# GATE ME 2018 SET-2

 Question 1
The Fourier cosine series for an even function $f(x)$ is given by
$f(x)=a_{0}+\sum_{n=1}^{\infty}a_{n}\cos(nx)$
The value of the coefficient $a_{2}$ for the function $f(x)=\cos ^{2}(x) in \left [ 0 , \pi \right ]$ is
 A -0.5 B 0 C 0.5 D 1
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} \cos ^{2} x &=\frac{1+\cos 2 x}{2} \\ f(x) &=\frac{1}{2}+\frac{\cos 2 x}{2} \\ f(x) &=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cdot \cos n x \\ a_{0} &=1 \\ a_{1} &=0 \\ a_{2} &=\frac{1}{2} \end{aligned}
 Question 2
The divergence of the vector field $\overrightarrow{u}=e^{x}\left ( \cos y\hat{i} + \sin y\hat{j} \right)$ is
 A 0 B $e^{x}\cos y + e^{x}\sin y$ C $2e^{x}\cos y$ D $2e^{x}\sin y$
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} \vec{u} &=e^{x} \cos y \hat{i}+e^{x} \cdot \sin y \hat{j} \\ \nabla \cdot \vec{u} &=\frac{\partial}{\partial x}\left(u_{1}\right)+\frac{\partial}{\partial y}\left(u_{2}\right) \\ &=\frac{\partial}{\partial x}\left(e^{x} \cdot \cos y\right)+\frac{\partial}{\partial y}\left(e^{x} \cdot \sin y\right) \\ &=e^{x} \cos y+e^{x} \cos y \\ \nabla \cdot \vec{u} &=2 e^{x} \cdot \cos y \end{aligned}
 Question 3
Consider a function u which depends on position x and time t. The partial differential equation
$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$
is known as the
 A Wave equation B Heat equation C Laplace's equation D Elasticity equation
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$is known as heat equation
 Question 4
If y is the solution of the differential equation
$y^{3}\frac{d y}{dx}+x^{3}=0,y(0)=1,$ the value of $y\left ( -1 \right )$ is
 A -2 B -1 C 0 D 1
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} y^{3} \frac{d y}{d x} &=-x^{3} \\ y^{3} d y &=-x^{3} d x \\ \int y^{3} d y &=-\int x^{3} d x \\ \frac{y^{4}}{4} &=\frac{-x^{4}}{4}+C \\ \frac{x^{4}+y^{4}}{4} &=C \\ y(0) &=1 \\ \frac{0+1}{4} &=C \\ C &=\frac{1}{4} \\ x^{4}+y^{4} &=1 \\ y^{4} &=1-x^{4} \\ y &=\sqrt[4]{1-x^{4}} \\ \text{When,}\quad x &=-1 \\ y &=0 \end{aligned}
 Question 5
The minimum axial compressive load, P, required to initiate buckling for a pinned-pinned slender column with bending stiffness EI and length L is
 A $P=\frac{\pi ^{2}EI}{4L^{2}}$ B $P=\frac{\pi ^{2}EI}{L^{2}}$ C $P=\frac{3\pi ^{2}EI}{4L^{2}}$ D $P=\frac{4\pi ^{2}EI}{L^{2}}$
Strength of Materials   Bending of Beams
Question 5 Explanation:
For both ends hinged buckling load,
$P=\frac{\pi^{2} E I}{L^{2}}$
 Question 6
A frictionless gear train is shown in the figure. The leftmost 12-teeth gear is given a torque of 100 N-m. The output torque from the 60-teeth gear on the right in N-m is
 A 5 B 20 C 500 D 2000
Theory of Machine   Gear and Gear Train
Question 6 Explanation:

\begin{aligned} \tau_{1}&=100 \mathrm{Nm}\\ \text{Let speed of 1 is }N_{1} \\ (1,2):\qquad N_{2}&=N_{1} \times \frac{T_{1}}{T_{2}}=N_{1} \times \frac{12}{48}=\frac{N_{1}}{4} \\ N_{3}=N_{2}&=\frac{N_{1}}{4}\\ (3,4) \qquad N_{4}&=N_{3} \times \frac{T_{3}}{T_{4}}=\frac{N_{1}}{4} \times \frac{12}{60} \\ N_{4}&=\frac{N_{1}}{20} \end{aligned}
By Power conservation
(Assume $\eta$ (efficiency) =1)
\begin{aligned} \tau_{1} \times N_{1} &=\tau_{4} \times N_{4} \\ 100 \times N_{1} &=\tau_{4} \times \frac{N_{1}}{20} \\ \tau_{4} &=2000 \mathrm{N}-\mathrm{m} \end{aligned}
 Question 7
In a single degree of freedom underdamped spring-mass-damper system as shown in the figure, an additional damper is added in parallel such that the system still remains underdamped. Which one of the following statements is ALWAYS true?
