List of practice Questions

Let \( M \) be any square matrix of arbitrary order \( n \) such that \( M^2 = 0 \) and the nullity of \( M \) is 6. Then the maximum possible value of \( n \) (in integer) is ________

Consider the usual inner product in \( \mathbb{R}^4 \). Let \( u \in \mathbb{R}^4 \) be a unit vector orthogonal to the subspace \[ S = \{(x_1, x_2, x_3, x_4)^T \in \mathbb{R}^4 \mid x_1 + x_2 + x_3 + x_4 = 0 \}. \] If \( v = (1, -2, 1, 1)^T \), and the vectors \( u \) and \( v - \alpha u \) are orthogonal, then the value of \( \alpha^2 \) (rounded off to two decimal places) is equal to ________

Let \( \{ B(t) \}_{t \geq 0} \) be a standard Brownian motion and let \( \Phi(\cdot) \) be the cumulative distribution function of the standard normal distribution. If \[ P\left( \left( B(2) + 2B(3) \right)>1 \right) = 1 - \Phi\left( \frac{1}{\sqrt{\alpha}} \right), \, \alpha>0, \] then the value of \( \alpha \) (in integer) is equal to ________

Let \( X \) and \( Y \) be two independent exponential random variables with \( E(X^2) = \frac{1}{2} \) and \( E(Y^2) = \frac{2}{9} \). Then \( P(X<2Y) \) (rounded off to two decimal places) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables from a normal distribution with mean 1 and variance 4. Let \( S_n = X_1^2 + X_2^2 + \dots + X_n^2 \). If \( \text{Var}(S_n) \) denotes the variance of \( S_n \), then the value of \[ \lim_{n \to \infty} \left( \frac{\text{Var}(S_n)}{n} - \left( \frac{E(S_n)}{n} \right)^2 \right) \quad \text{(in integer) is equal to} \, ________. \]

At a telephone exchange, telephone calls arrive independently at an average rate of 1 call per minute, and the number of telephone calls follows a Poisson distribution. Five time intervals, each of duration 2 minutes, are chosen at random. Let \( p \) denote the probability that in each of the five time intervals at most 1 call arrives at the telephone exchange. Then \( e^{10} p \) (in integer) is equal to ________

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with cumulative distribution function \( F(x) \). Let the empirical distribution function of the sample be \( F_n(x) \). The classical Kolmogorov-Smirnov goodness of fit test statistic is given by \[ T_n = \sqrt{n} D_n = \sqrt{n} \sup_{-\infty<x<\infty} | F_n(x) - F(x) |. \] Consider the following statements: \begin{enumerate} \item The distribution of \( T_n \) is the same for all continuous underlying distribution functions \( F(x) \). \item \( D_n \) converges to 0 almost surely, as \( n \to \infty \). \end{enumerate} Which one of the following statements is/are true?

Consider the following transition matrices \( P_1 \) and \( P_2 \) of two Markov chains: \[ P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}. \] Which of the following statements is correct?

Let \( X_1, X_2, \dots, X_{20} \) be a random sample of size 20 from \( N_6(\mu, \Sigma) \), with det(\(\Sigma\)) \(\neq 0\), and suppose both \(\mu\) and \(\Sigma\) are unknown. Let \[ \bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad S = \frac{1}{19} \sum_{i=1}^{20} (X_i - \bar{X})(X_i - \bar{X})^T. \] Consider the following two statements: \begin{enumerate} \item The distribution of \( 19S \) is \( W_6(19, \Sigma) \) (Wishart distribution of order 6 with 19 degrees of freedom). \item The distribution of \( (X_3 - \mu)^T S^{-1} (X_3 - \mu) \) is \( \chi^2_6 \) (Chi-square distribution with 6 degrees of freedom). \end{enumerate} Then which of the above statements is/are true?

Let \( X_1, X_2, \dots, X_{18} \) be a random sample from the distribution \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta} e^{-x^2/\theta}, & x>0,\\ 0, & x \leq 0. \end{cases} \] What is the maximum likelihood estimator (MLE) of \( \theta \)?

Let \( X_1, X_2, \dots, X_n \) be a random sample from a population \( f(x; \theta) \), where \( \theta \) is a parameter. Then which one of the following statements is NOT true?

A random sample \( X_1, X_2, \dots, X_6 \) is taken from a Bernoulli distribution with the parameter \( \theta \). The null hypothesis \( H_0: \theta = \frac{1}{2} \) is to be tested against the alternative hypothesis \( H_1: \theta>\frac{1}{2} \), based on the statistic \( Y = \sum_{i=1}^{6} X_i \). If the value of \( Y \) corresponding to the observed sample values is 4, then the p-value of the test is ________?

Let \(M\) be a 2 \(\times\) 2 real matrix such that \((I + M)^{-1} = I - \alpha M\), where \(\alpha\) is a non-zero real number and \(I\) is the 2 \(\times\) 2 identity matrix. If the trace of the matrix \(M\) is 3, then the value of \(\alpha\) is

Let \(\{X(t)\}_{t\geq 0}\) be a linear pure death process with death rate \(\mu_i = 5i\), \(i = 0, 1, \dots, N\), \(N \geq 1\). Suppose that \(p_i(t) = P(X(t) = i)\). Then the system of forward Kolmogorov’s equations is

Let \( S^2 \) be the variance of a random sample of size \( n>1 \) from a normal population with an unknown mean \( \mu \) and an unknown finite variance \( \sigma^2>0 \). Consider the following statements:
(I) \( S^2 \) is an unbiased estimator of \( \sigma^2 \), and \( S \) is an unbiased estimator of \( \sigma \).
(II) \( \frac{n-1}{n} S^2 \) is a maximum likelihood estimator of \( \sigma^2 \), and \( \sqrt{\frac{n-1}{n}} S \) is a maximum likelihood estimator of \( \sigma \).
Which of the above statements is/are true?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function defined by \[ f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Find the value of \( \frac{\partial f}{\partial x} \) at \( (0, 0) \).

