List of practice Questions

The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $2 \times 10^{-9}$ J per molecule is :

If 'f' denotes the ratio of the number of nuclei decayed Nd to the number of nuclei at t=0 N0 then for a collection of radioactive nuclei, the rate of change of 'f' with respect to time is given as: [$\lambda$ is the radioactive decay constant]

The truss shown is subjected to a force $ P $. All members of the truss have the same length $ L $. The reaction at A and force in member AB are

For a state of plane strain, the normal strains are given by $\epsilon_{xx} = 1000 \times 10^{-6}, \epsilon_{yy} = 200 \times 10^{-6}, $ and the maximum shear strain is $ \gamma_{\text{max}} = 1000 \times 10^{-6}. $The value of shear strain$ \, \gamma_{xy} \, $for this strain state is

A force $ F = 40 \, \text{kN} $ is applied on the hook as shown. The equivalent force-couple system at B is

The figure shows a structure with supports. The correct free body diagram when the supports are removed is

Match the different transformations/reactions in Column I with the most suitable information in Column II.

A magnet of mass 50 g has a magnetic moment of $ 4.2 \times 10^{-7} \, \text{A m}^2 $. The density of the magnet is 7.2 g/cm³. The intensity of magnetization in A/m is $\underline{\hspace{2cm}}$ (round off to 3 decimal places).

In the context of scanning electron microscopy, match the information in Column I with the most appropriate information in Column II.

Match the heat treatment processes given in Column I with the most suitable outcomes in Column II.

A co-joined cross-ply laminate composite, as shown in the figure, is distorted upon heating. What are the resultant shapes of edges XY and YZ?

Fe - 10 atom % C austenite (fcc), having no Fe vacancies, has a lattice parameter of 4 Å. The density of austenite in g/cm³ is $\underline{\hspace{2cm}}$ (round off to 2 decimal places).

An element transforms from $ \alpha $ to $ \beta $ at 773 K and 1 atm pressure with 912 J mol$^{-1}$ as enthalpy of transformation. The molar volumes of $ \alpha $ and $ \beta $ phases are 7.377 cm³ and 7.317 cm³, respectively. Assume that the difference in molar volumes of $ \alpha $ and $ \beta $ is independent of pressure. The pressure (in atm) required for $ \alpha $ to $ \beta $ transformation to occur at 723 K is $\underline{\hspace{2cm}}$ (round off to nearest integer).

The two-dimensional velocity field $ \mathbf{V} $ of a flow in a Cartesian coordinate system is given in dimensionless form by $ \mathbf{V} = (x^2 - axy) \hat{i} + \left( bxy - \frac{y^2}{2} \right) \hat{j} $. Here, $ \hat{i} $ and $ \hat{j} $ are the unit vectors along the $ x $ and $ y $ directions respectively, $ a $ and $ b $ are independent of $ x $, $ y $ and time. If the flow is incompressible, then the value of $ (a - b) $, up to one decimal place, is $\underline{\hspace{1cm}}$.

A two-dimensional Eulerian velocity field is given (in m/s) by $ \mathbf{V} = \left[ (\sqrt{5})x \right] \hat{i} - \left[ (\sqrt{12})y \right] \hat{j} $, where $ x $ and $ y $ are the coordinates (in meters) in a Cartesian coordinate system. The magnitude of the acceleration (in m/s$^2$, up to one decimal place) of a fluid particle at $ x = 1 \, \text{m} $ and $ y = -1 \, \text{m} $ is $\underline{\hspace{2cm}}$.

Let the transformation $ y(x) = e^x \nu(x) $ reduce the ordinary differential equation $ x \frac{d^2y}{dx^2} + 2(1 - x) \frac{dy}{dx} + (x - 2) y = 0 $, where $ \alpha, \beta, \gamma $ are real constants. Then, the arithmetic mean of $ \alpha, \beta, \gamma $ is $\underline{\hspace{2cm}}$ (round off to three decimal places).

