Mechanical Engineering Gate Yearwise

Mechanical Engineering GATE 2013 Set-3 Questions with Answer

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Ques 1 CAD/CAM

In a CAD package, mirror image of a 2D point P(5,10) is to be obtained about a line which passes through the origin and makes an angle of 45° counterclockwise with the X-axis. The coordinates of the transformed point will be

(7.5, 5)

(10, 5)

(7.5, -5)

(10, -5)

Ques 2 Engineering Mathematics

Choose the CORRECT set of functions, which are linearly dependent.

sin x, sin²x and cos²x

cos x, sin x and tan x

cos 2x, sin²x and cos²x

cos 2x, sin x and cos x

Ques 3 Engineering Mathematics

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

0.5

greater than zero and less than 0.5

greater than 0.5 and less than 1.0

1.0

Ques 4 Engineering Mathematics

The function f(t) satisfies the differential equation d²f/dt² + f = 0 and the auxiliary conditions, f(0)=0, df/dt(0)=4. The Laplace transform of f(t) is given by

2/(s+1)

4/(s+1)

4/(s²+1)

2/(s⁴+1)

Ques 5 Engineering Mathematics

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.
∫∫_S (1/4)(F.n)dA
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

2π

3π/4

4π

Ques 6 Engineering Mathematics

The value of the definite integral ∫₁^e √x ln(x) dx is

(4/9)√e³ + (2/9)

(2/9)√e³ - (4/9)

(2/9)√e³ + (4/9)

(4/9)√e³ - (2/9)

Ques 7 Engineering Mathematics

The solution to the differential equation d²u/dx² - k(du/dx) = 0 where k is a constant, subjected to the boundary conditions u(0)=0 and u(L)=U, is

u = U(x/L)

u = U((1-e^kx)/(1-e^kL))

u = U((1-e^-kx)/(1-e^-kL))

u = U((1+e^kx)/(1+e^kL))

Ques 8 Engineering Mathematics

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 probability of the guessed answer being correct is 1/4. Given that the student has answered the question correctly, the conditional probability that the student knows the correct answer is

2/3

3/4

5/6

8/9

Ques 9 Engineering Mathematics

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

P-2, Q-1, R-3

P-3, Q-2, R-1

P-1, Q-2, R-3

P-3, Q-1, R-2

Ques 10 Engineering Mathematics

The eigenvalues of a symmetric matrix are all

complex with non-zero positive imaginary part.

complex with non-zero negative imaginary part.

real.

pure imaginary.

Ques 11 Engineering Mathematics

The partial differential equation ∂u/∂t + u(∂u/∂x) = ∂²u/∂x² is a

linear equation of order 2

non-linear equation of order 1

linear equation of order 1

non-linear equation of order 2

Ques 12 Engineering Mechanics

A pin jointed uniform rigid rod of weight W and length L is supported horizontally by an external force F as shown in the figure below. The force F is suddenly removed. At the instant of force removal, the magnitude of vertical reaction developed at the support is

zero

W/4

W/2

Ques 13 Fluid Mechanics

In order to have maximum power from a Pelton turbine, the bucket speed must be

equal to the jet speed.

equal to half of the jet speed.

equal to twice the jet speed.

independent of the jet speed.

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