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.

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