List of practice Questions

For α, β ∈ R , α ≠ β, if −2 and 5 are the eigenvalues of the matrix 𝑀=[1−𝛼1+𝛽𝛽 𝛼+𝛽] and 𝑋 = [𝑥1 𝑥2 ] is an eigenvector of 𝑀 associated to −2, then\(M=\begin{bmatrix} 1-α& 1+β\\ β & a+B& \end{bmatrix}\)
and 𝑋 = \(\begin{bmatrix}  X_1\\ X_2 \end{bmatrix}\) is an eigenvector of 𝑀 associated to −2, then

A firm has production function 𝑦=𝐾 ^0.5𝐿 ^0.5 and faces wage rate 𝑤=4 and rental rate of capital 𝑟=4. The firm’s marginal cost is equal to ______ (in integer).

The plot of the function f(x) = ||x| - 1| is

Let 𝑦̂=5.5+3.2 𝑥 be an estimated regression equation using a large sample. The 95% confidence interval of the coefficient of 𝑥 is [0.26, 6.14] and 𝑅² = 0.26. The standard error of the estimated coefficient is _______ (round off to 1 decimal place).

Let 𝜋 be the proportion of a population vaccinated against a disease. An estimate ̂𝜋 = 0.64 is found using a sample of 100 individuals from the population. The 𝑧 test statistic for the null hypothesis 𝐻₀ : 𝜋 = 0.58 is ________ (round off to 2 decimal places).

A system has N spins, where each spin is capable of existing in 4 possible states. The difference in entropy of disordered states (where all possible spin configurations are equally probable) and ordered states is

An industry has 3 firms (1, 2 and 3) in Cournot competition. They have no fixed costs, and their constant marginal costs are respectively\(c_1=\frac{9}{30} , 𝑐_2 =\frac{ 10}{30} , 𝑐_3 = \frac{11}{30} .\)
They face an industry inverse demand function 𝑃=1−𝑄, where 𝑃 is the market price and 𝑄 is the industry output (sum of outputs of the 3 firms). Suppose that 𝑄 𝑐 is the industry output under Cournot-Nash equilibrium. Then (𝑄 ^𝑐 )⁻¹ is equal to _______ (in integer).

A consumer has utility function 𝑢(𝑥₁, 𝑥₂ ) = max {0.5 𝑥₁ , 0.5 𝑥₂} + min{𝑥₁ , 𝑥₂}.
She has some positive income 𝑦, and faces positive prices 𝑝₁, 𝑝₂ for goods 1 and 2 respectively. Suppose 𝑝₂ = 1. There exists a lowest price 𝑝̅̅₁ such that if 𝑝₁ > 𝑝̅̅₁ then the unique utility maximizing choice is to buy ONLY good 2. Then 𝑝̅̅₁ is _________ (in integer).

Temperature (T) dependence of the total specific heat (C_v) for a two dimensional metallic solid at low temperatures is

An economy has three firms: X, Y and Z. Every unit of output that X produces creates a benefit of INR 700 for Y and a cost of INR 300 for Z. Firm X’s cost curve is
\(𝐶(𝑄_𝑋) = 2𝑄^2_X + 10\)
where 𝐶 represents cost and 𝑄_𝑋 is the output. The market price for the output of X is INR 1600 per unit. The difference between the socially optimal output and private profit maximizing output of firm X (in INR) is _____ (in integer).

For the following circuit, choose the correct waveform corresponding to the output signal (V_out). Given V_in = 5 sin(200πt) V, forward bias voltage of the diodes (D and Z) = 0.7 V and reverse Zener voltage = 3 V.

Let ∫ sin⁹ 𝑥 cos(11𝑥) 𝑑𝑥 = cos(10𝑥) 𝑓(𝑥) + 𝑐 , where 𝑐 is a constant. If 𝑓 ′′ (\(\frac{\pi}{4}\)) − 𝑘𝑓 ′(\(\frac{ \pi}{4}\)) = 0, then 𝑘 is equal to ______ (in integer)

If the ground state energy of a particle in an infinite potential well of width L₁ is equal to the energy of the second excited state in another infinite potential well of width L₂, then the ratio \(\frac{L_1}{L_2}\) is equal to

Let 𝑀 = \(\begin{bmatrix} K & 1 & 1 \\ 1 & K & 1 \\ 1 & 1 & K \end{bmatrix}\)and I₃ be the identity matrix of order 3. If the rank of the matrix 10 𝐼₃ − 𝑀 is 2 then 𝑘 is equal to _____ (in integer).

