List of practice Questions

A cylindrical rod of length \(h\) and diameter \(d\) is placed inside a cubic enclosure of side length \(L\). \(S\) denotes the inner surface of the cube. The view-factor \(F_{S-S}\) is

A linear transformation maps a point \((x, y)\) in the plane to the point \((\hat{x}, \hat{y})\) according to the rule \[ \hat{x} = 3y, \hat{y} = 2x. \] Then, the disc \(x^2 + y^2 \le 1\) gets transformed to a region with an area equal to ................ (Rounded off to two decimals)
Use \(\pi = 3.14\).

The value of k that makes the complex-valued function \[ f(z) = e^{-kx} (\cos 2y - i \sin 2y) \] analytic, where \(z = x + iy\), is ................ (Answer in integer)

The braking system shown in the figure uses a belt to slow down a pulley rotating in the clockwise direction by the application of a force P. The belt wraps around the pulley over an angle \(\alpha = 270\) degrees. The coefficient of friction between the belt and the pulley is 0.3. The influence of centrifugal forces on the belt is negligible. During braking, the ratio of the tensions \(T_1\) to \(T_2\) in the belt is equal to ................ (Rounded off to two decimal places)
Take \(\pi = 3.14\).

Consider a counter-flow heat exchanger with the inlet temperatures of two fluids (1 and 2) being \(T_{1, in} = 300\) K and \(T_{2, in} = 350\) K. The heat capacity rates of the two fluids are \(C_1 = 1000\) W/K and \(C_2 = 400\) W/K, and the effectiveness of the heat exchanger is 0.5. The actual heat transfer rate is ............... kW. (Answer in integer)

Which one of the options given is the inverse Laplace transform of \(\frac{1}{s^3 - s}\)?
\(u(t)\) denotes the unit-step function.

Two surfaces P and Q are to be joined together. In which of the given joining operation(s), there is no melting of the two surfaces P and Q for creating the joint?

In a metal casting process to manufacture parts, both patterns and moulds provide shape by dictating where the material should or should not go. Which of the option(s) given correctly describe(s) the mould and the pattern?

The principal stresses at a point P in a solid are 70 MPa, -70 MPa and 0. The yield stress of the material is 100 MPa. Which prediction(s) about material failure at P is/are CORRECT?

Air (density = 1.2 kg/m³, kinematic viscosity = 1.5×10⁻⁵ m²/s) flows over a flat plate with a free-stream velocity of 2 m/s. The wall shear stress at a location 15 mm from the leading edge is \(\tau_w\). What is the wall shear stress at a location 30 mm from the leading edge?

Consider an isentropic flow of air (ratio of specific heats = 1.4) through a duct as shown in the figure. The variations in the flow across the cross-section are negligible. The flow conditions at Location 1 are given as follows:
\(P_1 = 100\) kPa, \(\rho_1 = 1.2\) kg/m³, \(u_1 = 400\) m/s
The duct cross-sectional area at Location 2 is given by \(A_2 = 2A_1\), where \(A_1\) denotes the duct cross-sectional area at Location 1. Which one of the given statements about the velocity \(u_2\) and pressure \(P_2\) at Location 2 is TRUE?

Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on

A heat engine extracts heat (\(Q_H\)) from a thermal reservoir at a temperature of 1000 K and rejects heat (\(Q_L\)) to a thermal reservoir at a temperature of 100 K, while producing work (\(W\)). Which one of the combinations of [\(Q_H\), \(Q_L\) and \(W\)] given is allowed?

A machine produces a defective component with a probability of 0.015. The number of defective components in a packed box containing 200 components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are

The minute-hand and second-hand of a clock cross each other ............... times between 09:15:00 AM and 09:45:00 AM on a day.

In a recently held parent-teacher meeting, the teachers had very few complaints about Ravi. After all, Ravi was a hardworking and kind student. Incidentally, almost all of Ravi's friends at school were hardworking and kind too. But the teachers drew attention to Ravi's complete lack of interest in sports. The teachers believed that, along with some of his friends who showed similar disinterest in sports, Ravi needed to engage in some sports for his overall development.
Based only on the information provided above, which one of the following statements can be logically inferred with certainty?

Consider the following inequalities \[ p^2 - 4q<4 \] \[ 3p + 2q<6 \] where p and q are positive integers. The value of \((p+q)\) is ................

Which one of the sentence sequences in the given options creates a coherent narrative?
(i) I could not bring myself to knock.
(ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.
(iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.
(iv) I decided I would rather wander down the passage.

He did not manage to fix the car himself, so he ............... in the garage.

Planting : Seed :: Raising : ............... (By word meaning)

Let \(y : (\sqrt{\frac{2}{3}}, ∞) → \R\) be the solution of
(2x − y)y' + (2y − x) = 0,
y(1) = 3.
Then

Let A and B be orthogonal and det A+det B=0. Then

Let \(GL_2(\mathbb{C})\) denote the group of \(2 \times 2\) invertible complex matrices with usual matrix multiplication. For \(S, T \in GL_2(\mathbb{C})\), \(\langle S, T \rangle\) denotes the subgroup generated by \(S\) and \(T\). Let
\[ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in GL_2(\mathbb{C}) \] and \(G_1, G_2, G_3\) be three subgroups of \(GL_2(\mathbb{C})\) given by
\[ G_1 = \langle S, T_1 \rangle, \text{ where } T_1 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}, \]
\[ G_2 = \langle S, T_2 \rangle, \text{ where } T_2 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \]
\[ G_3 = \langle S, T_3 \rangle, \text{ where } T_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. \]
Let \(Z(G_i)\) denote the center of \(G_i\) for \(i = 1, 2, 3\).
Which of the following statements is correct?

Let \(\ell^2 = \{(x_1, x_2, x_3, \dots) : x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^\infty x_n^2 < \infty\}\).
For a sequence \((x_1, x_2, x_3, \dots) \in \ell^2\), define \[ ||(x_1, x_2, x_3, \dots)||_2 = \left(\sum_{n=1}^\infty x_n^2\right)^{1/2}. \]
Let \(S: (\ell^2, ||.||_2) \to (\ell^2, ||.||_2)\) and \(T: (\ell^2, ||.||_2) \to (\ell^2, ||.||_2)\) be defined by
\[ S(x_1, x_2, x_3, \dots) = (y_1, y_2, y_3, \dots), \text{ where } y_n = \begin{cases} 0, & n=1 \\ x_{n-1}, & n \ge 2 \end{cases} \]
\[ T(x_1, x_2, x_3, \dots) = (y_1, y_2, y_3, \dots), \text{ where } y_n = \begin{cases} 0, & n \text{ is odd} \\ x_n, & n \text{ is even} \end{cases} \]
Then