equation because in linear equation, the product of f with d f d2 / _h2 _{is not allow.}

Therefore, it is a third order non-linear ordinary differential equation. SOL 2.31 Option (D) is correct.

Let I x dx 1 2 = + 3 3

-#

tan x1 = - 3₃ -6 @ [tan 1( ) tan 1( )] 3 3 = - + - - -2 2 p p _p = - -a k= tan-1(_-q)_=-tan-1( )_q SOL 2.32 Option (B) is correct.

Let, z _{= -1}3+₂4_ii

Divide and multiply z by the conjugate of (1-2i) to convert it in the form of

a+bi we have z = -₁3+₂4i_i #₁1₊+2₂i_i ( ) ( ) ( )( ) i i i 1 2 3 4 1 2 2 2 = -+ + i i i 1 4 3 10 8 2 2 = -+ + (i ) 1 4 3 10 8 = +_{- -} -i _i 5 5 10 _{1 2} = - + =- + z ₌ (_-1)2₊( )2 2 ₌ 5 _{a+ib = a}2_+b2

SOL 2.33 Option (C) is correct.

( ) y=f x 2 3 0 (2 3 ) if if if x x x x x 3 2 3 2 3 2 < > = -= - -Z [ \ ] ] ] ] ]]

Checking the continuity of the function. At x =2₃, Lf x( ) lim f ₃2 h h 0 = -" b l 2 lim2 _{3 3} h h 0 = - -" b l lim2 2 3h h 0 = - + " =0 and Rf x( ) lim f ₃2 h h 0 = + " b l lim3 3 h 2 ₂ h 0 = + -" b l lim2 3h 2 0 h 0 = + - = " Since Llimf x( ) h"0 =Rlimh"0f x( ) So, function is continuous 6 !x R

Now checking the differentiability : ( ) Lf xl limf h_h f h 0 3 2 3 2 = -_- -" ^ h ^ h lim2 3 _hh 0 h 0 3 2 = - _-- -" ^ h lim2 2_h 3h lim 3h_h 3 h 0 h 0 = - +_- = _- =-" " and Rf xl( ) limf h_h f h 0 3 2 3 2 = + -" ^ h ^ h lim3 h_h 2 0 lim2 3_hh 2 h 0 h 3 2 0 = + - - = + -" " ^ h 3 = Since _{Lf 3}lb l2 !_{Rf 3l}b l2 , f x( ) is not differentiable at x

3 2 = . SOL 2.34 Option (A) is correct.

GATE MCQ Mechanical Engineering (4-volumes)

And l1 and l2 are the eigen values of the matrix A.

The characteristic equation is written as

A-lI =0 2 1 2 3 1 0 0 1 l -> H > H ₌₀ 2 1 2 3 l l -- =0 ...(i) (2-l)(3-l)-2 =0 5 4 2 l - l+ =0 & l =1 & 4 Putting l =1 in equation (i),

x x 2 1 1 2 3 1 1 2 -> H H> 0 0 => H where x_x1 2 > H is eigen vector x x 1 1 2 2 1 2 > H H> => H0₀ x1+2x2 =0 or x1+2x2=0 Let x2 =K Then x1+2K =0 & x1=-2K

So, the eigen vector is

K K

2

-> H or >-2₁H

Since option A> H_-2₁ is in the same ratio of x1 and x2. Therefore option (A) is an eigen vector.

SOL 2.35 Option (A) is correct. ( )

f t is the inverse Laplace

So, f t( ) ( ) s s1 1 L 1 2 = + -; E ( ) s s2 1₊1 = A_s +_sB2 + +_sC ₁ ( ) ( ) ( ) s s As s B s Cs 1 1 1 2 2 = + + + + + ( ) ( ) ( ) s s s A C s A B B 1 2 2 = + + + + +

Compare the coefficients of s s2, _{and constant terms and we get}

A+C =0; A+B =0 and B =1

Solving above equation, we get A=-1, B=1 and C=1

Thus f t( ) _s s s 1 1 1 1 L 1 2 = - - + _{+ +} : D t e 1 t =- + + - _{= - +t 1 e}-t s 1a e L 1 at + = - _{: D}

-SOL 2.36 Option (C) is correct. The box contains : Number of washers = 2

Number of bolts = 4

Total objects = 2 + 3 + 4 = 9

First two washers are drawn from the box which contain 9 items. So the probability of drawing 2 washers is,

P1 ! ! 9! C C 7 2 1 9 2 2 2 = == ₉ 7 2₈! ! _{7! 9} 2 _{8 36}1 # # # = = = nC 1 n =

After this box contains only 7 objects and then 3 nuts drawn from it. So the probability of drawing 3 nuts from the remaining objects is,

P2 ! !! ! ! ! C C 4 37 1 7 64 35 4 351 7 3 3 3 # # # = = = =

After this box contain only 4 objects, probability of drawing 4 bolts from the box,

P3 _4CC ₁1 1

4 4

4

= = =

Therefore the required probability is,

P =P P P1 2 3 = ₃₆1 #₃₅1 #1=₁₂₆₀1 SOL 2.37 Option (B) is correct.

