# GATE ME 2010

 Question 1
The parabolic arc $y=\sqrt{x}, 1\leq x\leq 2$ is revolved around the x-axis. The volume of the solid of revolution is
 A $\frac{\pi }{4}$ B $\frac{\pi }{2}$ C $\frac{3\pi }{4}$ D $\frac{3\pi }{2}$
Engineering Mathematics   Calculus
Question 1 Explanation:
The volume of a solid generated by revolution about the x-axis, of the area bounded by curve $y=f(x)$,the x-axis and the ordinates
$\text{x=a, y=b is}$
$\text { Volume }=\int_{a}^{b} \pi y^{2} d x$
Here, $a=1, b=2$ and $y=\sqrt{x} \Rightarrow y^{2}=x$
\begin{aligned} \therefore Volume &=\int_{1}^{2} \pi \cdot x \cdot d x \\ &=\pi \cdot\left[\frac{x^{2}}{2}\right]_{1}^{2}=\frac{\pi}{2}\left[x^{2}\right]_{1}^{2} \\ &=\frac{\pi}{2}\left[2^{2}-1^{2}\right]=\frac{3}{2} \pi \end{aligned}
 Question 2
The Blasius equation, $\frac{\mathrm{d}^{3}f }{\mathrm{d} \eta ^{3}}+\frac{f}{2}\frac{\mathrm{d} ^{2}f }{\mathrm{d} \eta ^{2}}=0$ is a
 A Second order nonlinear ordinary differential equation B Third order nonlinear ordinary differential equation C Third order linear ordinary differential equation D Mixed order nonlinear ordinary differential equation
Engineering Mathematics   Differential Equations
Question 2 Explanation:
$\frac{d^{3} f}{d n^{3}}+\frac{f}{2} \frac{d^{2} f}{d n^{2}}=0$ is third order $\left(\frac{d^{3} f}{d n^{3}}\right)$ and it is non linear, since the product $f \times \frac{d^{2} f}{d n^{2}}$ is not allowed in linear differential equation.

 Question 3
The value of the integral $\int_{-\infty }^{\infty }\frac{\mathrm{d} x}{1+x^{2}}$ is
 A $-\pi$ B $\frac{-\pi }{2}$ C $\frac{\pi }{2}$ D $\pi$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \int_{-\infty}^{\infty} \frac{d x}{1+x^{2}} &=\left[\tan ^{-1} x\right]_{-\infty}^{\infty} \\ &=\tan ^{-1}(\infty)-\tan ^{-1}(-\infty) \\ &=\frac{\pi}{2}-\left[\frac{-\pi}{2}\right]=\pi \end{aligned}
 Question 4
The modulus of the complex number $\left ( \frac{3+4i}{1-2i} \right )$ is
 A 5 B $\sqrt{5}$ C $\frac{1}{\sqrt{5}}$ D $\frac{1}{5}$
Engineering Mathematics   Complex Variables
Question 4 Explanation:
\begin{aligned} Z &=\frac{3+4 i}{1-2 i}=\frac{(3+4 i)(1+2 i)}{(1-2 i)(1+2 i)} \\ &=\frac{-5+10 i}{5}=-1+2 i \\ |Z| &=\sqrt{(-1)^{2}+(2)^{2}}=\sqrt{5} \end{aligned}
 Question 5
The function $y=\left | 2-3x \right |$
 A is continuous $\forall x\in R$ and differentiable $\forall x\in R$ B is continuous $\forall x\in R$ and differentiable $\forall x\in R$ except at $x=\frac{3}{2}$ C is continuous $\forall x\in R$ and differentiable $\forall x\in R$ except at $x=\frac{2}{3}$ D is continuous $\forall x\in R$ except at $x=3$ and differentiable $\forall x\in R$
Engineering Mathematics   Calculus
Question 5 Explanation:
\begin{aligned} y=|2-3 x| &=2-3 x \quad 2-3 x \geq 0 \\ &=3 x-2 \quad 2-3 x \lt 0 \end{aligned}
Therefore
$y=2-3 x \quad x \leq \frac{2}{3}$
$=3 x-2 \quad x gt \frac{2}{3}$
since $2-3 x$ and $3 x-2$ are polynomials, these are continuous at all points. The only concern is at
$x=\frac{2}{3}$
Left limit at $x=\frac{2}{3} is 2-3 \times \frac{2}{3}=0$
Right limit at $x=\frac{2}{3} is 3 \times \frac{2}{3}-2=0$
$f\left(\frac{2}{3}\right)=2-3 \times \frac{2}{3}=0$
since, Left limit = Right limit $=f\left(\frac{2}{3}\right)$
Function is continuous at $\frac{2}{3}$
y is therefore continuous $\forall x \in R$
Now since $2-3 x$ and $3 x-2$ are polynomials,they are differentiable.
only concern is at $x =\frac{2}{3}$
Now, at $x=\frac{2}{3}$, L D= Left derivative =-3
R D= Right derivative =+3
$LD \neq RD$
$\therefore$ The function y is not differentiable at
$x=\frac{2}{3}$
So, we can say that y is differential $\forall \bar{x} \in R$, except at $x=\frac{2}{3}$

There are 5 questions to complete.