 A Transmissibility will increase. B Transmissibility will decrease. C Time period of free oscillations will increase. D Time period of free oscillations will decrease.
Theory of Machine   Vibration
Question 7 Explanation:
$\text { Tansmissibility, } \quad \in=\frac{\sqrt{1+\left(\frac{2 \xi \omega}{\omega_{n}}\right)^{2}}}{\sqrt{\left\{1-\left(\frac{\omega}{\omega_{n}}\right)^{2}\right\}^{2}+\left\{\frac{2 \xi \omega}{\omega_{n}}\right\}^{2}}}$
Then damping will increase.
But it is still under damped.
But $\xi$ will increase.
Here no unbalance force is there.
But $\omega_{d}=\sqrt{1-\xi^{2}} \cdot \omega_{n} \quad\left\{\omega_{n}=\sqrt{\frac{K}{M}}\right\}$
But as $\xi$ increases
then $\xi^{2}$ will increase
Then $\omega_{d}$ will decrease
Then $T_{d}=\frac{2 \pi}{\omega_{d}} \text { will increase }$
 Question 8
Pre-tensioning of a bolted joint is used to
 A strain harden the bolt head B decrease stiffness of the bolted joint C increase stiffness of the bolted joint D prevent yielding of the thread root
Machine Design   Bolted, Riveted and Welded Joint
Question 8 Explanation:
Pretension increase stiffness of system..
 Question 9
The peak wavelength of radiation emitted by a black body at a temperature of 2000 K is 1.45 $\mu$m. If the peak wavelength of emitted radiation changes to 2.90 $\mu$m, then the temperature (in K) of the black body is
 A 500 B 1000 C 4000 D 8000
\begin{aligned} \lambda_{M} T &=\text { constant } \\ \lambda_{M 1} T_{1} &=\lambda_{M 2} T_{2} \\ 1.45 \times 2000 &=\lambda_{M 2} T_{2}=2.90 \times T_{2} \\ \therefore \quad T_{2} &=\left(\frac{1.45}{2.90} \times 2000\right)=1000 \mathrm{K} \end{aligned}
For an ideal gas with constant properties undergoing a quasi-static process, which one of the following represents the change of entropy $(\Delta s)$ from state 1 to 2?
 A $\Delta s=C_{p}ln\left ( \frac{T_{2}}{T_{1}} \right )-R ln\left ( \frac{p_{2}}{p_{1}} \right )$ B $\Delta s=C_{v}ln\left ( \frac{T_{2}}{T_{1}} \right )-C_{p} ln\left ( \frac{v_{2}}{v_{1}} \right )$ C $\Delta s=C_{p}ln\left ( \frac{T_{2}}{T_{1}} \right )-C_{v} ln\left ( \frac{p_{2}}{p_{1}} \right )$ D $\Delta s=C_{v}ln\left ( \frac{T_{2}}{T_{1}} \right )+R ln\left ( \frac{v_{1}}{v_{2}} \right )$
$\begin{array}{l}\mathrm{Tds}=\mathrm{dH}-\mathrm{VdP}\\ \mathrm{dS}=\frac{\mathrm{dH}}{\mathrm T}-\frac{\mathrm V}{\mathrm T}\mathrm{dP}\\ \mathrm{For}\;\mathrm{an}\;\mathrm{ideal}\;\mathrm{gas}\\ \mathrm{PV}=\mathrm{RT}\\ \frac{\mathrm V}{\mathrm T}=\frac{\mathrm R}{\mathrm P}\\ \mathrm{dS}={\mathrm C}_{\mathrm P} \frac{\mathrm{dT}}{\mathrm T}-\frac{\mathrm R}{\mathrm p}\mathrm{dP}\\ {\mathrm S}_2-{\mathrm S}_1={\mathrm C}_\mathrm P\ln\frac{{\mathrm T}_2}{{\mathrm T}_1}-\mathrm{Rln}\frac{{\mathrm P}_2}{{\mathrm P}_1}\\\\\end{array}$