Question 1 |

Multiplication of real valued square matrices of same dimension is

associative | |

commutative | |

always positive definite | |

not always possible to compute |

Question 1 Explanation:

Matrix multiplication is associative.

Question 2 |

The value of \lim_{x \to 1} \left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right ) is

c | |

c+1 | |

\frac{c}{c+1} | |

\frac{c+1}{c} |

Question 2 Explanation:

Applying L Hospital rule

\lim_{x \to 1}\left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )=\lim_{x \to 1}\left ( \frac{1-e^{-c+cx}}{-x(ce^{-c+x})-(e^{-c+cx})} \right ) =\frac{-c}{-c-1}=\frac{c}{c+1}

\lim_{x \to 1}\left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )=\lim_{x \to 1}\left ( \frac{1-e^{-c+cx}}{-x(ce^{-c+x})-(e^{-c+cx})} \right ) =\frac{-c}{-c-1}=\frac{c}{c+1}

Question 3 |

The Laplace transform of a function f(t) is L(f)=\frac{1}{s^2+\omega ^2}. Then f(t) is

f(t)=\frac{1}{\omega ^2}(1-\cos \omega t) | |

f(t)=\frac{1}{\omega} \cos \omega t | |

f(t)=\frac{1}{\omega} \sin \omega t | |

f(t)=\frac{1}{\omega ^2}(1-\sin \omega t) |

Question 3 Explanation:

L(t)=\frac{1}{s^{2}+\omega ^{2}}

f(t)=L^{-1}\left \{ \frac{1}{s^{2}+\omega ^{2}} \right \}=\frac{1}{\omega }\sin \omega t

f(t)=L^{-1}\left \{ \frac{1}{s^{2}+\omega ^{2}} \right \}=\frac{1}{\omega }\sin \omega t

Question 4 |

Which of the following function f(z), of the complex variable z, is NOT analytic at all the points of the complex plane?

f(z)=z^2 | |

f(z)=e^z | |

f(z)=\sin z | |

f(z)=\log z |

Question 4 Explanation:

logz is not analytic at all points.

Question 5 |

The members carrying zero force (i.e. zero-force members) in the truss shown in the
figure, for any load P \gt 0 with no appreciable deformation of the truss (i.e. with no
appreciable change in angles between the members), are

BF and DH only | |

BF, DH and GC only | |

BF, DH, GC, CD and DE only | |

BF, DH, GC, FG and GH only |

Question 5 Explanation:

If at any joint three forces are acting out of which
two of them are collinear then force in third member
must be zero.

For member ED look at joint E.

Similarity look for other members.

For member ED look at joint E.

Similarity look for other members.

Question 6 |

A single-degree-of-freedom oscillator is subjected to harmonic excitation F(t) = F_0 \cos (\omega t) as shown in the figure.

The non-zero value of \omega, for which the amplitude of the force transmitted to the ground will be F_0, is

The non-zero value of \omega, for which the amplitude of the force transmitted to the ground will be F_0, is

\sqrt{\frac{k}{2m}} | |

\sqrt{\frac{k}{m}} | |

\sqrt{\frac{2k}{m}} | |

2 \sqrt{\frac{k}{m}} |

Question 6 Explanation:

Given,

F_{1}=F_{0}

then transmissibility

\epsilon =\frac{F_{T}}{F_{0}}=\frac{F_{0}}{F_{0}}=1

\begin{aligned} \frac{\sqrt{1+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}}{\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}=1 \\ 1+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2=\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2\\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =\pm 1 \\ \text{ Taking (-ve) sign,} \\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =-1 \\ \frac{\omega }{\omega _n} =\sqrt{2}\\ \omega=\sqrt{2}\omega_n=\sqrt{\frac{2k}{m}} \end{aligned}

F_{1}=F_{0}

then transmissibility

\epsilon =\frac{F_{T}}{F_{0}}=\frac{F_{0}}{F_{0}}=1

\begin{aligned} \frac{\sqrt{1+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}}{\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}=1 \\ 1+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2=\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2\\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =\pm 1 \\ \text{ Taking (-ve) sign,} \\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =-1 \\ \frac{\omega }{\omega _n} =\sqrt{2}\\ \omega=\sqrt{2}\omega_n=\sqrt{\frac{2k}{m}} \end{aligned}

Question 7 |

The stress state at a point in a material under plane stress condition is equi-biaxial tension
with a magnitude of 10 MPa. If one unit on the \sigma -\tau plane is 1 MPa, the Mohr's circle representation of the state-of-stress is given by

a circle with a radius equal to principal stress and its center at the origin of the
\sigma -\tau plane | |

a point on the \sigma axis at a distance of 10 units from the origin | |

a circle with a radius of 10 units on the \sigma -\tau plane | |

a point on the \tau axis at a distance of 10 units from the origin |

Question 7 Explanation:

The given state of stress is represented by a point on \sigma -\tau graph which is located on \sigma-axis at a distance of 10 units from origin.

Question 8 |

A four bar mechanism is shown below

For the mechanism to be a crank-rocker mechanism, the length of the link PQ can be

For the mechanism to be a crank-rocker mechanism, the length of the link PQ can be

80 mm | |

200 mm | |

300 mm | |

350 mm |

Question 8 Explanation:

For Crank-Rocker mechanism, shortest link must be crank and adjacent to fixed as well as Grashoff's law must be satisfied.

If l = 80 mm then shortest will be = 80 mm

as well as (80 + 600) \lt(400 + 300)

680 \lt 700

Therefore law is satisfied.

\Rightarrow l = 80 mm

Question 9 |

A helical gear with 20^{\circ} pressure angle and 30^{\circ} helix angle mounted at the mid-span of
a shaft that is supported between two bearings at the ends. The nature of the stresses
induced in the shaft is

normal stress due to bending only | |

normal stress due to bending in one plane and axial loading; shear stress due to
torsion | |

normal stress due to bending in two planes and axial loading; shear stress due to
torsion | |

normal stress due to bending in two planes; shear stress due to torsion |

Question 10 |

The crystal structure of \gamma iron (austenite phase) is

BCC | |

FCC | |

HCP | |

BCT |

Question 10 Explanation:

Austenite has a cubic-close packed crystal structure,
also referred to as a face-centred cubic structure
with an atom at each corner and in the centre of
each face of the unit cell.

There are 10 questions to complete.