Question 1 |

The product of eigenvalues of the matrix P is

P=\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}

P=\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}

-6 | |

2 | |

6 | |

-2 |

Question 1 Explanation:

Product of Eigen value = determinant value

\begin{array}{l} =2(3-6)+1(8-0) \\ =2(-3)+8=-6+8=2 \end{array}

\begin{array}{l} =2(3-6)+1(8-0) \\ =2(-3)+8=-6+8=2 \end{array}

Question 2 |

The value of lim _{x \to 0}\: \frac{x^{3}-\sin(x)}{x} is

0 | |

3 | |

1 | |

-1 |

Question 2 Explanation:

\begin{aligned} \lim _{x \rightarrow 0} \frac{x^{3}-\sin x}{x} &=\left(\frac{0}{0} \mathrm{form}\right) \\ \lim _{x \rightarrow 0} \frac{3 x^{2}-\cos x}{1} &=0-\cos 0 \\ &=0-1=-1 \end{aligned}

Question 3 |

consider the following partial differential equation for u(x,y) with the constant c\gt 1:

\frac{\partial y}{\partial x}+c \: \frac{\partial u}{\partial x} =0

solution of this equations is

\frac{\partial y}{\partial x}+c \: \frac{\partial u}{\partial x} =0

solution of this equations is

u(x,y)=f(x+cy) | |

u(x,y)=f(x-cy) | |

u(x,y)=f(cx+y) | |

u(x,y)=f(cx-y) |

Question 3 Explanation:

\small \begin{aligned} u &=f(x-c y) \\ \frac{\partial u}{\partial x} &=f^{\prime}(x-c y)(1) \\ \frac{\partial u}{\partial y} &=f^{\prime}(x-c y)(-c) \\ &=-c \cdot f^{\prime}(x-c y)\\ &=-c \cdot \frac{\partial u}{\partial x} \\ \therefore \quad \frac{\partial u}{\partial y} & +c \frac{\partial u}{\partial x} =0 \end{aligned}

Question 4 |

The differential equation \frac{\mathrm{d^{2}}y}{\mathrm{d} x^{2}}+16y=0 for y(x) with the two boundary conditions

\left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=0}=1 and \left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=\frac{\pi}{2}}=-1 has

\left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=0}=1 and \left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=\frac{\pi}{2}}=-1 has

no solution | |

exactly two solutions | |

exactly one solutions | |

infinitely many solutions |

Question 4 Explanation:

\begin{aligned} \left(d^{2}+16\right) y &=0 \\ A E \text { is } m^{2}+16 &=0 \\ m &=\pm 4 i \end{aligned}

Solution is

y=c_{1} \cos 4 x+c_{2} \sin 4 x

\begin{aligned} y' &=-4 c_{1} \sin 4 x+4 c_{2} \cos 4 x \\ y'(0) &=1 \\ 1 &=4 c_{2} \\ c_{2} &=1 / 4 \\ y'(\pi / 2) &=-1 \\ -1 &=-4 c_{1} \sin 2 \pi+4 c_{2} \cos 2 \pi \\ -1 &=0+4 c_{2} \\ c_{2} &=-1 / 4 \end{aligned}

Therefore the given differential equation has no solution.

Solution is

y=c_{1} \cos 4 x+c_{2} \sin 4 x

\begin{aligned} y' &=-4 c_{1} \sin 4 x+4 c_{2} \cos 4 x \\ y'(0) &=1 \\ 1 &=4 c_{2} \\ c_{2} &=1 / 4 \\ y'(\pi / 2) &=-1 \\ -1 &=-4 c_{1} \sin 2 \pi+4 c_{2} \cos 2 \pi \\ -1 &=0+4 c_{2} \\ c_{2} &=-1 / 4 \end{aligned}

Therefore the given differential equation has no solution.

Question 5 |

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is ________

3.5 | |

3 | |

2.5 | |

1 |

Question 5 Explanation:

\begin{aligned} \text{mean} &=E(x)=\Sigma x \cdot P(x) \\ &=1(1 / 6)+2(1 / 6)+3(1 / 6)+4 \\ &(1 / 6)+5(1 / 6)+6(1 / 6) \\ &=\frac{1}{6}(1+2+3+4+5+6) \\ &=\frac{21}{6}=3.5 \end{aligned}

Question 6 |

For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter, the average velocity in the fully devloped region is constant. Which one of the following statement about the average velocity in the devloping region is true?

