# GATE ME 2015 SET-1

 Question 1
If any two columns of a determinant P= $\begin{vmatrix} 4 & 7 & 8\\ 3 & 1 & 5\\ 9& 6 & 2 \end{vmatrix}$ are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?
 A Absolute value remains unchanged but sign will change. B Both absolute value and sign will change. C Absolute value will change but sign will not change D Both absolute value and sign will remain unchanged
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Property of determinant: If any two row or column are interchanged, then mangnitude of determinant remains same but sign changes.
 Question 2
Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?
 A I B II C III D IV
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
We know that probability density function of normal distribution is given by
$F(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left ( \frac{x-\mu }{\sigma } \right )^2}$
For having lowest variance $(\sigma ^2)$, standard deviation $(\sigma )$ will be lowest. From the density function $f(x)$, we can say that as σ decreases $F(x)$ will increase, so curve having highest peak has lowest standard deviation and variance.
 Question 3
Simpson's $1/3$ rule is used to integrate the function $\frac{3}{5}x^{2}+\frac{9}{5}$ between x = 0 and x = 1 using the least number of equal sub -intervals. The value of the integral is _____________
 A 1 B 2 C 3 D 4
Engineering Mathematics   Numerical Methods
Question 3 Explanation:
$f(x)=\frac{3}{5} x^{2}+\frac{9}{5}$
$\begin{array}{|c|c|c|c|}\hline x & 0 & 0.5 & 1 \\\hline f(x) & 1.8 & 1.95 & 2.4 \\\hline\end{array}\\ \Rightarrow\int_{0}^{1} f(x)=\frac{h}{3}\left[y_{0}+4 y_{1}+y_{2}\right]$
$=\frac{0.5}{3}[1.8+4(1.95)+2.4]=2$
 Question 4
The value of $\lim_{x\rightarrow 0}\frac{1-cos(x^{2})}{2x^{4}}$ is
 A 0 B $1/2$ C $1/4$ D undefined
Engineering Mathematics   Calculus
Question 4 Explanation:
$\lim _{x \rightarrow 0} \frac{1-\cos \left(x^{2}\right)}{2 x^{4}} \quad$ putting the x $\rightarrow 0$
we get $\frac{0}{0}$ form
Applying L' Hospital rule
$\Rightarrow \lim _{x \rightarrow 0} \frac{2 x \sin \left(x^{2}\right)}{8 x^{3}} \quad$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)}{4 x^{2}}$
$\Rightarrow \frac{1}{4} \lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)}{x^{2}}$
$\Rightarrow \frac{1}{4} \lim _{x^{2} \rightarrow 0} \frac{\sin \left(x^{2}\right)}{x^{2}}=\frac{1}{4} \times 1=\frac{1}{4}$
 Question 5
Given two complex numbers $z_{1}=5+(5\sqrt{3})i$ and $z_{2}=2/\sqrt{3}+2i$, the argument of $z_{1}/z_{2}$ in degrees is
 A 0 B 30 C 60 D 90
Engineering Mathematics   Complex Variables
Question 5 Explanation:
\begin{aligned} z_{1} &=5+(5 \sqrt{3}) i \\ z_{2} &=\frac{2}{\sqrt{3}}+2 i \\ \arg \left(z_{1}\right) &=\theta_{1}=\tan ^{-1}\left(\frac{5 \sqrt{3}}{5}\right) \\ \theta_{1} &=60^{\circ} \\ \arg \left(z_{2}\right) &=\theta_{2}=\tan ^{-1}\left(\frac{2}{2 \sqrt{3}}\right) \\ \theta_{2} &=60^{\circ} \\ \arg \left(\frac{z_{1}}{z_{2}}\right) &=\arg \left(z_{1}\right)-\arg \left(z_{2}\right) \\ &=60-60=0^{\circ} \end{aligned}
 Question 6
Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of
 A 4 B 16 C 32 D 64
Fluid Mechanics   Flow Through Pipes
Question 6 Explanation:
\begin{aligned} h_{f}&=\frac{f L V^{2}}{2 g d}\\ &\text{Mass flow rate,}\\ m&=\rho A V=\rho \times \frac{\pi}{4} d^{2} V\\ or\quad V&=\frac{4 m}{\rho \pi d^{2}} \\ \therefore \quad h_{f}&=\frac{f L}{2 g d} \times \frac{16 m^{2}}{\rho^{2} \pi^{2} d^{4}}\\ &=\frac{8 f m^{2}}{\rho^{2} \pi^{2}} \times \frac{L}{d^{5}} \\ &=C \times \frac{L}{d^{5}} \text { where } C=\frac{8 f m^{2}}{\rho^{2} \pi^{2}} \\ h_{f_{1}}&=C \times \frac{L}{d^{5}}\\ and\quad h_{2} &=C \times \frac{2 L}{(d / 2)^{5}}=\frac{C \times 2 L}{\frac{d^{5}}{2^{5}}} \\ &=\frac{C \times 2 \times 2^{5} L}{d^{5}}=\frac{C \times 64 L}{d^{5}} \\ h_{1 / 2} &=64 h_{11} \end{aligned}
The head loss due to friction will increase by a factor of 64
 Question 7
The Blasius equation related to boundary layer theory is a
 A third-order linear partial differential equation B third-order nonlinear partial differential equation C second-order nonlinear ordinary differential equation D third-order nonlinear ordinary differential equation
Heat Transfer   Heat Transfer in Flow Over Plates and Pipes
Question 7 Explanation:
$2 \frac{d^{3} f}{d \eta^{3}}+f \frac{d^{2} f}{d \eta^{2}}=0$
third-order non-linear differential equation.
 Question 8
For flow of viscous fluid over a flat plate, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is
 A thinner than the velocity boundary layer B thicker than the velocity boundary layer C of the same thickness as the velocity boundary layer D not formed at all
Heat Transfer   Heat Transfer in Flow Over Plates and Pipes
Question 8 Explanation:
For the flow of viscous fluid over a flat plate if the fluid temperature is the same as the plate temperature the thermal boundary layer is not formed at all because boundary layer formation took place if there is some difference in fluid property of no slip layer and remaining fluid.
As here given that fluid is viscous and flowing over a flat plate. Here, definitely a kinematic boundary layer will be formed but as there is no temperature difference so no formation of thermal boundary layer.
 Question 9
For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy,
 A it is sufficient to know only the temperature B both temperature and pressure are required to be known C both temperature and volume are required to be known D both temperature and mass are required to be known
Thermodynamics   First Law, Heat, Work and Energy
Question 9 Explanation:
For constant valves of specific heats

