List of practice Questions

Omkar and Shiva were partners in a firm. Omkar was entitled to a salary of \rupee 20,000 p.a. while Shiva was entitled to a salary of \rupee 50,000 p.a. Net profit for the year ended 31\textsuperscript{st March, 2023, after charging the salaries of Omkar and Shiva, was \rupee 5,60,000. The total amount credited to Omkar’s capital account will be:}

If for the matrix \( A = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix} \), \( A + A' = 2\sqrt{3}I \), then the value of \( x \in \left[ 0, \frac{\pi}{2} \right] \) is:

Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:

The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) is:

If \( A = [a_{ij}] \) is a \( 3 \times 3 \) matrix, where \( a_{ij} = i - 3j \), then which of the following is false?

Show that \( f(x) = e^x - e^{-x} + x - \tan^{-1} x \) is strictly increasing in its domain.

If \( M \) and \( m \) denote the local maximum and local minimum values of the function \( f(x) = x + \frac{1}{x} \) (\( x \neq 0 \)) respectively, find the value of \( M - m \).

Solve the following system of equations using matrices:
\[ \frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4, \quad \frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1, \quad \frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2, \] where \( x, y, z \neq 0 \).

(a) If \( x^{30} y^{20} = (x + y)^{50} \), prove that

\[ \frac{dy}{dx} = \frac{y}{x}. \]

The number of points of discontinuity of \( f(x) = \begin{cases} |x| + 3, & \text{if } x \leq -3, \\ -2x, & \text{if } -3 < x < 3, \\ 6x + 2, & \text{if } x \geq 3 \end{cases} \) is:

If \( \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:

Find \( P(A \,|\, E_1) + P(A \,|\, E_2) \):

Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h.
\includegraphics[width=\linewidth]{latex.png} The relation between fuel consumption \( F \) (liters per 100 km) and speed \( V \) (km/h) under some constraints is given as: \[ F = \frac{V^2}{500} - \frac{V}{4} + 14. \] On the basis of the above information, answer the following questions: \begin{itemize} [(i)] Find \( F \), when \( V = 40 \, \text{km/h} \). [(ii)] Find \( \frac{dF}{dV} \). [(iii)] (a) Find the speed \( V \) for which fuel consumption \( F \) is minimum.
\hspace*{0.8cm} OR
(b) Find the quantity of fuel required to travel \( 600 \, \text{km} \) at the speed \( V \) at which \( \frac{dF}{dV} = -0.01 \). \end{itemize}

Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]

The position vectors of vertices of \( \Delta ABC \) are \( A(2\hat{i} - \hat{j} + \hat{k}) \), \( B(\hat{i} - 3\hat{j} - 5\hat{k}) \), and \( C(3\hat{i} - 4\hat{j} - 4\hat{k}) \). Find all the angles of \( \Delta ABC \).

Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.}

A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as: \[ (a, b) \, R \, (c, d) \iff a - c = b - d. \] Show that \( R \) is an equivalence relation.

Find the probability that the airplane will not crash.

If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).

Find: \[ \int \frac{e^{4x} - 1}{e^{4x} + 1} \, dx. \]

A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).

Find: \[ \int x^2 \log(x^2 - 1) \, dx. \]