Question 1 |

The sum of two normally distributed random variables X and Y is

always normally distributed | |

normally distributed, only if X and Y are independent | |

normally distributed, only if X and Y have the same standard deviation | |

normally distributed, only if X and Y have the same mean |

Question 1 Explanation:

\begin{aligned} X_{1} &\sim N\left(\mu_{1}, \sigma_{1}\right)\\ \text{and }\quad X_{2} &\sim N\left(\mu_{2}, \sigma_{2}\right)\\ \text{then }\quad X_{1}+X_{2} &\sim N\left(\mu_{1}+\mu_{2}, \sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\right) \end{aligned}

Always normally distributed.

Always normally distributed.

Question 2 |

A matrix P is decomposed into its symmetric part S and skew symmetric part V. If

S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}, V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix},

Then matrix P is

S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}, V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix},

Then matrix P is

A | |

B | |

C | |

D |

Question 2 Explanation:

\begin{array}{l} S=\frac{P+P^{T}}{2} \\ V=\frac{P-P^{T}}{2} \\ P=S+V=\left(\begin{array}{ccc} -4 & 2 & 5 \\ 6 & 3 & 7 \\ -1 & 0 & 2 \end{array}\right) \end{array}

Question 3 |

Let I=\int_{x=0}^{1}\int_{y=0}^{x^2}xy^2dydx then, I may also be expressed as

I=\int_{y=0}^{1}\int_{x=0}^{\sqrt{y}}xy^2dxdy | |

I=\int_{y=0}^{1}\int_{x=\sqrt{y}}^{1}yx^2dxdy | |

I=\int_{y=0}^{1}\int_{x=\sqrt{y}}^{1}xy^2dxdy | |

I=\int_{y=0}^{1}\int_{x=0}^{\sqrt{y}}yx^2dxdy |

Question 3 Explanation:

I=\int_{0}^{1} \int_{0}^{x^{2}} x y^{2} d y d x

Change on rules, I=\int_{y=0}^{1} \int_{x=\sqrt{y}}^{1} x y^{2} d x d y

Change on rules, I=\int_{y=0}^{1} \int_{x=\sqrt{y}}^{1} x y^{2} d x d y

Question 4 |

The solution of

\frac{d^2y}{dt^2}-y=1,

which additionally satisfies y|_{t=0}=\left.\begin{matrix} \frac{dy}{dt} \end{matrix}\right|_{t=0}=0 in the Laplace s-domain is

\frac{d^2y}{dt^2}-y=1,

which additionally satisfies y|_{t=0}=\left.\begin{matrix} \frac{dy}{dt} \end{matrix}\right|_{t=0}=0 in the Laplace s-domain is

\frac{1}{s(s+1)(s-1)} | |

\frac{1}{s(s+1)} | |

\frac{1}{s(s-1)} | |

\frac{1}{s-1} |

Question 4 Explanation:

\begin{aligned} y^{\prime\prime}-y &=1 \\ y(0) &=1 \\ y^{\prime}(0) &=1 \\ L\left\{y^{\prime\prime}-y\right\} &=L\{1\} \\ s^{2} Y(s)-s y(0)-y^{\prime}(0)-y(s) &=\frac{1}{s} \\ y(s) &=\frac{1}{s\left(s^{2}-1\right)}\\ & =\frac{1}{s(s+1)(s-1)} \end{aligned}

Question 5 |

An attempt is made to pull a roller of weight W over a curb (step) by applying a horizontal
force F as shown in the figure.

The coefficient of static friction between the roller and the ground (including the edge of the step) is \mu. Identify the correct free body diagram (FBD) of the roller when the roller is just about to climb over the step.

The coefficient of static friction between the roller and the ground (including the edge of the step) is \mu. Identify the correct free body diagram (FBD) of the roller when the roller is just about to climb over the step.

A | |

B | |

C | |

D |

Question 5 Explanation:

Weigh = W

Note:

(i) When the cylinder is about to make out of the curb, it will loose its contact at point A, only contact will be at it B.

(ii) At verge of moving out of curb, Roller will be in equation under W, F and contact force from B and these three forces has to be concurrent so contact force from B will pass through C.

(iii) Even the surfaces are rough but there will be no friction at B for the said condition.

