List of practice Questions

For a quantum particle confined inside a cubic box of side \( L \), the ground state energy is given by \( E_0 \). The energy of the first excited state is
In a heat engine based on the Carnot cycle, heat is added to the working substance at constant
Isothermal compressibility is given by
For using a transistor as an amplifier, choose the correct option regarding the resistances of base-emitter \( (R_{BE}) \) and base-collector \( (R_{BC}) \) junctions.
A thin lens of refractive index \( \frac{3}{2} \) is kept inside a liquid of refractive index \( \frac{4}{3} \). If the focal length of the lens in air is 10 cm, then its focal length inside the liquid is
The Fermi-Dirac distribution function \( [n(\epsilon)] \) is \[ n(\epsilon) = \frac{1}{e^{(\epsilon - \epsilon_F) / k_B T} + 1} \] where \( k_B \) is the Boltzmann constant, \( T \) is the temperature and \( \epsilon_F \) is the Fermi energy.
If \( \phi(x,y,z) \) is a scalar function which satisfies the Laplace equation, then the gradient of \( \phi \) is
The function \( f(x) = \frac{8x}{x^2 + 9} \) is continuous everywhere except at
Let the Taylor polynomial of degree 20 for \( \frac{1}{(1-x)^3} \) at \( x = 0 \) be

\[ \sum_{n=0}^{20} a_n x^n \]

Then \( a_{15} \) equals
If

\[ P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{and} \quad 6P^{-1} = aI_3 + bP - P^2, \quad \text{then the ordered pair} \quad (a,b) \quad \text{is} \]

Let \( X_1, X_2, \dots, X_n \) be a random sample from a continuous distribution with the probability density function \( f(x) \). To test the hypothesis \( H_0: f(x) = e^{-x^2} \) against \( H_1: f(x) = e^{-2|x|} \), the rejection region of the most powerful size \( \alpha \) test is of the form, for some \( c>0 \),
Let \( P \) be a \( 2 \times 2 \) real matrix such that every non-zero vector in \( \mathbb{R}^2 \) is an eigenvector of \( P \). Suppose that \( \lambda_1 \) and \( \lambda_2 \) denote the eigenvalues of \( P \) and \( P \left[ \begin{array}{c} \sqrt{2} \\ \sqrt{3} \end{array} \right] = \left[ \begin{array}{c} 2 \\ t \end{array} \right] \) for some \( t \in \mathbb{R} \). Which of the following statements is (are) TRUE?
Let \( X \) be a random variable with the cumulative distribution function

\[ F(x) = \begin{cases} 0, & x < 0 \\ 1 + x^2, & 0 \leq x < 1 \\ \frac{10}{3} + x^2, & 1 \leq x < 2 \\ 1, & x \geq 2 \end{cases} \]

Which of the following statements is (are) TRUE?
Let \( X \) and \( Y \) be i.i.d. \( \text{Exp}(\lambda) \) random variables. If \( Z = \max\{X - Y, 0\} \), then which of the following statements is (are) TRUE?

Let the discrete random variables \( X \) and \( Y \) have the joint probability mass function 

Which of the following statements is (are) TRUE? 
 

Let \( X_1, X_2, \dots \) be a sequence of i.i.d. continuous random variables with the probability density function 

If \( S_n = X_1 + X_2 + \dots + X_n \) and \( \bar{X}_n = \frac{S_n}{n} \), then the distributions of which of the following sequences of random variables converge(s) to a normal distribution with mean 0 and a finite variance? 
 

Let \( X_1, X_2, \dots, X_n \) be a random sample from a continuous distribution with the probability density function for \( \lambda>0 \) 

 To test the hypothesis \( H_0: \lambda = \frac{1}{2} \) against \( H_1: \lambda = \frac{3}{4} \) at the level \( \alpha \) (with \( 0<\alpha<1 \)), which of the following statements is (are) TRUE? 
 

Let \( f: [0, 2] \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq |x - y|^{4/3} \) for all \( x, y \in [0, 2] \). If \( \int_0^2 f(x) \, dx = \frac{2}{3} \), then

 

If

\[ \begin{pmatrix} \frac{\sqrt{5}}{3} & -\frac{2}{3} & c \\ \frac{2}{3} & \frac{\sqrt{5}}{3} & d \\ a & b & 1 \end{pmatrix} \]

is a real orthogonal matrix, then \( a^2 + b^2 + c^2 + d^2 \) equals ...............
Let \( 0, 1, 0, 0, 1 \) be the observed values of a random sample of size five from a discrete distribution with the probability mass function \( P(X = 1) = 1 - P(X = 0) = 1 - e^{-\lambda} \), where \( \lambda>0 \). The method of moments estimate (round off to 2 decimal places) of \( \lambda \) equals .............
For any real number \( y \), let \( \lfloor y \rfloor \) be the greatest integer less than or equal to \( y \) and let \( \{ y \} = y - \lfloor y \rfloor \). For \( n = 1, 2, \dots \), and for \( x \in \mathbb{R} \), let

\[ f_{2n}(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases} \quad \text{and} \quad f_{2n-1}(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases} \]

Then

\[ \lim_{x \to 0} \sum_{k=1}^{100} f_k(x) \text{ equals ............}. \]

If \( ad - bc = 2 \) and \( ps - qr = 1 \), then the determinant of

\[ \begin{pmatrix} a & b & 0 & 0 \\ 3 & 10 & 2p & q \\ c & d & 0 & 0 \\ 2 & 7 & 2r & s \end{pmatrix} \]

equals ............
Let \( X \) be a continuous random variable with the probability density function

\[ f(x) = \begin{cases} ax^2, & 0 < x < 1 \\ bx^{-4}, & x \geq 1 \\ 0, & \text{otherwise} \end{cases} \]

where \( a \) and \( b \) are positive real numbers. If \( E(X) = 1 \), then \( E(X^2) \) equals ................
Let \( X_1, X_2, X_3 \) be a random sample from a continuous distribution with the probability density function

\[ f(x) = \begin{cases} e^{-(x-\mu)}, & x \geq \mu \\ 0, & \text{otherwise} \end{cases} \]

Let \( X_{(1)} = \min\{X_1, X_2, X_3\} \) and \( c > 0 \) be a real number. Then \( (X_{(1)} - c, X_{(1)}) \) is a 97% confidence interval for \( \mu \), if \( c \) (round off to 2 decimal places) equals ................
Let \( X \) be a discrete random variable with the probability mass function

\[ P(X = n) = \begin{cases} \frac{-2c}{n}, & n = -1, -2 \\ d, & n = 0 \\ \frac{cn}{n}, & n = 1, 2 \\ 0, & \text{otherwise} \end{cases} \]

where \( c \) and \( d \) are positive real numbers. If \( P(|X| \leq 1) = \frac{3}{4} \), then \( E(X) \) equals