List of practice Questions

Let V be a nonzero subspace of the complex vector space 𝑀₇(ℂ) such that every nonzero matrix in 𝑉 is invertible. Then, the dimension of V over ℂ is

Which of the following is/are true ?

The structure of Q in the following reaction scheme is.

Which of the following is/are linear transformations ?

The major product of the reaction is

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

The rate of addition of 1-hexyl radical to the given molecules follows the order

The major product of the reaction is

The diagram that best describes the variation of viscosity (𝜂) of water with temperature at 1 atm is

The system with the lowest zero-point energy when it is confined to a one dimensional box of length 𝐿 is

The metal ion present in human carbonic anhydrase is

The oxoacid of sulfur that has S-O-S bond is

An alkaline (NaOH) solution of a compound produces a yellow colored solution on addition of NaBO₃. The compound is

Let T : \(P_2(\R) → P_4(\R)\) be the linear transformation given by T(p(x)) = p(x²). Then the rank of T is equal to __________.

The correct relation(s) for an ideal gas in a closed system is/are

If y is the solution of
y" - 2y' + y = e^x,
y(0) = 0, y'(0) = −1/2,
then y(1) is equal to __________. (rounded off to two decimal places)

The molecule(s) that follow(s) 𝐼_𝑎< 𝐼_𝑏=𝐼𝑐 (𝐼_𝑎, 𝐼_𝑏, and 𝐼_𝑐 are the principal moments of inertia) is/are

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)

The role(s) of fluorspar in the electrolytic reduction of Al₂O₃ is/are to

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

The correct statement(s) about the complexes I (K3[CoF₆]) and II (K₃[RhF₆]) is/are

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

The diatomic molecule(s) that has/have bond order of one is/are

Let f(x) = \(\sqrt[3]{x}\) for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then \(\lim\limits_{h→0} θ(h)\) is equal to _________. (rounded off to two decimal places)