# Mechanical Engineering - GATE 2017 Paper Solution

Question No. 1

The product of eigenvalues of matrix P is

$\mathrm P=\begin{bmatrix}2&0&1\\4&-3&3\\0&2&-1\end{bmatrix}$

Question No. 2

The value of $\lim\nolimits_{x\rightarrow0}\frac{x^3-\sin(x)}x\mathrm{is}$

Question No. 3

Consider the following partial differential equation for $\style{font-family:'Times New Roman'}{u(x\mathit,y)}$ with the constant >1:

$\style{font-family:'Times New Roman'}{\frac{\partial u}{\partial y}+C\frac{\partial u}{\partial x}=0}$

Solution of this equation is

Question No. 4

The differential equation $\style{font-family:'Times New Roman'}{\frac{d^2y}{dx^2}+16y=0}$ for $\style{font-family:'Times New Roman'}{y\left(x\right)}$ with the two boundary conditions $\style{font-family:'Times New Roman'}{{\left.\frac{dy}{dx}\right|}_{x-0}=1\;\mathrm{and}\;{\left.\frac{dy}{dx}\right|}_{x-\frac{\mathrm\pi}2}=-1}$ has

Question No. 5

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is ______________

Question No. 6

For steady flow of a viscos incompressible fluid through a circular pipe of constant diameter, the averavge velocity in the fully developed region is constant. Which one of the following statements about the average velocity in the developing region is TRUE?

Question No. 7

Consider the two-dimensional velocity filed given by $\style{font-family:'Times New Roman'}{\overrightarrow V=(5+a_1x+b_1y)\widehat i+(4+a_2x+b_2y)\widehat j,}$ where a1, b1, a2 and b2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

Question No. 8

Water (density = 1000 kg/m3) at ambient temperature flows through a horizontal pipe of uniform cross section at the rate of 1 kg/s. If the pressure drop across the pipe is 100 kPa, the minimum power required to pump the water across the pipe, in watts, is __________

Question No. 9

Which one of the following is NOT a rotating machine?

Question No. 10

Saturated steam at $\style{font-family:'Times New Roman'}{100^\circ\mathrm C}$ condenses on the outside of a tube. Cold fluid enters the tube at $\style{font-family:'Times New Roman'}{20^\circ\mathrm C}$ and exits at $\style{font-family:'Times New Roman'}{50^\circ\mathrm C}$. The value of the Log Mean Temperature Difference (LMTD) is _____________$°\mathrm{C}$.

Question No. 11

The molar specific heat at constant volume of an ideal gas is equal to 2.5 times the universal gas constant (8.314 J/mol K). When the temperature increases by 100 K, the change in molar specific enthalpy is ____________ J/mol.

Question No. 12

A heat pump absorbs 10 kW of heat from outside environment at 250 K While absorbing 15 kW of work. It delivers the heat to a room that must be kept warm at 300 K. The Cofficient of Performance (COP) of the heat pump is ______________

Question No. 13

The Poisson's ratio for a perfectly incompressible linear elastic  material is

Question No. 14

A partical of unit mass is moving on a plane. Its trajectory, in polar coordinates, is given by $\style{font-family:'Times New Roman'}{r(t)=t^2,\theta(t)\;=t}$, where t is time. The kinetic energy of the partical at time t=2 is

Question No. 15

A motor driving a solid circulr steel shaft transmits 40 kW of power at 500 rpm. If the diameter of the shaft is 400 mm, the maximum shear stress in the shaft is _________ MPa.

Question No. 16

Consider a beam with circular cross-section of diameter d. The ratio of the second moment of area about the natural axis to the section moduls of the area is

Question No. 17

The following figure shows the velocity-time plot for a particle traveling along a staright line. The distance coverd by the particle from t = 0 to t = 5 s is ____________ m,

Question No. 18

The damping ratio for a viscously damped spring mass system, governed by the relationship $\style{font-family:'Times New Roman'}{m\frac{\mathrm d^2x}{\mathrm dt^2}+c\frac{\mathrm dx}{\mathrm dt}+kx=F(t)}$, is given by

Cylindrical pins of diameter $\style{font-family:'Times New Roman'}{15^{\pm0.020}\mathrm{mm}}$ are being produced on a machine. Statistical quality control tests show a mean of 14.995 mm and standard deviation of 0.004 mm. The process capability index Cp is