# GATE CE 2018 SET-1

 Question 1
Which one of the following matrices is singular?
 A $\begin{bmatrix} 2 &5 \\ 1 & 3 \end{bmatrix}$ B $\begin{bmatrix} 3 &2 \\ 2 & 3 \end{bmatrix}$ C $\begin{bmatrix} 2 &4 \\ 3 &6 \end{bmatrix}$ D $\begin{bmatrix} 4 &3 \\ 6 &2 \end{bmatrix}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Option (A): $\left | A \right |=6-5=1$
Option (B): $\left | A \right |=9-4=5$
Option (C): $\left | A \right |=12-12=0$
Option (D): $\left | A \right |=8-18=-10$
Hence matrix (C) is singular.
 Question 2
For the given orthogonal matrix Q,
$Q=\begin{bmatrix} 3/7 &2/7 &6/7\\ -6/7 &3/7 &2/7\\ 2/7 &6/7 &-3/7 \end{bmatrix}$
The inverse is
 A $\begin{bmatrix} 3/7 &2/7 &6/7\\ -6/7 &3/7 &2/7\\ 2/7 &6/7 &-3/7 \end{bmatrix}$ B $\begin{bmatrix} -3/7 &-2/7 &-6/7\\ 6/7 &-3/7 &-2/7\\ -2/7 &-6/7 &3/7 \end{bmatrix}$ C $\begin{bmatrix} 3/7 &-6/7 &2/7\\ 2/7 &3/7 &6/7\\ 6/7 &2/7 &-3/7 \end{bmatrix}$ D $\begin{bmatrix} -3/7 &6/7 &-2/7\\ -6/7 &-3/7 &-6/7\\ -2/7 &-2/7 &3/7 \end{bmatrix}$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} \left | Q \right |&=\frac{3}{7}\left ( -\frac{9}{49}-\frac{12}{49} \right )-\frac{2}{7}\left ( \frac{18}{49}-\frac{4}{49} \right )+\frac{6}{7}\left ( \frac{-36}{49}-\frac{6}{49} \right ) \\ &=-1 \\ Adj. \; Q&=\begin{bmatrix} -\frac{21}{49} & \frac{42}{49} &-\frac{14}{49} \\ -\frac{14}{49}& -\frac{21}{49} & -\frac{42}{49}\\ -\frac{42}{49} & -\frac{14}{49} & \frac{21}{42} \end{bmatrix} \\ \therefore\;\; Q^{-1}&=\frac{Adj\: Q}{\left | Q \right |}=\begin{bmatrix} \frac{3}{7} & -\frac{6}{7} & \frac{2}{7}\\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7}\\ \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \end{bmatrix} \end{aligned}
Or $\because$ Q is orthogonal
$\therefore \;\; Q^{-1}=Q^{T}$
 Question 3
At the point x= 0, the function $f(x)=x^{3}$ has
 A local maximum B local minimum C both local maximum and minimum D neither local maximum nor local minimum
Engineering Mathematics   Calculus
Question 3 Explanation:
$f\left ( x \right )=x^{3}$ at $x=0$

