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\geq0, 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
\gt1
B
\lt0.5
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.

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