It increases until the flow is fully developed. | |

It is constant and is eaual to the average velocity in the fully devloped region. | |

It decreases until the flow is fully developed | |

It is constant but is always lower than the average velocity in the fully devloped region. |

Question 6 Explanation:

The average velocity in pipe flow always be same either for developing flow or fully developed flow.

Question 7 |

Consider the two-dimensional velocity field given by \overrightarrow{\mathrm{v}}=(5+a_{1}x+b_{1}y)\hat{i}\: + \: (4+a_{2}x+b_{2}y)\hat{j} ,where a_1,b_1,a_2 text{ and }b_2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

a_{1}+\, b_{1}=0 | |

a_{1}+\, b_{2}=0 | |

a_{2}+\, b_{2}=0 | |

a_{2}+\, b_{1}=0 |

Question 7 Explanation:

Given that the flow is incompressible.

\therefore \quad \operatorname{div} \bar{V}=0

\frac{\partial}{\partial x}\left(5+a_{1} x+b_{1} y\right)+\frac{\partial}{\partial y}\left(8+a_{2} x+b_{2} y\right)=0

a_{1}+b_{2}=0

\therefore \quad \operatorname{div} \bar{V}=0

\frac{\partial}{\partial x}\left(5+a_{1} x+b_{1} y\right)+\frac{\partial}{\partial y}\left(8+a_{2} x+b_{2} y\right)=0

a_{1}+b_{2}=0

Question 8 |

Water (density=1000 kg/k^{3} ) at ambient temperature flows through a horizontral pipe of uniform cross section at the rate of kg/s. If the pressure drop across the pipe is 100 kPa, the minimum power required to pump the water across the pipe, in watts. is_________

10 | |

100 | |

1000 | |

10000 |

Question 8 Explanation:

\begin{aligned} \rho &=1000 \mathrm{kg} / \mathrm{m}^{3} \\ m &=1 \mathrm{kg} / \mathrm{s} \\ \Delta p &=100 \mathrm{kPa}=100 \times 10^{3} \mathrm{Pa} \\ P &=\rho Q g h_{f} \\ &=Q \Delta p \\ &=\frac{m}{\rho} \Delta p=\frac{1}{1000} \times 100 \times 10^{3} \\ &=100 \mathrm{W} \end{aligned}

Question 9 |

Which one of the following is Not a rotating machine?

Centrifugal pump | |

Gear pump | |

Jet pump | |

vane |

Question 9 Explanation:

In the given options all the pumps have rotating machine elements except Jet pump.

Centrifugal pump: It has rotating part eg.,impeller, Vane.

Gear Pump: In this pump there is gear mechanism which is rotating part.

Jet Pump: Jet pump utilizes ejector principle which have nozzle and diffuser not rotating parts.

Vane Pump: It consist of rotating disc which called as rotor in which number of radial slots are there where sliding vanes is inserted.

Centrifugal pump: It has rotating part eg.,impeller, Vane.

Gear Pump: In this pump there is gear mechanism which is rotating part.

Jet Pump: Jet pump utilizes ejector principle which have nozzle and diffuser not rotating parts.

Vane Pump: It consist of rotating disc which called as rotor in which number of radial slots are there where sliding vanes is inserted.

Question 10 |

Saturated steam at 100^{\circ}C condenses on the outside of a tube.cold fluid enters the tube at 20^{\circ}C and exits at 50^{\circ}C the value of the log mean Temperature Difference (LMTD) is _________^{\circ}C.

30 | |

80 | |

64 | |

84 |

Question 10 Explanation:

Temperature profiles are

\begin{aligned} \Delta T_{i}=100-20=80^{\circ} \mathrm{C} \\ \Delta T_{e}=100-50=50^{\circ} \mathrm{C} \end{aligned} (LMTD) (parallel or counter HE)

\begin{aligned} =\frac{\Delta T_{i}-\Delta T_{e}}{\ln \frac{\Delta T_{i}}{\Delta T_{e}}}=\frac{80-50}{\ln \frac{80}{50}} \\ =63.82^{\circ} \mathrm{C} \end{aligned}

\begin{aligned} \Delta T_{i}=100-20=80^{\circ} \mathrm{C} \\ \Delta T_{e}=100-50=50^{\circ} \mathrm{C} \end{aligned} (LMTD) (parallel or counter HE)

\begin{aligned} =\frac{\Delta T_{i}-\Delta T_{e}}{\ln \frac{\Delta T_{i}}{\Delta T_{e}}}=\frac{80-50}{\ln \frac{80}{50}} \\ =63.82^{\circ} \mathrm{C} \end{aligned}

There are 10 questions to complete.