Given : h =60c- =0 60c

h =60#₁₈₀p = p₃ =1 047. radians From the table, we have

y0=0, y1=1066, y2=-323, y3=0, y4=323, y5=-355 and y6=0 From the Simpson’s 1/3rd rule the flywheel Energy is,

E =h y y₃6( 0+ 6)+4(y1+y3+y5)+2(y2+y4)@ Substitute the values, we get

E =1 047 0 0.₃ 6( + )+4 1066( + -0 355)+2(-323+323)@

. _{( )}

3

1 047 4 711 2 0#

= 6 + @ =993 Nm rad (Joules/cycle)

SOL 2.38 Option (A) is correct.

Given : M _x53 54

5 3 => H

And [ ]MT _{=[ ]M} -1

We know that when ₆A_@T ₌₆A_@-1

then it is called orthogonal matrix.

M T 6 @ MI =_{6 @} M T M 6 @ 6 @ =I

Substitute the values of M and MT_{, we get}

x x 5 3 5 4 5 3 5 3 5 4 5 3

.

>

H

> H =>1_{0 1}0H x x x 5 3 5 3 5 4 5 3 5 3 5 3 5 4 5 3 5 4 54 53 53 2 # # # # # + + + + b b b b b l l l l l R S S S S V W W W W 1 1 0 0 => H

GATE MCQ Mechanical Engineering (4-volumes)

x x x 1 259 2 25 12 5 3 25 12 5 3 + + + > H =>1_{0 1}0H

Comparing both sides a12 element,

x 25 12 5 3 + =0 "x 25 12 3 5 5 4 # =-

=-SOL 2.39 Option (C) is correct.

Let, V ₌3xzi₊2xyj_-yz2k

We know divergence vector field of V is given by (4 :V)

So, 4:V xi yj zk : 3xzi 2xyj yz2k 2 2 2 2 2 2 =_c + + m ^ + - h V 4: =3z+2x-2yz At point P 1 1 1( , , ) (4 :V)P 1 1 1( , , )=3#1+2#1-2#1#1=3 SOL 2.40 Option (C) is correct.

Let f s( ) s s 1 L 1 2₊ = - _{; E}

First, take the function

s s

1

2₊ and break it by the partial fraction, 1 s2₊s =_{s s(} 1₊₁₎= -_s1 _(s+1₁₎ ( ) Solve by s+1 1 = As + +sB1

*

4

So, s 1 s L 1 2₊ -c m =L-1;1_s -_(s+1₁₎E ₌L 1 1_{s -}L 1 _s 1 ₁ + - _{: D} - _{: D= -1 e}-t SOL 2.41 Option (D) is correct.

Total number of cases ₌23₌8

& Possible cases when coins are tossed simultaneously. H H H T H T T T H H T H T H T T H T H H T T H T

From these cases we can see that out of total 8 cases 7 cases contain at least one head. So, the probability of come at least one head is = ₈7

SOL 2.42 Option (C) is correct.

Given : z = +x iy is a analytic function

( )

f z =u x y( , )+iv x y( , )

u =xy ..(i)

Analytic function satisfies the Cauchy-Riemann equation.

x u 2 2 y v 2 2 = and u_{y x}v 2 2 2 2 =-So from equation (i),

x u 2 2 _{=y &} y v _y 2 2 = y u 2 2 _{=x &} x v _x 2 2 =-Let v x y( , ) be the conjugate function of u x y( , )

dv x v dx y v dy 2 2 2 2 = + = -( x dx) +( )y dy

Integrating both the sides,

dv

#

=-

#

xdx+

#

ydy

v =-x₂2 +y₂2+k _{= 2}1(y2_-x2)₊k

SOL 2.43 Option (A) is correct.

Given _xdxdy +y ₌x4

dx dy

x y1

+b l ₌x3 _...(i)

It is a single order differential equation. Compare this with _dxdy +Py=Q and we get

P =_x1 Q₌x3 Its solution will be

( . .)

y I F =

#

Q I F dx( . .) +C

. .

I F ₌e#Pdx₌e #_{x dx}1 =_elog xe =_x

Complete solution is given by,

yx x3 xdx C # =

#

+ ₌

#

x dx4 ₊C x _C 5 5 = + ...(ii) and y 1( )= ₅6 at x=1 & y 5 6

= From equation (ii),

5 6 ₁ # = +₅1 C & C 5 6 5 1 ₁ = - = Then, from equation (ii), we get

yx =x₅5+1 & y x

x

5 1 4

= +

SOL 2.44 Option (B) is correct.

The equation of circle with unit radius and centre at origin is given by,

x2₊y2₌1

Finding the integration of (x₊y)2 _{on path AB traversed in counter-clockwise} sense So using the polar form