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

\frac{1}{72} | |

\frac{1}{55} | |

\frac{1}{36} | |

\frac{1}{27} |

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

1 | |

2 | |

3 | |

4 |

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

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 =

f(\xi)(b-a) | |

f(b)(\xi-a) | |

f(a)(b-\xi) | |

0 |

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?

F(z)=iz+kRe(z)+ i \; lm(z)

For what value of k will F(z) satisfy the Cauchy-Riemann equations?

0 | |

1 | |

-1 | |

y |

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

The tension of the strings are

T_{1} = 100 N \; and \; T_{2} = 0 N | |

T_{1} = 0 N \; and \; T_{2} = 100 N | |

T_{1} = 75 N \; and \; T_{2} = 25 N | |

T_{1} = 25 N \; and \; T_{2} = 75 N |

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

\frac{\sigma _{1} + \sigma _{3}}{2} | |

\frac{\sigma _{1} - \sigma _{3}}{2} | |

\sqrt{\frac{\sigma _{1} + \sigma _{3}}{2}} | |

\sqrt{\frac{\sigma _{1} - \sigma _{3}}{2}} |

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

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

\sqrt{\frac{M}{K}} | |

\frac{1}{2\pi}\sqrt{\frac{K}{M}} | |

2\pi\sqrt{\frac{K}{M}} | |

\sqrt{\frac{K}{M}} |

Question 7 Explanation:

M \ddot{x}+K x=f \cos (\omega t)

Resonance is when \quad \omega=\omega_{n}=\sqrt{\frac{K}{M}}

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.

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.

I and III | |

I and IV | |

II and III | |

II and IV |

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

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?

It satisfies continuity equation. | |

It is unidirectional when x\rightarrow 0 and y\rightarrow \infty. | |

Its streamlines are given by x = y. | |

It is irrotational |

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}

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?

Its pressure is less than the saturation pressure at a given temperature. | |

Its temperature is less than the saturation temperature at a given pressure. | |

Its volume is less than the volume of the saturated vapour at a given temperature. | |

Its enthalpy is less than the enthalpy of the saturated vapour at a given pressure. |

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.