In the given circuit, with an ideal op-amp for what value of \(\frac{R_1}{R_2}\) the output of the amplifier V_out = V₂ - V₁ ?

In a two period model, a consumer is maximizing the present discounted utility
𝑊𝑡 = ln(𝑐_𝑡) +\(\frac{ 1}{ 1 + }\) ln(𝑐_𝑡+1) 
with respect to 𝑐𝑡 and 𝑐𝑡+1 and subject to the following budget constraint 
\(𝑐_𝑡 +\frac{ 𝑐_𝑡+1}{ 1 + 𝑟} ≤ 𝑦_𝑡 +\frac{ 𝑦_𝑡+1 }{1 + 𝑟 }\)
where 𝑐𝑖 and 𝑦𝑖 are the consumption and income in period 𝑖 (𝑖 = 𝑡,𝑡 + 1) respectively, 𝜃 ∈ [0, ∞) is the time discount rate and 𝑟 ∈ [0, ∞) is the rate of interest. Suppose, consumer is in the interior equilibrium and 𝜃 = 0.05 and 𝑟 = 0.08. In equilibrium, the ratio \(\frac{𝑐_𝑡+1}{ 𝑐_𝑡}\) is equal to _____ (round off to 2 decimal places).

A projectile of mass m is moving in the vertical x-y plane with the origin on the ground and y-axis pointing vertically up. Taking the gravitational potential energy to be zero on the ground, the total energy of the particle written in planar polar coordinates (r, θ) is (here g is the acceleration due to gravity)

The portfolio of an investment firm comprises of two risky assets, 𝑆 and 𝑇, whose returns are denoted by random variables 𝑅_𝑠 and 𝑅_𝑇 respectively. The mean, the variance and the covariance of the returns are
𝐸(𝑅_𝑠 ) = 0.08, 𝑉𝑎𝑟(𝑅_𝑠 ) = 0.07, 
𝐸(𝑅_𝑇 ) = 0.05, 𝑉𝑎𝑟(𝑅_𝑇 ) = 0.05, 𝐶𝑜𝑣(𝑅𝑠 , 𝑅_𝑇 ) = 0.04. 
Let 𝑤 be the proportion of assets allotted to 𝑆 so that the return from the portfolio is 𝑅 = 𝑤𝑅_𝑠 + (1 − 𝑤)𝑅_𝑇 . The value of 𝑤 which minimizes 𝑉𝑎𝑟(𝑅) is _____(round off to 2 decimal places)

A small bar magnet is dropped through different hollow copper tubes with same length and inner diameter but with different outer diameter. The variation in the time (t) taken for the magnet to reach the bottom of the tube depends on its wall thickness (d) as

A number 𝑥 is randomly chosen from the set of the first 100 natural numbers. The probability that 𝑥 satisfies the condition \(𝑥 + \frac{300}{𝑥} > 65\) is ______ (round off to 2 decimal places).

Two digital inputs A and B are given to the following circuit. For A = 1, B = 0, the values of X and Y are :

For K∈R, let 𝑓(𝑥)=𝑥⁴+2𝑥³+𝑘𝑥²−𝑘, X∈R. If 𝑥=\(\frac{3}{2}\) is a point of local minima of 𝑓 and 𝑚 is the global minimum value of 𝑓 then 𝑓(0) − 𝑚 is equal to _______ (in integer).

If (𝑥^∗ , 𝑦^∗) is the optimal solution of the problem
maximize 𝑓(𝑥, 𝑦) = 100−𝑒^−𝑥−𝑒 ^−𝑦 
subject to 𝑒𝑥+𝑦=\(\frac{𝑒}{𝑒−1},\) 𝑥 ≥ 0, 𝑦 ≥ 0. 
Then \(\sqrt{\frac{y^*}{x^*}}\) is equal to ________ (round off to 2 decimal places).

For a cubic unit cell, the dashed arrow in which of the following figures represents the direction [220] ?

Which of the following statements about the viscosity of a dilute ideal gas is correct ?