# 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,
$P=\frac{\pi^{2} E I}{L^{2}}$

There are 5 questions to complete.