# GATE ME 2016 SET-1

 Question 1
The solution to the system of equations
$\begin{bmatrix} 2&5 \\ -4&3 \end{bmatrix} \begin{Bmatrix} x\\y \end{Bmatrix} = \begin{Bmatrix} 2\\-30 \end{Bmatrix}$
 A 6,2 B -6,2 C -6,-2 D 6,-2
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
${\left[\begin{array}{cc}2 & 5 \\-4 & 3\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{c}2 \\-30\end{array}\right]}$
${\left[\begin{array}{ccc}2 & 5 & 2 \\-4 & 3 & -30\end{array}\right]}$
$R_{2}+2 R_{1}$
${\left[\begin{array}{ccc}2 & 5 & 2 \\0 & 13 & -26\end{array}\right]}$
\begin{aligned} 13 y&=-26 \\ \text { or } \qquad y&=-2 \\ 2 x+5 y&=2 \\ 2 x+5(-2)&=2 \\ 2 x=2+10&=12 \\ \text{or }\qquad x&=6 \end{aligned}
 Question 2
If $f ( t )$ is a function defined for all t$\geq$0, its Laplace transform F(s) is defined as
 A $\int_{0}^{\infty }e^{st}f( t )dt$ B $\int_{0}^{\infty }e^{-st}f( t )dt$ C $\int_{0}^{\infty }e^{ist}f( t )dt$ D $\int_{0}^{\infty }e^{-ist}f( t )dt$
Engineering Mathematics   Calculus
Question 2 Explanation:
$L(f(t))=\int_{0}^{\infty} e^{-s t} f(t) d t$
 Question 3
$f( z )=u( x ,y )+iv(x,y )$ is an analytic function of Complex Variables $z=x+iy$ where $i=\sqrt{-1}$ If $u(x,y)=2xy$, then $v(x,y)$ may be expressed as
 A $-x^{2}+y^{2}+constant$ B $x^{2}-y^{2}+constant$ C $x^{2}+y^{2}+constant$ D $-(x^{2}+y^{2})+constant$
Engineering Mathematics   Complex Variables
Question 3 Explanation:
$u=2 x y$
$u_{x}=2 y \quad u_{y}=2 x$
In option (A)
$V_{x}=-2 x \quad u_{y}=-V_{x}$
$V_{y}=2 y$
C-R equation are satisfied only in option A
 Question 4
Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is $\mu$. The standard deviation for this distribution is given by
 A $\sqrt{\mu }$ B $\mu ^{2}$ C $\mu$ D $1/\mu$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
In poisson distribution mean = Variance
Given that mean = Variance =m
Standard deviation $=\sqrt{\text { Variance }}=\sqrt{\mu}$
 Question 5
Solve the equation $x=10\cos( x )$ using the Newton-Raphson method. The initial guess is $x=\pi /4$ The value of the predicted root after the first iteration, up to second decimal, is ________
 A 1.5639 B 2.569 C 9.241 D 7.586
Engineering Mathematics   Numerical Methods
Question 5 Explanation:
\begin{aligned} f(x) &=x-10 \cos x f\left(\frac{\pi}{4}\right) \\ &=\frac{\pi}{4}-\frac{10}{\sqrt{2}}=-6.286 \\ f^{\prime}(x) &=1+10 \sin x f^{\prime}\left(\frac{\pi}{4}\right) \\ &=1+\frac{10}{\sqrt{2}}=8.07 \\ x_{1} &=x_{0}-\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}=\frac{\pi}{4}-\left(\frac{-6.286}{8.07}\right) \\ &=\frac{\pi}{4}+\frac{6.286}{8.07}=1.5639 \end{aligned}
 Question 6
A rigid ball of weight 100 N is suspended with the help of a string. The ball is pulled by a horizontal force F such that the string makes an angle of $30^{\circ}$ with the vertical. The magnitude of force F (in N) is __________ A 57.74 B 40.59 C 65.35 D 98.26
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 6 Explanation: Applying Lami's theorem
\begin{aligned} \frac{100}{\sin 120^{\circ}} &=\frac{F}{\sin \left(60^{\circ}+90^{\circ}\right)}=\frac{T}{\sin 90^{\circ}} \\ F &=\frac{100 \sin 150^{\circ}}{\sin 120^{\circ}} \\ &=57.74 \mathrm{N} \\ \end{aligned}
 Question 7
A point mass M is released from rest and slides down a spherical bowl (of radius R) from a height H as shown in the figure below. The surface of the bowl is smooth (no friction). The velocity of the mass at the bottom of the bowl is A $\sqrt{gH}$ B $\sqrt{2gR}$ C $\sqrt{2gH}$ D 0
Engineering Mechanics   Impulse and Momentum, Energy Formulations
Question 7 Explanation: Loss of P.E. = Gain in K.E.