At $x=0$, the function $y=x^{3}$ has neither minima nor maxima.
 Question 4
A column of height h with a rectangular cross-section of size ax2a has a buckling load of P. If the cross-section is changed to 0.5a x 3a and its height changed to 1.5h, the buckling load of the redesigned column will be
 A P/12 B P/4 C P/2 D 3P/4
RCC Structures   Footing, Columns, Beams and Slabs
Question 4 Explanation:
\begin{aligned} \text { For column, } & P=\frac{\pi^{2} E I_{\min }}{L^{2}} \\ &=\frac{\pi^{2} E\left(\frac{2 a \times a^{3}}{12}\right)}{h^{2}}=\frac{\pi^{2} E a^{4}}{6 h^{2}}\\ \text{For new column, }P&=\frac{\pi^{2} E\left[\frac{3 a \times(0.5 a)^{3}}{12}\right]}{(1.5 h)^{2}} \\ &=\frac{1}{12} \times \frac{\pi^{2} E a^{4}}{6 h^{2}}=\frac{P}{12} \end{aligned}
 Question 5
A steel column of ISHB 350 @72.4 kg/m is subjected to a factored axial compressive load of 2000 kN. The load is transferred to a concrete pedestal of grade M20 through a square base plate. Consider bearing strength of concrete as 0.45$f_{ck}$, where $f_{ck}$ is the characteristic strength of concrete. Using limit state method and neglecting the self weight of base plate and steel column, the length of a side of the base plate to be provided is
 A 39 cm B 42 cm C 45 cm D 48 cm
Design of Steel Structures   Beams
Question 5 Explanation:
\begin{aligned} &\text{Area required for base plate}\\ &=\frac{\text{Factored load}}{\text{Bearing capacity of concrete}}\\ &=\frac{2000\times 10^{3}}{0.45\times 20}=222222.222mm^{2} \end{aligned}
So, side of base plate $=\sqrt{\text{Area}}=471.4 mm= 47.14 cm$
Since, provided area must be more than required
So, answer should be 48 cm.
 Question 6
The Le Chatelier apparatus is used to determine
 A compressive strength of cement B fineness of cement C setting time of cement D soundness of cement
Construction Materials and Management
Question 6 Explanation:
Le Chatelier Apparatus is used to detemine the soundness of cement as per IS code4031 (part 3); this cement testing procedure is called Le Chatelier test for determining the unsoundness properties of cement due to presence of "free lime".
 Question 7
 A creep B hydration C segregation D shrinkage
RCC Structures   Working Stress and Limit State Method
Question 7 Explanation:
 Question 8
A solid circular beam with radius of 0.25 m and length of 2 m is subjected to a twisting moment of 20 kNm about the z-axis at the free end, which is the only load acting as shown in the figure. The shear stress component $\tau_{xy}$ at Point 'M' in the cross-section of the beam at a distance of 1 m from the fixed end is
 A 0.0 MPa B 0.51 MPa C 0.815 MPa D 2.0 MPa
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 8 Explanation:

The only non-zero stresses are $\tau_{\theta z}=\tau_{z \theta}=\tau$
if $\theta$ is $90^{\circ}$ then $\theta=y$
\begin{aligned} \text{Hence}\quad\tau_{z y} &=\tau_{y z}=\tau_{\max } \\ &=16 T / \pi \mathrm{d}^{3}=0.815 \mathrm{MPa} \end{aligned}
But in rest of the planes shear stresses are zero, hence, $\tau_{x y}=\tau_{y x}=0$
 Question 9
Two rectangular under-reinforced concrete beam sections X and Y are similar in all aspects except that the longitudinal compression reinforcement in section Y is 10% more. Which one of the following is the correct statement?
 A Section X has less flexural strength and is less ductile than section Y B Section X has less flexural strength but is more ductile than section Y C Sections X and Y have equal flexural strength but different ductility D Sections X and Y have equal flexural strength and ductility
RCC Structures   Footing, Columns, Beams and Slabs
Question 9 Explanation:

Due to presence of more compression steel in section Y, NA of section of Y is above than as of X. It means Y is more under-reinforced than X so ductility of Y is more.
Since compression steel of Y is more so flexure resistance of X is less than as of Y.
 Question 10
The percent reduction in the bearing capacity of a strip footing resting on sand under flooding condition (water level at the base of the footing) when compared to the situation where the water level is at a depth much greater than the width of footing, is approximately
 A 0 B 25 C 50 D 100
Geotechnical Engineering   Shallow Foundation and Bearing Capacity
Question 10 Explanation:
For strip footing on sand (c=0)
$q_{u}=\gamma D_{f} N_{q}+0.5 \mathrm{B} \gamma N_{\gamma}$
In flooding condition water level rises to base of footing hence IIIrd term unit weight of soil will change and IInd term unit weight will be unaffected.
$\begin{array}{ll} \therefore & q_{u}=\gamma D_{f} N_{q}+0.5 B \gamma N_{\gamma} \\ \because & \gamma \simeq \frac{1}{2} \gamma_{s a t} \end{array}$
Hence third term reduced and second term will be same thereby percentage reduction will not be 50%.
According to option approach answer should be 25%
Note: If water table rises to ground level then both $\gamma$ will reduce to $\gamma$. Hence, percentage reduction would be approximately 50%.
There are 10 questions to complete.