# 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
 Question 6
An element is subjected to biaxial normal tensile strains of 0.0030 and 0.0020. The normal strain in the plane of maximum shear strain is
 A Zero B 0.001 C 0.0025 D 0.005
Solid Mechanics   Properties of Metals, Stress and Strain
Question 6 Explanation:
$\varepsilon _x=0.0030$
$\varepsilon _y=0.0020$
Normal strain in the plane of maximum shear strain
$\varepsilon _{avg}=\frac{\varepsilon _x+\varepsilon _y}{2}=\frac{0.0030+0.0020}{2}=0.0025$
 Question 7
Consider the pin-jointed plane truss shown in the figure. Let $R_P,R_Q$ and $R_R$ denote the vertical reactions (upward positive) applied by the supports at P, Q, and R, respectively, on the truss. The correct combination of ($R_P,R_Q,R_R$)is represented by
 A (30, -30, 30) kN B (20, 0, 10) kN C (10, 30, -10) kN D (0, 60, -30) kN
Structural Analysis   Trusses
Question 7 Explanation:

Adopting method of sections and taking LHS of the section
\begin{aligned} &\Sigma F_y= 0\\ &R_P=30kN \\ &\text{For complete truss,} \\ &\Sigma M_R =0 \\ &9R_P-30 \times 6 -R_Q \times 3=0 \\ &R_Q=30kN(\downarrow ) \\ &\text{Taking RHS of section,} \\ &\Sigma F_y=0\Rightarrow R_R=-R_Q \\ &\text{Thus},\;\; R_Q= 30kN(\downarrow )\\ &R_R=30kN(\uparrow ) \end{aligned}
 Question 8
Assuming that there is no possibility of shear buckling in the web, the maximum reduction permitted by IS 800-2007 in the (low-shear) design bending strength of a semi-compact steel section due to high shear is
 A Zero B 25% C 50% D governed by the area of the flange
Design of Steel Structures   Beams
Question 8 Explanation:
As per IS 800 : 2007
For semi compact section
(i) In low shear case ($V \leq 0.6 V_d$)
$M_d = Z_ef_y/\gamma _{mo}$
(ii) In high shear case ($V \gt 0.6 V_d$)
$M_d = Z_ef_y/\gamma _{mo}$
So reduction is zero.
 Question 9
In the reinforced beam section shown in the figure, the nominal cover provided at the bottom of the beam as per IS 456-2000, is
 A 30 mm B 36 mm C 42 mm D 50 mm
RCC Structures   Working Stress and Limit State Method
Question 9 Explanation:
Nominal cover = Effective cover $-\frac{\phi _m}{2}-\phi _{st}$
$=50-\frac{16}{2}-12=30mm$
Nominal cover is the distance from extreme concrete fbre to the surface of stirrup.
 Question 10
The interior angles of four triangles are given below:

Which of the triangles are ill-conditioned and should be avoided in Triangulation surveys?
 A Both P and R B Both Q and R C Both P and S D Both Q and S
Geomatics Engineering   Theodolites, Compass and Traverse Surveying
Question 10 Explanation:
For an ill conditioned traingle in triangulation survey, any angle can be less than $38^{\circ}$, and can be greater than $38^{\circ}$.
For traingles Q and S, the above condition is valid.
There are 10 questions to complete.