# 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}

There are 5 questions to complete.