# GATE ME 2018 SET-1

 Question 1
Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled out of the box at random one after another without replacement. The probability that all the three balls are red is
 A $\frac{1}{72}$ B $\frac{1}{55}$ C $\frac{1}{36}$ D $\frac{1}{27}$
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:

Probability that all the three balls are red is
$\begin{array}{l} =\mathrm{R} \cdot \mathrm{R} \cdot \mathrm{R} \\ =\frac{4}{12} \times \frac{3}{11} \times \frac{2}{10}=\frac{24}{1320}=\frac{1}{55} \end{array}$
 Question 2
The rank of the matrix $\begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix}$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} &\left[\begin{array}{ccc}-4 & 1 & -1 \\-1 & -1 & -1 \\7 & -3 & 1\end{array}\right]\\ R_{1} \longleftrightarrow R_{2} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\-4 & 1 & -1 \\7 & -3 & 1\end{array}\right]} \\ R_{2}-4 R_{1}, R_{3}+7 R_{1} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & -10 & -6\end{array}\right]} \\ R_{3}+2 R_{2} \qquad &{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & 0 & 0\end{array}\right]} \end{aligned}
No. of non zero rows =2
rank=2
 Question 3
According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value $\xi$ in this interval such that $\int_{a}^{b}f(x)dx$ =
 A f($\xi$)(b-a) B f(b)($\xi$-a) C f(a)(b-$\xi$) D 0
Engineering Mathematics   Calculus
Question 3 Explanation:
$\int_{a}^{b} f(\xi) d x=f(\xi)(b-a)$
 Question 4
F(z) is a function of the complex variable z = x + iy given by
$F(z)=iz+kRe(z)+ i \; lm(z)$
For what value of k will F(z) satisfy the Cauchy-Riemann equations?
 A 0 B 1 C -1 D y
Engineering Mathematics   Complex Variables
Question 4 Explanation:
\begin{aligned} F(z) &=i z+k \operatorname{Re}(z)+i \operatorname{Im}(z) \\ u+i v &=i(x+i y)+k x+i y \\ u+i v &=k x-y+i(x+y) \\ u &=k x-y, v=x+y \\ u_{x} &=k, u_{y}=-1 \\ V &=x+y \\ v_{x} &=1 \\ v_{y} &=1 \\ u_{x} &=v_{y} \\ k &=1 \end{aligned}
 Question 5
A bar of uniform cross section and weighing 100 N is held horizontally using two massless and inextensible strings S1 and S2 as shown in the figure.

The tension of the strings are
 A $T_{1} = 100 N \; and \; T_{2} = 0 N$ B $T_{1} = 0 N \; and \; T_{2} = 100 N$ C $T_{1} = 75 N \; and \; T_{2} = 25 N$ D $T_{1} = 25 N \; and \; T_{2} = 75 N$
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 5 Explanation:

\begin{aligned} T_{1}+T_{2} &=100 \mathrm{N} \qquad \ldots(i)\\ \Sigma M_{A} &=0 \\ T_{2} \cdot \frac{L}{2} &=100 \times \frac{L}{2} \\ \therefore \qquad T_{2} &=100 \mathrm{N} \\ T_{1} &=0 \mathrm{N} \end{aligned}
 Question 6
If $\sigma _{1}$ and $\sigma _{3}$ are the algebraically largest and smallest principal stresses respectively, the value of the maximum shear stress is
 A $\frac{\sigma _{1} + \sigma _{3}}{2}$ B $\frac{\sigma _{1} - \sigma _{3}}{2}$ C $\sqrt{\frac{\sigma _{1} + \sigma _{3}}{2}}$ D $\sqrt{\frac{\sigma _{1} - \sigma _{3}}{2}}$
Strength of Materials   Mohr's Circle
Question 6 Explanation:
Maximum shear stress $=\frac{\sigma_{1}-\sigma_{3}}{2}$
 Question 7
The equation of motion for a spring-mass system excited by a harmonic force is
$M\ddot{x} + Kx = Fcos(\omega t)$
where M is the mass, K is the spring stiffness, F is the force amplitude and is the angular frequency of excitation. Resonance occurs when is equal to
 A $\sqrt{\frac{M}{K}}$ B $\frac{1}{2\pi}\sqrt{\frac{K}{M}}$ C $2\pi\sqrt{\frac{K}{M}}$ D $\sqrt{\frac{K}{M}}$
Theory of Machine   Vibration
Question 7 Explanation:
$M \ddot{x}+K x=f \cos (\omega t)$
Resonance is when $\quad \omega=\omega_{n}=\sqrt{\frac{K}{M}}$
 Question 8
For an Oldham coupling used between two shafts, which among the following statements are correct?
I. Torsional load is transferred along shaft axis.
II. A velocity ratio of 1:2 between shafts is obtained without using gears.
III. Bending load is transferred transverse to shaft axis.
IV. Rotation is transferred along shaft axis.
 A I and III B I and IV C II and III D II and IV
Strength of Materials   Torsion of Shafts
Question 8 Explanation:
Oldham coupling is used to connect two shafts which are not on the same axis means they are not aligned to the same axis
So,
(1) Torsional load is transferred along shaft axis as both shafts are rotating member.
So, that statement is correct.
(2) A velocity ratio 1:1 between shafts is obtained using gears.

So, this statement is wrong
(3) Bending load is not transferred transverse to shaft axis as there is no transverse load.
(4) Rotation is transferred along shaft axis
So, this statement is correct
 Question 9
For a two-dimentional incompressible flow field given by $\vec{u}=A(x\hat{i}-y\hat{j})$ , where $A > 0$ which one of the following statements is FALSE?
 A It satisfies continuity equation. B It is unidirectional when $x\rightarrow 0$ and $y\rightarrow \infty$. C Its streamlines are given by x = y. D It is irrotational
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 9 Explanation:
C is the false statement
2D incompressible flow continuity equation.
\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} &=0 \\ \frac{\partial(A x)}{\partial x}+\frac{\partial(-A y)}{\partial y} &=0 \end{aligned}
A-A=0 it satisfies continuity equation.
$\Rightarrow A S \quad \vec{V}=A x \hat{i}-A y \hat{j}$
As $y \rightarrow \infty$ velocity vector field will not be defined along y axis.
So flow will be along x-axis i.e. 1 -D flow.
$\Rightarrow$ Stream line equation for $2 \mathrm{D}$
\begin{aligned} \frac{d x}{u} &=\frac{d y}{v} \\ \frac{d x}{A x} &=\frac{d y}{-A y} \\ \ln x &=-\ln y+\ln c \\ \ln x y &=\ln c \\ x y &=c \rightarrow \text { streamline equation } \end{aligned}
 Question 10
Which one of the following statements is correct for a superheated vapour?
 A Its pressure is less than the saturation pressure at a given temperature. B Its temperature is less than the saturation temperature at a given pressure. C Its volume is less than the volume of the saturated vapour at a given temperature. D Its enthalpy is less than the enthalpy of the saturated vapour at a given pressure.
Thermodynamics   Availability and Irreversibility
Question 10 Explanation:

$P_{\text {sat }} @ T_{1} \rightarrow$ saturation pressure at $T_{1}$ temperature $P_{1} \rightarrow$ pressure of superheated vapour at state 1
$P_{1} \lt P_{\text {sat }} @_{T_{1}}$
There are 10 questions to complete.