# GATE CE 2019 SET-1

 Question 1
Which one of the following is correct?
 A $\lim_{x\rightarrow 0}\left ( \frac{sin4x}{sin 2x} \right )=2$ and $\lim_{x\rightarrow 0}\left ( \frac{tanx}{x} \right )=1$ B $\lim_{x\rightarrow 0}\left ( \frac{sin4x}{sin 2x} \right )=1$ and $\lim_{x\rightarrow 0}\left ( \frac{tanx}{x} \right )=1$ C $\lim_{x\rightarrow 0}\left ( \frac{sin4x}{sin 2x} \right )= \infty$ and $\lim_{x\rightarrow 0}\left ( \frac{tanx}{x} \right )=1$ D $\lim_{x\rightarrow 0}\left ( \frac{sin4x}{sin 2x} \right )=2$ and $\lim_{x\rightarrow 0}\left ( \frac{tanx}{x} \right )= \infty$
Engineering Mathematics   Calculus
Question 1 Explanation:
$\lim_{x \to 0}\left ( \frac{\sin 4x}{\sin 2x} \right )=\lim_{x \to 0}\left ( \frac{\frac{\sin 4x}{x}}{\frac{\sin 2x}{x}} \right )=\frac{4}{2}=2$ and $\lim_{x \to 0}\left ( \frac{\tan x}{x} \right )=1$
 Question 2
Consider a two-dimensional flow through isotropic soil along x direction and z direction. If h is the hydraulic head, the Laplace's equation of continuity is expressed as
 A $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial z}=0$ B $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial x}\frac{\partial h}{\partial z}+\frac{\partial h}{\partial z}=0$ C $\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial z^2}=0$ D $\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial x \partial z}+\frac{\partial^2 h}{\partial z^2}=0$
Engineering Mathematics   Partial Differential Equation
Question 2 Explanation:
The Laplace's equation of continuity for two dimensional flow in a soil is expressed as:
$k_x\frac{\partial^2 h}{\partial x^2}+k_z\frac{\partial^2 h}{\partial z^2}=0$... for anisotropic soil $[k_x\neq k_z]$
:
$\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial z^2}=0$... for isotropic soil $[k_x = k_z]$

 Question 3
A simple mass-spring oscillatory system consists of a mass m, suspended from a spring of stiffness k. Considering z as the displacementof the system at any time t, the equation of motion for the free vibration of the system is $m\ddot{z}+kz=0$. The natural frequency of the system is
 A $\frac{k}{m}$ B $\sqrt{\frac{k}{m}}$ C $\frac{m}{k}$ D $\sqrt{\frac{m}{k}}$
Engineering Mechanics
Question 3 Explanation:
\begin{aligned} m\ddot{z}+kz&=0 \\ \ddot{z}+\frac{k}{m}z&=0 \\ \text{Comparing with} \\ \ddot{z}+\omega _n^2 z&=0 \\ \text{We get}\;\; \omega _n&=\sqrt{\frac{k}{m}} \end{aligned}
 Question 4
For a small value of h, the Taylor series expansion for $f(x+h)$ is
 A $f(x)+hf'(x)+\frac{h^2}{2!}f''(x)+\frac{h^3}{3!}f'''(x)+...\infty$ B $f(x)-hf'(x)+\frac{h^2}{2!}f''(x)-\frac{h^3}{3!}f'''(x)+...\infty$ C $f(x)+hf'(x)+\frac{h^2}{2}f''(x)+\frac{h^3}{3}f'''(x)+...\infty$ D $f(x)-hf'(x)+\frac{h^2}{2}f''(x)-\frac{h^3}{3}f'''(x)+...\infty$
Engineering Mathematics   Calculus
Question 4 Explanation:
We know that Taylor series for small h of f(x + h) is,
$f(x+h)=f(x)+hf'(x)+\frac{h^2}{2!}f''(x)+\frac{h^3}{3!}f'''(x)+...$
 Question 5
A plane truss is shown in the figure

Which one of the options contains ONLY zero force members in the truss?
 A FG, FI, HI, RS B FG, FH, HI, RS C FI, HI, PR, RS D FI, FG, RS, PR
Structural Analysis   Trusses
Question 5 Explanation:
So zero force members are FI, FG, RS, PR

There are 5 questions to complete.