# GATE CE 2016 SET-2

 Question 1
The spot speeds (expressed in km/hr) observed at a road section are 66, 62, 45, 79, 32, 51, 56, 60, 53, and 49. The median speed (expressed in km/hr) is ________.
(Note: answer with one decimal accuracy)
 A 54.5 B 51.5 C 53.5 D 56
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:
Median speed is the speed at the middle value in series of spot speeds that are arranged in ascending order. 50% of speed values will be greater than the median 50% will be less than the median.
Ascending order order of spot speed studies are 32, 39, 45, 51, 53, 56, 60, 62, 66, 79
Median speed=$\frac{53+56}{2}$=54.5 km/hr
 Question 2
The optimum value of the function $f(x)=x^{2}-4x+2$ is
 A 2 (maximum) B 2 (minimum) C $-2$ (maximum) D $-2$ (minimum)
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} {f}'&=0 \\ \Rightarrow \;\; 2x-4&=0 \\ \Rightarrow \;\; x&=2 \text{ (stationary point)}\\ {f}''\left ( x \right )&=2 \gt 0 \\ \Rightarrow\;\; f(x)& \text{ is minimum at } x=2\end{aligned}
i.e., $\left ( 2 \right )^{2}-4\left ( 2 \right )+2=-2$
$\therefore$ The optimum value of $f(x)$ is $-2$ (minimum).

 Question 3
The Fourier series of the function,
$\begin{matrix} f(x) & =0 & -\pi \lt x \leq 0 \\ f(x) &=\pi-x & 0 \lt x \lt \pi \end{matrix}$

in the interval $[-\pi ,\pi ]$ is

$f(x)=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\cdots\: \cdots \ \cdots \right ]$ $+ \left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\cdots \: \cdots\: \cdot \right ]$

The convergence of the above Fourier series at x = 0 gives
 A $\sum_{n=1}^{\infty }\frac{1}{n^{2}}=\frac{\pi ^{2}}{6}$ B $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi ^{2}}{12}$ C $\sum_{n=1}^{\infty }\frac{1}{(2n-1)^{2}}=\frac{\pi ^{2}}{8}$ D $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{(2n-1)}=\frac{\pi}{4}$
Engineering Mathematics   Partial Differential Equation
Question 3 Explanation:
The function is $f(x)=0$
$-p\lt x\leq 0$
$=p-x,\, 0 \lt x \lt \pi$
And Fourier series is,
$f\left ( x \right )=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\frac{\cos 5x}{5^{2}}+... \right ]+\left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+... \right ] ...\left ( i \right )$
At x=0, (a point of discontinuity), the fourier series converges to $\frac{1}{2}\left [ f\left ( 0^{-1} \right )+f\left ( 0^{+} \right ) \right ]$
where $f\left ( 0^{-} \right )=\lim_{x\rightarrow 0}\left ( \pi -x \right )=\pi$
Hence, eq. (i), we get,
$\frac{\pi }{2}=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{1}{1^{2}}+\frac{1}{3^{2}}+... \right ]$
$\Rightarrow \;\; \frac{1}{1}+\frac{1}{3^{2}}+\frac{1}{5^{2}}+...\frac{\pi ^{2}}{8}$
 Question 4
X and Y are two random independent events. It is known that $P(X)=0.40$ and $P(X\cup Y^{C})=0.7$. Which one of the following is the value of $P(X\cup Y)$ ?
 A 0.7 B 0.5 C 0.4 D 0.3
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
$\; \; \; \; P\left ( X\: \cup \: Y^{c} \right )=0.7$
$\Rightarrow \; \; P\left ( X \right )+P\left ( Y^{c} \right )-P\left ( X \right )P\left ( Y^{c} \right )=0.7$
(Since X, Y are independent events)
$\Rightarrow \; \; P\left ( X \right )+1-P\left ( Y \right )-P\left ( X \right )\left \{ 1-P\left ( Y \right ) \right \}=0$
$\Rightarrow \; \; P\left ( X \right )-P\left ( X\: \cap \: Y \right )=0.3\; \; \; \; \; \; ...\left ( i \right )$
$\; \; \; \; P\left ( X\: \cup \: Y \right )=P\left ( X \right )+P\left ( Y \right )-P\left ( X\: \cap \: Y \right )$
$\; \; \; \; =0.4+0.3=0.7$
 Question 5
What is the value of $\lim_{\begin{matrix} x\rightarrow 0\\ y\rightarrow 0 \end{matrix}} \frac{xy}{x^{2}+y^{2}}$ ?
 A 1 B -1 C 0 D Limit does not exist
Engineering Mathematics   Calculus
Question 5 Explanation:
(i) $\lim_{x\rightarrow \infty }\frac{xy}{x^{2}+y^{2}}\lim_{y\rightarrow \infty }\left ( \frac{0}{0^{2}+y^{2}} \right )=0$
(i.e., put $x=0$ and then $y=0$)
(ii) $\lim_{x\rightarrow 0 y\rightarrow 0}\frac{xy}{x^{2}+y^{2}}\lim_{x\rightarrow 0}\left ( \frac{0}{x^{2}+0} \right )=0$
( i.e., put $y=0$ and then $x=0$)
(iii)$\lim_{x\rightarrow 0 y\rightarrow 0}\frac{xy}{x^{2}+y^{2}}\lim_{x\rightarrow 0}\frac{x\left ( mx \right )}{x^{2}+m^{2}x^{2}}$
(i.e., put $y=mx$)
$\lim_{x\rightarrow \infty }\left ( \frac{m}{1+m^{2}} \right )=\frac{m}{1+m^{2}}$
which depends on m.

There are 5 questions to complete.