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.

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