In a steam power plant based on Rankine cycle, steam is initially expanded in a high-pressure turbine. The steam is then reheated in a reheater and finally expanded in a low-pressure turbine. The expansion work in the high-pressure turbine is 400 kJ/kg and in the low-pressure turbine is 850 kJ/kg, whereas the pump work is 15 kJ/kg. If the cycle efficiency is 32%, the heat rejected in the condenser is ________________ kJ/kg (round off to 2 decimal places).

An engine running on an air standard Otto cycle has a displacement volume of 250 cm\(^3\) and a clearance volume of 35.7 cm\(^3\). The pressure and temperature at the beginning of the compression process are 100 kPa and 300 K, respectively. Heat transfer during constant-volume heat addition process is 800 kJ/kg. The specific heat at constant volume is 0.718 kJ/kg.K and the ratio of specific heats at constant pressure and constant volume is 1.4. Assume the specific heats to remain constant during the cycle. The maximum pressure in the cycle is ________________ kPa (round off to the nearest integer).

A steady two-dimensional flow field is specified by the stream function \[ \psi = k x^3 y, \] where \( x \) and \( y \) are in meter and the constant \( k = 1 \, \text{m}^2 \text{s}^{-1} \). The magnitude of acceleration at a point \( (x, y) = (1 \, \text{m}, 1 \, \text{m}) \) is ________________ m/s² (round off to 2 decimal places).

Consider a solid slab (thermal conductivity, \( k = 10 \, \text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1} \)) with thickness 0.2 m and of infinite extent in the other two directions as shown in the figure. Surface 2, at 300 K, is exposed to a fluid flow at a free stream temperature (\( T_{\infty} \)) of 293 K, with a convective heat transfer coefficient (\( h \)) of 100 W\( \cdot\text{m}^{-2}\cdot\text{K}^{-1} \). Surface 2 is opaque, diffuse and gray with an emissivity (\( \varepsilon \)) of 0.5 and exchanges heat by radiation with very large surroundings at 0 K. Radiative heat transfer inside the solid slab is neglected. The Stefan-Boltzmann constant is \( 5.67 \times 10^{-8} \, \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4} \). The temperature \( T_1 \) of Surface 1 of the slab, under steady-state conditions, is ________________ K (round off to the nearest integer).

During open-heart surgery, a patient’s blood is cooled down to 25°C from 37°C using a concentric tube counter-flow heat exchanger. Water enters the heat exchanger at 4°C and leaves at 18°C. Blood flow rate during the surgery is 5 L/minute.
Use the following fluid properties: \[ \begin{array}{|c|c|c|} \hline \text{Fluid} & \text{Density (kg/m}^3\text{)} & \text{Specific heat (J/kg-K)}
\hline \text{Blood} & 1050 & 3740
\text{Water} & 1000 & 4200
\hline \end{array} \] Effectiveness of the heat exchanger is ________________ (round off to 2 decimal places).

A thin-walled cylindrical pressure vessel has mean wall thickness of \( t \) and nominal radius of \( r \). The Poisson's ratio of the wall material is \( \frac{1}{3} \). When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became \( \bar{t} \). The corresponding change in the wall thickness of the cylindrical portion, i.e. \( 100 \times \frac{\bar{t} - t}{t} \), is ________________% (round off to 3 decimal places).

A schematic of an epicyclic gear train is shown in the figure. The sun (gear 1) and planet (gear 2) are external, and the ring gear (gear 3) is internal. Gear 1, gear 3 and arm OP are pivoted to the ground at O. Gear 2 is carried on the arm OP via the pivot joint at P, and is in mesh with the other two gears. Gear 2 has 20 teeth and gear 3 has 80 teeth. If gear 1 is kept fixed at 0 rpm and gear 3 rotates at 900 rpm counter clockwise (ccw), the magnitude of angular velocity of arm OP is ________________ rpm (in integer).

Under orthogonal cutting condition, a turning operation is carried out on a metallic workpiece at a cutting speed of 4 m/s. The orthogonal rake angle of the cutting tool is \(5^\circ\). The uncut chip thickness and width of cut are 0.2 mm and 3 mm, respectively. In this turning operation, the resulting friction angle and shear angle are \(45^\circ\) and \(25^\circ\), respectively. If the dynamic yield shear strength of the workpiece material under this cutting condition is 1000 MPa, then the cutting force is ________________ N (round off to one decimal place).

A 1 mm thick cylindrical tube, 100 mm in diameter, is orthogonally turned such that the entire wall thickness of the tube is cut in a single pass. The axial feed of the tool is 1 m/minute and the specific cutting energy (\( u \)) of the tube material is 6 J/mm\(^3\). Neglect contribution of feed force towards power. The power required to carry out this operation is ________________ kW (round off to one decimal place).