GATE ME 2017 SET-1

Question 1
The product of eigenvalues of the matrix P is
P=\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}
A
-6
B
2
C
6
D
-2
Engineering Mathematics   Linear Algebra
Question 1 Explanation: 
Product of Eigen value = determinant value
\begin{array}{l} =2(3-6)+1(8-0) \\ =2(-3)+8=-6+8=2 \end{array}
Question 2
The value of lim _{x \to 0}\: \frac{x^{3}-\sin(x)}{x} is
A
0
B
3
C
1
D
-1
Engineering Mathematics   Calculus
Question 2 Explanation: 
\begin{aligned} \lim _{x \rightarrow 0} \frac{x^{3}-\sin x}{x} &=\left(\frac{0}{0} \mathrm{form}\right) \\ \lim _{x \rightarrow 0} \frac{3 x^{2}-\cos x}{1} &=0-\cos 0 \\ &=0-1=-1 \end{aligned}
Question 3
consider the following partial differential equation for u(x,y) with the constant c\gt 1:
\frac{\partial y}{\partial x}+c \: \frac{\partial u}{\partial x} =0
solution of this equations is
A
u(x,y)=f(x+cy)
B
u(x,y)=f(x-cy)
C
u(x,y)=f(cx+y)
D
u(x,y)=f(cx-y)
Engineering Mathematics   Differential Equations
Question 3 Explanation: 
\small \begin{aligned} u &=f(x-c y) \\ \frac{\partial u}{\partial x} &=f^{\prime}(x-c y)(1) \\ \frac{\partial u}{\partial y} &=f^{\prime}(x-c y)(-c) \\ &=-c \cdot f^{\prime}(x-c y)\\ &=-c \cdot \frac{\partial u}{\partial x} \\ \therefore \quad \frac{\partial u}{\partial y} & +c \frac{\partial u}{\partial x} =0 \end{aligned}
Question 4
The differential equation \frac{\mathrm{d^{2}}y}{\mathrm{d} x^{2}}+16y=0 for y(x) with the two boundary conditions
\left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=0}=1 and \left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=\frac{\pi}{2}}=-1 has
A
no solution
B
exactly two solutions
C
exactly one solutions
D
infinitely many solutions
Engineering Mathematics   Differential Equations
Question 4 Explanation: 
\begin{aligned} \left(d^{2}+16\right) y &=0 \\ A E \text { is } m^{2}+16 &=0 \\ m &=\pm 4 i \end{aligned}
Solution is
y=c_{1} \cos 4 x+c_{2} \sin 4 x
\begin{aligned} y' &=-4 c_{1} \sin 4 x+4 c_{2} \cos 4 x \\ y'(0) &=1 \\ 1 &=4 c_{2} \\ c_{2} &=1 / 4 \\ y'(\pi / 2) &=-1 \\ -1 &=-4 c_{1} \sin 2 \pi+4 c_{2} \cos 2 \pi \\ -1 &=0+4 c_{2} \\ c_{2} &=-1 / 4 \end{aligned}
Therefore the given differential equation has no solution.
Question 5
A six-face fair dice is rolled a large number of times. The mean value of the outcomes is ________
A
3.5
B
3
C
2.5
D
1
Engineering Mathematics   Probability and Statistics
Question 5 Explanation: 


\begin{aligned} \text{mean} &=E(x)=\Sigma x \cdot P(x) \\ &=1(1 / 6)+2(1 / 6)+3(1 / 6)+4 \\ &(1 / 6)+5(1 / 6)+6(1 / 6) \\ &=\frac{1}{6}(1+2+3+4+5+6) \\ &=\frac{21}{6}=3.5 \end{aligned}
Question 6
For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter, the average velocity in the fully devloped region is constant. Which one of the following statement about the average velocity in the devloping region is true?
A
It increases until the flow is fully developed.
B
It is constant and is eaual to the average velocity in the fully devloped region.
C
It decreases until the flow is fully developed
D
It is constant but is always lower than the average velocity in the fully devloped region.
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 6 Explanation: 
The average velocity in pipe flow always be same either for developing flow or fully developed flow.
Question 7
Consider the two-dimensional velocity field given by \overrightarrow{\mathrm{v}}=(5+a_{1}x+b_{1}y)\hat{i}\: + \: (4+a_{2}x+b_{2}y)\hat{j} ,where a_1,b_1,a_2 text{ and }b_2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?
A
a_{1}+\, b_{1}=0
B
a_{1}+\, b_{2}=0
C
a_{2}+\, b_{2}=0
D
a_{2}+\, b_{1}=0
Fluid Mechanics   Fluid Kinematics
Question 7 Explanation: 
Given that the flow is incompressible.
\therefore \quad \operatorname{div} \bar{V}=0
\frac{\partial}{\partial x}\left(5+a_{1} x+b_{1} y\right)+\frac{\partial}{\partial y}\left(8+a_{2} x+b_{2} y\right)=0
a_{1}+b_{2}=0
Question 8
Water (density=1000 kg/k^{3} ) at ambient temperature flows through a horizontral pipe of uniform cross section at the rate of kg/s. If the pressure drop across the pipe is 100 kPa, the minimum power required to pump the water across the pipe, in watts. is_________
A
10
B
100
C
1000
D
10000
Fluid Mechanics   Flow Through Pipes
Question 8 Explanation: 
\begin{aligned} \rho &=1000 \mathrm{kg} / \mathrm{m}^{3} \\ m &=1 \mathrm{kg} / \mathrm{s} \\ \Delta p &=100 \mathrm{kPa}=100 \times 10^{3} \mathrm{Pa} \\ P &=\rho Q g h_{f} \\ &=Q \Delta p \\ &=\frac{m}{\rho} \Delta p=\frac{1}{1000} \times 100 \times 10^{3} \\ &=100 \mathrm{W} \end{aligned}
Question 9
Which one of the following is Not a rotating machine?
A
Centrifugal pump
B
Gear pump
C
Jet pump
D
vane
Fluid Mechanics   Turbines and Pumps
Question 9 Explanation: 
In the given options all the pumps have rotating machine elements except Jet pump.

Centrifugal pump: It has rotating part eg.,impeller, Vane.

Gear Pump: In this pump there is gear mechanism which is rotating part.

Jet Pump: Jet pump utilizes ejector principle which have nozzle and diffuser not rotating parts.

Vane Pump: It consist of rotating disc which called as rotor in which number of radial slots are there where sliding vanes is inserted.
Question 10
Saturated steam at 100^{\circ}C condenses on the outside of a tube.cold fluid enters the tube at 20^{\circ}C and exits at 50^{\circ}C the value of the log mean Temperature Difference (LMTD) is _________^{\circ}C.
A
30
B
80
C
64
D
84
Heat Transfer   Heat Exchanger
Question 10 Explanation: 
Temperature profiles are

\begin{aligned} \Delta T_{i}=100-20=80^{\circ} \mathrm{C} \\ \Delta T_{e}=100-50=50^{\circ} \mathrm{C} \end{aligned} (LMTD) (parallel or counter HE)
\begin{aligned} =\frac{\Delta T_{i}-\Delta T_{e}}{\ln \frac{\Delta T_{i}}{\Delta T_{e}}}=\frac{80-50}{\ln \frac{80}{50}} \\ =63.82^{\circ} \mathrm{C} \end{aligned}
There are 10 questions to complete.

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