List of practice Questions

Let p(x) = x⁵⁷ + 3x¹⁰ - 21x³ + x² + 21 and
\(q(x)=p(x)+\sum\limits_{j=1}^{57}p^{(j)}(x) \ \text{for all }x \in \R,\)
where p^(j)(x) denotes the j^th derivative of p(x). Then the function q admits

Let R₁ and R₂ be the radii of convergence of the power series \(\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}\) and \(\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}\), respectively. Then

Let S and T be non-empty subsets of \(\R^2\), and W be a non-zero proper subspace of \(\R^2\). Consider the following statements :
I. If span(S) = \(\R^2\) , then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

Let S be the triangular region whose vertices are (0, 0), \((0,\frac{\pi}{2})\) and \((\frac{\pi}{2},0)\). The value of \(\iint\limits_S\sin(x)\cos(y)dx\ dy\) is equal to ________. (rounded off to two decimal places)

Let V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

Let W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

Let y : (1, ∞) → \(\R\) be the solution of the differential equation
\(y"-\frac{2y}{(1-x)^2}=0\)
satisfying y(2) = 1 and\(\lim\limits_{x→∞}y(x) = 0\). Then y(3) is equal to __________. (rounded off to two decimal places)

The limit
\(\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)\)
is

The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.

The sum of the series \(\sum\limits_{n=1}^{\infin}\frac{2n+1}{(n^2+1)(n^2+2n+2)}\) is equal to _________.(rounded off to two decimal places)

The value of
\(\int^1_0\int_0^{1-x}\cos(x^3+y^2)dy\ dx-\int^1_0\int_0^{1-x}\cos(x^3+y^2)dx\ dy\)
is

The value of \(\lim\limits_{n \rightarrow \infin}\left(1+\frac{1}{2^n}+\frac{1}{3^n}+...+\frac{1}{(2023)^n}\right)^\frac{1}{n}\) is equal to _____________ . (rounded off to two decimal places)

The value of
\(\lim\limits_{n\rightarrow \infin}\left(n\int\limits^1_0\frac{x^n}{x+1}dx\right)\)
is equal to __________ . (rounded off to two decimal places)

Which of the following functions is/are Riemann integrable on [0, 1] ?

Which of the following is/are linear transformations ?

Which of the following is/are true ?

A competitive firm can sell any output at price 𝑃 = 1. Production depends on capital alone, and the production function 𝑦 = 𝑓(𝐾) is twice continuously differentiable, with
𝑓(0) = 0, 𝑓 ′ > 0, 𝑓 ′′ < 0, \(lim\\_{ 𝐾→0 }\) 𝑓 ′ (𝐾) = ∞ , \(lim\\_{ 𝐾→∞}\) 𝑓 ′ (𝐾) = 0.
The firm has positive capital stock 𝐾̅  to start with, and can buy and sell capital at price 𝑟 per unit of capital. If the firm is maximizing profit then which of the following statements is NOT CORRECT?

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).

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).

A firm is operating in a perfectly competitive environment. A change in the market condition leads to an increase in the firm’s profit by an amount K. Which of the following describes the change in the Producer’s Surplus due to the above change in the market condition?

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).

A production function at time 𝑡 is given by
\(Y_t=A_tK^a_tL^{1-a}_t\) , 𝛼 ∈ (0, 1), 𝛼 ≠ 0.5, 
where 𝑌 is output, 𝐾 is capital, 𝐿 is labour and 𝐴 is the level of Total Factor Productivity. Define per capita output as \(𝑦_𝑡 ≡ \frac{𝑌_𝑡}{ 𝐿_𝑡} \)and capital-output ratio as \(𝑘_𝑡 ≡\frac{ 𝐾_𝑡}{ 𝑌_𝑡}\) . For any variable 𝑥𝑡 , denote \(\frac{𝑑𝑥_𝑡}{ 𝑑𝑡}\) by 𝑥̇ . The per capita output growth rate is

Among the following statements which one is CORRECT?
S1: 𝑥²+𝑦²=6 is a level curve of 𝑓(𝑥, 𝑦)=\(\sqrt{𝑥^2+𝑦^2}\)−𝑥²−𝑦²+2 
S2: 𝑥²−𝑦2=−3 is a level curve of 𝑔(𝑥, 𝑦) =\(𝑒−{^𝑥}^{2} 𝑒{^y}^{2}\)+𝑥⁴−2−2𝑥²𝑦²+y⁴