List of practice Questions

For a $2 \times 2$ matrix $A$, whose elements are real numbers, denote by $A^m$ the product $AA\dots A$ ($m$ times), where $m$ is a positive integer. Define $x_0 = 0$, $x_1 = 1$, $x_n = x_{n-1} + x_{n-2}$, for all $n \ge 2$ and \[ A_n = \begin{bmatrix} x_{n+1} & x_n\\ x_n & x_{n-1} \end{bmatrix}, \text{ for all } n \ge 1. \] Which of the following statements is TRUE for all $m \ge 3$?

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by \[ f(x) = \begin{cases} -ax - b ;& \text{if } x \\ 5x + 1 ;& \text{if } -1 \le x \le 1, \\ a^2x + 3b ;& \text{if } x > 1 . \end{cases} \] How many pairs $(a, b)$ are there for which $f$ is continuous at every point of $\mathbb{R}$?

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin^2(7x) - \sin^2(5x)$. Which of the following statements is NOT TRUE?

Consider the data of scores obtained by students in an examination. If the score of every student is increased by 2 marks, then which of the following statements is TRUE?

What is the value of $\int_{-1}^{2} \min\{1 - x, 1 - x^3\} \, dx$?

Let $\mathcal{C}$ be the set of all the circles in a plane. If \[ \mathcal{R} = \{(C_1, C_2) \in \mathcal{C} \times \mathcal{C} \mid C_1 \text{ and } C_2 \text{ intersect}\} , \] then which of the following statements is TRUE?

Let $r, l$ be two integers such that $r \ge l \ge 3$. What is the total number of functions \[ f : \{1, 2, \dots, r\} \to \{1, 2, \dots, r\} \] such that $f(1), f(2), \dots, f(l)$ are all distinct?

Let $l_1$ be the line joining $(1, 1, 1)$ and $(3, 1, 3)$ and let $l_2$ be the line joining $(0, 2, -1)$ and $(2, 0, 3)$. What is the angle between $l_1$ and $l_2$?

Consider the points $A(4\hat{i} + \hat{j} + 3\hat{k})$, $B(2\hat{j})$ and $C(-4\hat{i} + 3\hat{j} - 3\hat{k})$. Which of the following statements is TRUE?

Consider the following sets of points in the complex plane \[ A = \left\{ \cos \left( \frac{2n\pi}{5} \right) + i \sin \left( \frac{2n\pi}{5} \right) : n \in \mathbb{Z} \right\} \text{ and} \] \[ B = \left\{ \cos \left( \frac{2n}{5} \right) + i \sin \left( \frac{2n}{5} \right) : n \in \mathbb{Z} \right\} . \] Which of the following statements is TRUE?

Let $p(x)$ be a quadratic polynomial such that $p(1) = p(-1) = 0$. What is the coefficient of $x$ in $p(x)$?

The rate constant of a reaction at 600 K with an activation energy of 191.47 kJ mol$^{-1}$ is 5.0 $\times$ 10$^{-5}$ s$^{-1}$. What is the temperature at which the half-life of the reaction becomes 152 s? [Consider pre-exponential factor and activation energy to be independent of temperature. R = 8.314 J K$^{-1}$mol$^{-1}$]}

For two pure volatile liquids X and Y, attractive intermolecular interactions of both X-X and Y-Y are weaker than those of X-Y. The total vapour pressure of an equimolar solution of X and Y is p$_{\text{total}$. The vapour pressure of pure X and pure Y are p$^0_{\text{X}}$ and p$^0_{\text{Y}}$, respectively. Which one of the following relations is correct?

What is the ratio of the velocity of an electron in the fourth orbit of Be$^{3+}$ to the velocity of the electron in the second orbit of He$^{+}$?

An ideal gas goes through a reversible isothermal expansion (solid line) followed by a reversible adiabatic expansion (dashed line). Which of the following diagram(s) closely depict(s) the entire process?

100 mL of 1.0 M aqueous NaOH solution was diluted to 1.0 L by adding water. Half of this solution was discarded. A new 100 mL of 0.5 M aqueous NaOH solution was added to the remaining solution. What is the concentration of the final aqueous NaOH solution?

What is the order of pK$_b$ for the following molecules in an aqueous medium?

What are X and P in the following reaction sequence?

Consider the following silica-gel based thin-layer chromatogram of compounds A and B. Which one of the following statements is correct?

Which one of the following molecules is chiral?

Which one is an INCORRECT statement with regard to the following reaction?

What is the order of bond energy between C=S & C=Te, and between Cl–Cl & F–F?

Metal-ligand $\pi$-bond formation in Mn$_2$(CO)$_{10}$ and [MnO$_4$]$^-$ requires electron-pair donation between metal and ligand orbitals. Which one of the following represents the direction of electron-pair donation?

Which one of the following molecules shows an increase in bond order after loss of an electron from the highest occupied molecular orbital?

What are the numbers of protons (H$^+$) and electrons (e$^-$), respectively, required for the reduction of [Cr$_2$O$_7$]$^{2-}$ to Cr$^{3+}$ under an aqueous acidic condition?