FBD

Question 6 |

A circular disk of radius r is confined to roll without slipping at P and Q as shown in
the figure.

If the plates have velocities as shown, the magnitude of the angular velocity of the disk is

If the plates have velocities as shown, the magnitude of the angular velocity of the disk is

\frac{v}{r} | |

\frac{v}{2r} | |

\frac{2v}{3r} | |

\frac{3v}{2r} |

Question 6 Explanation:

For pure rolling

\begin{array}{l} v_{P}=v=(P R) \omega \quad \ldots(i)\\ v_{Q}=2 v=(Q R) \omega \quad \ldots(ii) \end{array}

Divide by (ii) to (i),

2=\frac{Q R}{P R} \Rightarrow Q R=2(P R)

\begin{aligned} P R+Q R &=2 r \\ P R+2(P R) &=2 r \\ P R &=\frac{2}{3} r \\ \text{From equation(i) }\quad v &=\left(\frac{2}{3} r\right) \omega \Rightarrow \omega=\frac{3 v}{2 r} \end{aligned}

Question 7 |

The equation of motion of a spring-mass-damper system is given by

\frac{d^2x}{dt^2}+3\frac{dx}{dt}+9x=10 \sin (5t)

The damping factor for the system is

\frac{d^2x}{dt^2}+3\frac{dx}{dt}+9x=10 \sin (5t)

The damping factor for the system is

0.25 | |

0.5 | |

2 | |

3 |

Question 7 Explanation:

\frac{d^{2} x}{d t^{2}}+3\left(\frac{d x}{d t}\right)+9 x=10 \sin 5 t

Comparing with standard equation:

\ddot{x}+\left(2 \xi \omega_{n}\right) \dot{x}+\left(\omega_{n}^{2}\right) x=\left(\frac{F_{o}}{m}\right) \sin \omega t

\begin{aligned} 2 \xi \omega_{n} &=3 \\ 2 \xi \times 3 &=3 &\left[\begin{array}{l} \omega_{n}^{2}=9 \\ \omega_{n}=3 \end{array}\right] \\ \xi &=\frac{1}{2}=0.5 \end{aligned}

Comparing with standard equation:

\ddot{x}+\left(2 \xi \omega_{n}\right) \dot{x}+\left(\omega_{n}^{2}\right) x=\left(\frac{F_{o}}{m}\right) \sin \omega t

\begin{aligned} 2 \xi \omega_{n} &=3 \\ 2 \xi \times 3 &=3 &\left[\begin{array}{l} \omega_{n}^{2}=9 \\ \omega_{n}=3 \end{array}\right] \\ \xi &=\frac{1}{2}=0.5 \end{aligned}

Question 8 |

The number of qualitatively distinct kinematic inversions possible for a Grashof chain with four revolute pairs is

1 | |

2 | |

3 | |

4 |

Question 8 Explanation:

They are:

1. Double crank mechanism

2. Crank-rocker mechanism

3. Double rocker mechanism

1. Double crank mechanism

2. Crank-rocker mechanism

3. Double rocker mechanism

Question 9 |

The process, that uses a tapered horn to amplify and focus the mechanical energy for machining of glass, is

electrochemical machining | |

electrical discharge machining | |

ultrasonic machining | |

abrasive jet machining |

Question 9 Explanation:

In Ultrasonic machining, the function of horn (also
called concentrator, it is a tapered metal bar) is to
amplify and focus vibration of the transducer to an
adequate intensity for driving the tool to fulfll the
cutting operation.

Question 10 |

Two plates, each of 6 mm thickness, are to be butt-welded. Consider the following
processes and select the correct sequence in increasing order of size of the heat affected
zone

1. Arc welding

2. MIG welding

3. Laser beam welding

4. Submerged arc welding

1. Arc welding

2. MIG welding

3. Laser beam welding

4. Submerged arc welding

1-4-2-3 | |

3-4-2-1 | |

4-3-2-1 | |

3-2-4-1 |

Question 10 Explanation:

Processes with low rate of heat input (slow heating) tend to produce high total heat
constant within the metal, slow cooling rates, and large heat-affected zones. high heat
input process, have low total heats, fast cooling rates and small heat affected zones

There are 10 questions to complete.