GATE Mechanical Engineering 2019 SET-1

 Question 1
Consider the matrix
$P=\begin{bmatrix} 1 & 1 &0 \\ 0&1 &1 \\ 0& 0 & 1 \end{bmatrix}$
The number of distinct eigenvalues of P is
 A 0 B 1 C 2 D 3
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
$\text { Given: } A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$
It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
$\therefore$Number of distinct eigen values = 1
Hence, option (B) is correct.
 Question 2
A parabola $x=y^2 \; with \; 0\leq x\leq 1$ is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360$^{\circ}$ around the x-axis is
 A $\frac{\pi}{4}$ B $\frac{\pi}{2}$ C $\pi$ D $2 \pi$
Engineering Mathematics   Calculus
Question 2 Explanation:
$\text { Given: } y^{2}=x, 0 \leq x \leq 1$
The value of solid obtained by rotating the area bounded by the curve
\begin{aligned} \mathrm{y}^{2}&=\mathrm{x}, 0 \leq \mathrm{x} \leq 1 \text{ about x-axis is}\\ V &=\int_{a}^{b} \pi y^{2} d x \\ V &=\int_{0}^{1} \pi x d x \\ &=\left(\frac{\pi \mathrm{x}^{2}}{2}\right)_{0}^{1} \\ &=\frac{\pi}{2} \end{aligned}

 Question 3
For the equation $\frac{dy}{dx} + 7x^2 y=0$, if y(0)=3/7, then the value of y(1)is
 A $\frac{7}{3}e^{-7/3}$ B $\frac{7}{3}e^{-3/7}$ C $\frac{3}{7}e^{-7/3}$ D $\frac{3}{7}e^{-3/7}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
Given $\frac{d y}{d x}+7 x^{2} y=0 \ldots(1)$
With $y(0)=\frac{3}{7} \ldots(2)$
Now,(1) is written as
$\Rightarrow \int \frac{1}{y} d y+\int 7 x^{2} d x=C$
$\Rightarrow \log y+\frac{7 x^{3}}{3}=C$
$\Rightarrow y=e^{\frac{7 x^{3}}{3}+c} \; \; ...(3)$
Using (2),(3) becomes $\frac{3}{7}=\mathrm{e}^{0} \cdot \mathrm{e}^{\mathrm{C}}(\mathrm{or}) \mathrm{e}^{\mathrm{C}}=\frac{3}{7} \; \; ...(4)$
$\therefore$ The solution of (1) with (3) &(4) is given by
$y=y(x)=e^{\frac{-7 x^{3}}{3}+c}=e^{\frac{-7 x^{3}}{3}} \cdot e^{c}=\frac{3}{7} \cdot e^{\frac{-7 x^{3}}{3}}$
Hence, $y(1)=y=\frac{3}{7} \cdot e^{-\frac{7}{3}}$
 Question 4
The lengths of a large stock of titanium rods follow a normal distribution with a mean ($\mu$) of 440 mm and a standard deviation ($\sigma$) of 1mm. What is the percentage of rods whose lengths lie between 438 mm and 441 mm?
 A 81.85% B 68.40% C 99.75% D 86.64%
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:

$\begin{array}{l} \mathrm{Z}=\frac{x-\mu}{\sigma} \\ \mathrm{Z}(\mathrm{x}=438)=\frac{438-440}{1}=-2 \\ \mathrm{P}(\mathrm{Z}=-2)=2.28 \% \\ \mathrm{Z}(\mathrm{x}=441)=\frac{441-440}{1}=1 \\ \mathrm{P}(\mathrm{Z}=1)=84.13 \% \\ \end{array}$
The percentage of rods whose lengths lie between 438 mm and 441 mm =
$\begin{array}{c} =\mathrm{P}(\mathrm{Z}=1)-\mathrm{P}(\mathrm{Z}=-2) \\ =84.13 \%-2.28 \%=81.85 \% \end{array}$
 Question 5
A flat-faced follower is driven using a circular eccentric cam rotating at a constant angular velocity $\omega$. At time t=0, the vertical position of the followeris y(0)=0, and the system is in the configuration shown below.

The vertical position of the follower face, y(t) is given by
 A $e \sin \omega t$ B $e(1+ \cos 2\omega t)$ C $e(1- \cos \omega t)$ D $e \sin 2 \omega t$
Theory of Machine   Cams
Question 5 Explanation:

\begin{aligned} \mathrm{AA}_{1}=\mathrm{y} &=\mathrm{e}(1-\cos \theta) \\ &=\mathrm{e}(1-\cos \omega \mathrm{t}) \end{aligned}

There are 5 questions to complete.