# GATE ME 2017 SET-2

 Question 1
Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is _______
 A 0.5 B 0.65 C 0.75 D 0.8
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:
Total four possibilities {HH. HT, TH, TT}
The probability of getting at least one head is $\frac{3}{4}$.
 Question 2
The divergence of the vector -yi+xj______
 A 0 B 1 C 2 D 0.5
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} \vec { F } & = - y \bar { i } + x \bar { j } \\ \nabla \cdot \bar { F } & = \frac { \partial } { \partial x } ( - y ) + \frac { \partial } { \partial y } ( x ) \\ & = 0 + 0 = 0 \end{aligned}
 Question 3
The determinant of a 2x2 matrix is 50. If one eigenvalue of the matrix is 10, the other eigenvalue is _____.
 A 5 B 6 C 7 D 8
Engineering Mathematics   Linear Algebra
Question 3 Explanation:
The product of eigen value of always equal to
the determinant value of the matrix.
\begin{aligned} \lambda_{1} &=10 \quad \lambda_{2}=\text { unknown } \quad|A|=50 \\ \lambda_{1} \cdot \lambda_{2} &=50 \\ 10\left(\lambda_{2}\right) &=50\\ \therefore \qquad \lambda_{2}&=5 \\ \end{aligned}
 Question 4
A sample of 15 data is follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The mode of the data is
 A 4 B 13 C 17 D 20
Industrial Engineering   PERT and CPM
Question 4 Explanation:
Mode means highest number of observations or occurrence of data most of the time as data 17, occurs four times, i.e., highest time. So mode is 17.
 Question 5
The Laplace transform of $te^{t}$ is
 A $\frac{s}{(s+1)^{2}}$ B $\frac{1}{(s-1)^{2}}$ C $\frac{1}{(s+1)^{2}}$ D $\frac{s}{s-1}$
Engineering Mathematics   Calculus
Question 5 Explanation:
$f ( t ) = t e ^ { t }$
$L ( t ) = \frac { 1 } { s ^ { 2 } }$
By first shifting rule
$L\left( t e ^ { t } \right) = \frac { 1 } { ( s - 1 ) ^ { 2 } }$
 Question 6
A mass m is attached to two identical springs having spring constant k as shown in the figure. The natural frequency $\omega$ of this single degree of freedom system is A $\sqrt{\frac{2k}{m}}$ B $\sqrt{\frac{k}{m}}$ C $\sqrt{\frac{k}{2m}}$ D $\sqrt{\frac{4k}{m}}$
Theory of Machine   Vibration
Question 6 Explanation: Equivalent stiffness
$k_{e q}=k+k=2 k$
Natural frequency is
$\omega_{n}=\sqrt{\frac{k_{\mathrm{eq}}}{m}}=\sqrt{\frac{2 \mathrm{k}}{\mathrm{m}}}$
 Question 7
The state of stress at a point is $\sigma _{x}=\sigma _{y}=\sigma _{z}=t_{xz}=t_{zx}=t_{yz}=t_{zy}=0$ and $t_{xy}=t_{yx}=50MPa$ . The maximum normal stress (in MPa) at that point is_____.
 A 49 B 50 C 55 D 60
Strength of Materials   Mohr's Circle
Question 7 Explanation:
Given state of stress condition indicates pure shear state of stress.
For pure shear state of stress,
Max. tensile stress = Max. comp. stress = Max. Shear stress
$=\tau_{X Y}=50 \mathrm{MPa}$ Hence, Max. normal stress $=50 \mathrm{MPa}$
 Question 8
For a loaded cantilever beam of uniform cross-section, the bending moment (in N.mm) along the length is $M(x)=5x^{2}+10x$ , where x is the distance (in mm) measured from the free end of the beam. The magnitude of shear force (in N) in the cross-section at x=10 mm is
 A 100 B 105 C 110 D 115
Strength of Materials   Shear-force and Bending Moment Diagrams
Question 8 Explanation:
$(\mathrm{BM})_{x-x}=M_{x-x}=5 x^{2}+10 x$ $\text { (S.F) }_{x-x}=F_{x-x}=?$
We know that at section X-X
\begin{aligned} F_{X-X} &=\frac{d}{d x}\left[5 x^{2}+10 x\right] \\ F_{X-X} &=10 x+10 \\ \left(F_{X-X}\right)_{x=10 \mathrm{mm}} &=10(10)+10=110 \mathrm{N} \end{aligned}
 Question 9
A cantilever beam of length L and flexural modulus EI is subjected to a point load P at the free end. The elastic strain energy stored in the beam due to bending (neglecting transverse shear) is
 A $\frac{P^{2}L^{3}}{6EI}$ B $\frac{P^{2}L^{3}}{3EI}$ C $\frac{PL^{3}}{3EI}$ D $\frac{PL^{3}}{6EI}$
Strength of Materials   Strain Energy and Thermal Stresses
Question 9 Explanation:
Let U is the S.E. in the beam due to B.M. (M) $U=\int_{0}^{L}\frac{(M_{x-x})^2 dx}{2(EI)_{x-x}}=\int_{0}^{L}\frac{(Px)^2 dx}{2EI_{NA}}$
$U=\frac{P^2}{2EI_{NA}}\int_{0}^{L}x^2dx=\frac{P^2}{2EI}\left ( \frac{L^3}{3} \right )$
$U=\frac{P^2L^3}{6EI_{NA}}$
 Question 10
A steel bar is held by two fixed supports as shown in the figure and is subjected to an increases of temperature $\Delta T=100^{\circ}C$.If the coefficent of thermal exapnasion and young's modules of elasticity of steel are $11\, \times \, 10^{-6} \, /^{\circ}C$ and 200GPa, respectively, the magnitude of thermal stress (in MPa) induced in the bar is _____ A 220 B 225 C 230 D 235
Strength of Materials   Strain Energy and Thermal Stresses
Question 10 Explanation:
We know that for completely restricted expansion, thermal stress developed in bar is given by
\begin{aligned} \sigma_{t h} &=\alpha(\Delta T) E \\ &=11 \times 10^{-6} \times 100 \times 200 \times 10^{3}\\ &=220 \mathrm{MPa}\text{(comp. in native)} \end{aligned}
There are 10 questions to complete. 