dh = F(T) only

So for calculation of specific enthalpy it is sufficient to know only the temperature.
 Question 10
A Carnot engine (CE-1) works between two temperature reservoirs A and B, where $T_{A}$ = 900 K and $T_{B}$ = 500 K. A second Carnot engine (CE-2) works between temperature reservoirs B and C, where $T_{C}$ = 300 K. In each cycle of CE-1 and CE-2, all the heat rejected by CE-1 to reservoir B is used by CE-2. For one cycle of operation, if the net Q absorbed by CE-1 from reservoir A is 150 MJ, the net heat rejected to reservoir C by CE-2 (in MJ) is ______________
 A 20 B 30 C 50 D 60
Thermodynamics   Second Law, Carnot Cycle and Entropy
Question 10 Explanation:

For Carnot engine (CE-2)
\begin{aligned} \eta_{2}&=1-\frac{T_{C}}{T_{B}}\\ \text{also}\qquad \eta_{2} &=1-\frac{Q_{C}}{Q_{B}} \\ \therefore \quad 1-\frac{T_{C}}{T_{B}} &=1-\frac{Q_{C}}{Q_{B}}\\ \text{or}\qquad\frac{T_{C}}{T_{B}} &=\frac{Q_{C}}{Q_{B}} \\ \frac{300}{500} &=\frac{Q_{C}}{\frac{5 \times 150}{9}}\\ \text{or}\qquad Q_{c}&=50 \mathrm{MJ} \end{aligned}
For Carnot engine (CE-2)
\begin{aligned} \eta_{2}&=1-\frac{T_{C}}{T_{B}}\\ \text{also}\qquad \eta_{2} &=1-\frac{Q_{C}}{Q_{B}} \\ \therefore \quad 1-\frac{T_{C}}{T_{B}} &=1-\frac{Q_{C}}{Q_{B}}\\ \text{or}\qquad\frac{T_{C}}{T_{B}} &=\frac{Q_{C}}{Q_{B}} \\ \frac{300}{500} &=\frac{Q_{C}}{\frac{5 \times 150}{9}}\\ \text{or}\qquad Q_{c}&=50 \mathrm{MJ} \end{aligned}
There are 10 questions to complete.