\begin{aligned} M g h &=\frac{1}{2} M v^{2} \\ v &=\sqrt{2 g H} \end{aligned}
 Question 8
The cross sections of two hollow bars made of the same material are concentric circles as shown in the figure. It is given that $r_{3}>r_{1}$ and $r_{4}>r_{2}$ and that the areas of the cross-sections are the same.J1 and J2 are the torsional rigidities of the bars on the left and right, respectively. The ratio J2/J1 is A $\gt$1 B $\lt$0.5 C 1 D between 0.5 and 1
Strength of Materials   Torsion of Shafts
Question 8 Explanation:
C.S. area of both is same
$r_{2}^{2}-r_{1}^{2}=r_{4}^{2}-r_{3}^{2}$
Torsional rigidity
$\begin{array}{l} J_{1}=G I_{p 1}=G \frac{\pi}{2}\left(r_{2}^{4}-r_{1}^{4}\right) \\ J_{2}=G I_{p 2}=G \frac{\pi}{2}\left(r_{4}^{4}-r_{3}^{4}\right) \\ \therefore \qquad \frac{J_{2}}{J_{1}}=\frac{r_{4}^{4}-r_{3}^{4}}{r_{2}^{4}-r_{1}^{4}}=\frac{\left(r_{4}^{2}-r_{3}^{2}\right)\left(r_{4}^{2}+r_{3}^{2}\right)}{\left(r_{2}^{2}-r_{1}^{2}\right)\left(r_{2}^{2}+r_{1}^{2}\right)} \\ \therefore \qquad \frac{J_{2}}{J_{1}}=\frac{r_{4}^{2}+r_{3}^{2}}{r_{2}^{2}+r_{1}^{2}}\\ \therefore \qquad r_{4}>r_{2}, r_{3}>r_{1}\\ \therefore \qquad \frac{J_{2}}{J_{1}} \gt 1 \end{array}$
 Question 9
A cantilever beam having square cross-section of side a is subjected to an end load. If a is increased by 19%, the tip deflection decreases approximately by
 A 19% B 29% C 41% D 50%
Strength of Materials   Bending of Beams
Question 9 Explanation: \begin{aligned} \Delta_{B} &=\frac{P L^{3}}{3 E I} \\ \therefore \quad \Delta & \propto \frac{1}{I}\\ \text{For square c-s} \\ I &=\frac{a^{4}}{12} \\ \frac{\Delta_{1}}{\Delta_{2}} &=\frac{I_{2}}{I_{1}}=2.005 \\ \therefore \quad \frac{\Delta_{1}}{\Delta_{2}} &=\frac{I_{1}}{I_{2}}=\frac{(1.19 a)^{4}}{a^{4}} \\ \therefore \quad \Delta_{2} &=\frac{\Delta_{1}}{2.005}=0.5 \Delta_{1} \end{aligned}
 Question 10
A car is moving on a curved horizontal road of radius 100 m with a speed of 20 m/s. The rotating masses of the engine have an angular speed of 100 rad/s in clockwise direction when viewed from the front of the car. The combined moment of inertia of the rotating masses is 10 kg-$m^{2}$. The magnitude of the gyroscopic moment (in N-m) is __________
 A 100 B 200 C 300 D 400
Theory of Machine   Gyroscope
Question 10 Explanation:
\begin{aligned} R&=100\;m \\ v&=20\;m/sec \\ \omega _p &=\frac{v}{R} \\ &= 0.2\; rad/sec\\ \omega _s &=100\; rad/sec \\ I&= 10\; Kg-m^2\\ &\text{Gyroscopic moment,}\\ I\omega _s\omega _p &=10 \times 0.2 \times 100\; N-m\\ &=200\; N-m \end{aligned}
There are 10 questions to complete.