# GATE CE 2012

 Question 1
The estimate of $\int_{0.5}^{0.5}\frac{dx}{x}$ obtained using Simpson's rule with three-point function evaluation exceeds the exact value by
 A 0.235 B 0.068 C 0.024 D 0.012
Engineering Mathematics   Numerical Methods
Question 1 Explanation:
Exact value of $\int_{0.5}^{1.5}\frac{dx}{x}=\left [ \log x \right ]_{0.5}^{1.5} = \log (1.5)- \log(0.5) =1.0986$
Approximate value by Simpson's rule with 3 point is,
\begin{aligned} I&=\frac{h}{3}\left ( f\left ( 0 \right )+4f\left ( 1 \right )+f\left ( 2 \right ) \right ) \\ n_{i}&=n_{pt}-1=3-1=2 \end{aligned}
($n_{pt}$ is the number of pts and $n_{i}$ is the number of intervals)
Here $h=\frac{b-a}{n_{i}}=\frac{1.5-0.5}{2}=0.5$
The table is

$I=\frac{0.5}{3}\left ( \frac{1}{0.5}+4 \times 1+\frac{1}{1.5}\right ) = 1.1111$
So the estimate exceeds the exact value by,
Approximate value-Exact value
$=0.012499 \approx 0.012499$
 Question 2
The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is
 A $\lt$50% B 50% C 75% D 100%
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
The annual precipitation is normally distributed with $\mu =1000\: mm$ and $\sigma =200\: mm$
\begin{aligned} p\left ( x\gt 1200 \right ) &=p\left ( z\gt \frac{1200-1000}{200} \right ) \\ &=p\left ( z\gt 1 \right ) \end{aligned}
Where z is the standard normal variate.
In normal distribution
Now, since
$p\left ( -1\lt z\lt 1 \right )\approx 0.68$
($\simeq 68%$ of data is within one standard deviation of mean)
\begin{aligned} p\left ( 0\lt z\lt 1 \right ) &=\frac{0.68}{2}&=0.34 \\ \text{So} p\left ( z\gt 1 \right ) &=0.5-0.34 \\ &=0.16\simeq 16\% \end{aligned}
which is $\lt 50\%$
 Question 3
The infinite series $1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$ corresponds to
 A sec x B $e^{x}$ C cos x D $1+sin^{2}x$
Engineering Mathematics   Partial Differential Equation
Question 3 Explanation:
$e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}...$
(By McLaurin's series expansion)
 Question 4
The Poisson's ratio is defined as
 A $|\frac{\text{Axial tress}}{\text{Lateral stress}}|$ B $|\frac{\text{Lateral strain}}{\text{Axial strain}}|$ C $|\frac{\text{Lateral stress}}{\text{Axial stress}}|$ D $|\frac{\text{Axial strain}}{\text{Lateral strain}}|$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 4 Explanation:
$\mu =\left | \frac{\text{Lateral strain}}{\text{Axial strain}} \right |$
 Question 5
The following statements are related to bending of beams:

I The slope of the bending moment diagram is equal to the shear force.
II The slope of the shear force diagram is equal to the load intensity.
III The slope of the curvature is equal to the flexural rotation.
IV The second derivative of the deflection is equal to the curvature.

The only FALSE statement is
 A I B II C III D IV
Solid Mechanics   Shear Force and Bending Moment
Question 5 Explanation:
We know that
\begin{aligned} \frac{d S}{d x} &=W ; \frac{d M}{d X}=S_{x} \\ E 1 \cdot \frac{d^{2} y}{d x^{2}} &=M \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{M}{E I} \\ \text{Also}\quad \frac{M}{I} &=\frac{f}{y}=\frac{E}{R} \quad \therefore \frac{M}{E I}=\frac{1}{R} \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{1}{R} \end{aligned}
 Question 6
If a small concrete cube is submerged deep in still water in such a way that the pressure exerted on all faces of the cube is p, then the maximum shear stress developed inside the cube is
 A 0 B p/2 C p D 2p
Solid Mechanics   Principal Stress and Principal Strain
Question 6 Explanation:

Maximum shear stress,
$\tau=\frac{\sigma_{1}-\sigma_{2}}{2}=\frac{P-P}{2}=0$
 Question 7
As per IS 456:2000, in the Limit State Design of a flexural member, the strain in reinforcing bars under tension at ultimate state should not be less than
 A $\frac{f_{y}}{E_{s}}$ B $\frac{f_{y}}{E_{s}}+0.002$ C $\frac{f_{y}}{1.15E_{s}}$ D $\frac{f_{y}}{1.15E_{s}}+0.002$
RCC Structures   Working Stress and Limit State Method
Question 7 Explanation:

 Question 8
Which one of the following is categorised as a long-term loss of prestress in a prestressed concrete member?
 A Loss due to elastic shortening B Loss due to friction C Loss due to relaxation of strands D Loss due to anchorage slip
RCC Structures   Prestressed Concrete Beams
 Question 9
In a steel plate with bolted connections, the rupture of the net section is a mode of failure under
 A tension B compression C flexure D shear
Design of Steel Structures   Structural Fasteners
 Question 10
The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends, is equal to
 A 0.5 B 1 C 2 D 4
Solid Mechanics   Theory of Columns and Shear Centre
Question 10 Explanation:
\begin{aligned} P_{e}&=\frac{\pi^{2}EI}{I^{2}_{\text{eft}}}\\ \frac{P_{e1}}{P_{e2}}=\left[\frac{(I_{\text{elf}})_{2}}{(I_{\text{elf}})_{1}} \right ]^{2}&=\left[\frac{I}{1/2}\right ]^{2}=4 \end{aligned}