Let $ I $ denote the identity matrix of order 7, and $ A $ be a $ 7 \times 7 $ real matrix having characteristic polynomial $ C_A(\lambda) = \lambda^2 (\lambda - 1)^\alpha (\lambda + 2)^\beta $, where $ \alpha $ and $ \beta $ are positive integers. If $ A $ is diagonalizable and $ \text{rank}(A) = \text{rank}(A + 2I) $, then $ \text{rank}(A - I) $ is $\underline{\hspace{2cm}}$ (in integer).

A person, who speaks the truth 3 out of 4 times, throws a fair die with six faces and informs that the outcome is 5. The probability that the outcome is really 5 is $\underline{\hspace{2cm}}$ (round off to three decimal places).

Let $ C_1 $ be the line segment from $ (0, 1) $ to $ \left( \frac{4}{5}, \frac{3}{5} \right) $, and let $ C_2 $ be the arc of the circle $ x^2 + y^2 = 1 $ from $ (0, 1) $ to $ \left( \frac{4}{5}, \frac{3}{5} \right) $. If \[ \alpha = \int_{C_1} \left( \frac{2x}{y} \hat{i} + \frac{1 - x^2}{y^2} \hat{j} \right) \cdot d\vec{r} \] and \[ \beta = \int_{C_2} \left( \frac{2x}{y} \hat{i} + \frac{1 - x^2}{y^2} \hat{j} \right) \cdot d\vec{r}, \] where $ \vec{r} = x \hat{i} + y \hat{j} $, then the value of $ \alpha^2 + \beta^2 $ is $\underline{\hspace{2cm}}$ (round off to two decimal places).

Water discharges steadily from a large reservoir through a long pipeline, as shown in the figure. The Darcy friction factor in the pipe is 0.02. The pipe diameter is 20 cm and the discharge of water is 360 m$^3$/h. Water level in the reservoir is 10 m and acceleration due to gravity is $ g = 10 \, \text{m/s}^2 $. If minor losses are negligible, the length $ L $ (in meters, up to one decimal place) of the pipeline is $\underline{\hspace{2cm}}$.
\includegraphics[width=0.75\linewidth]{imager21.png}

Water is flowing with a flow rate $ Q $ in a horizontal circular pipe. Due to the low pressure created at the venturi section (Section-1 in the figure), water from a reservoir is drawn upward using a connecting pipe as shown in the figure. Take acceleration due to gravity $ g = 10 \, \text{m/s}^2 $. The flow rate $ Q = 0.1 \, \text{m}^3/\text{s} $, $ D_1 = 8 \, \text{cm} $, and $ D_2 = 20 \, \text{cm} $. The maximum height $ h $ (in meters, up to one decimal place) of the venturi from the reservoir just sufficient to raise the liquid up to Section-1 is $\underline{\hspace{1cm}}$. \includegraphics[width=0.75\linewidth]{image22.png}

Let \[ S = \left\{ AX : A = \begin{bmatrix} 2 & -4 \\ 1 & 1 \\ 1 & -1 \end{bmatrix}, \, X = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right\}. \] If \[ \begin{bmatrix} -1 \\ \alpha \\ 1 \end{bmatrix} \in S, \text{ then the value of } \alpha \text{ is} \]

Let $ C $ be the boundary of the region $ R : 0 \leq x \leq \pi, 0 \leq y \leq \sin x $ in the xy-plane and $ \alpha $ be the area of the region $ R $. If $ C $ traverses once in the counterclockwise direction, then the value of the line integral \[ \int_C (2y \, dx + 5x \, dy) \] is equal to

Given that $ i = \sqrt{-1} $. The value of \[ \lim_{z \to e^{i \pi/3}} \frac{z^3 + 1}{z^4 + z^2 + 1} \] is

Let $ f(x) $ be a non-negative continuous function of real variable $ x $. If the area under the curve $ y = f(x) $ from $ x = 0 $ to $ x = a $ is $ \frac{a^2}{2} + \frac{a}{2} \sin a + \frac{\pi}{2} \cos a - \frac{\pi}{2} $, then the value of $ f \left( \frac{\pi}{2} \right) $ is $\underline{\hspace{2cm}}$ (round off to one decimal place).