List of practice Questions

Relation \( R \) on \( A = \{1,2,3,4,5,6,7\} \):
\[ R = \{(a,b) : a \text{ and } b \text{ are both odd or both even} \} \] Show that \( R \) is an equivalence relation and find the equivalence class \([1]\).
Evaluate: \[ \int_{1}^{3} (|x - 1| + |x - 2|) dx. \]
Find the particular solution of the differential equation:
\[ \frac{dy}{dx} = \frac{xy}{x^2 + y^2} \] given that \( y = 1 \) when \( x = 0 \).
Find the particular solution of: \[ (1 + x^2) \frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}, y = 0 \text{ when } x = 1. \]
Two bags: A (2W, 3R), B (4W, 5R). One red drawn. Find probability it came from B.
50 people, 20 always tell truth.Two selected. Find probability distribution of number of truth-tellers.
Find the integral:
\[ I = \int \frac{\cos \theta \, d\theta}{\sqrt{3 - 3\sin \theta - \cos^2 \theta}} \]
Find: \[ I = \int \frac{2x^2 + 1}{x^2(x^2 + 4)} \, dx \]
Given: \[ A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix} \]
Find \( A^{-1} \) and hence solve the following system of equations:
\[ \begin{cases} 3x + 2y + z = 2000 \\ 4x + y + 3z = 2500 \\ x + y + z = 900 \end{cases} \]

Simplify:
\( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \)

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto
Find the intervals in which the function
\( f(x) = x^4 - 4x^3 + 4x^2 + 15 \) is increasing .
If \( |\vec{a}| = 7 \), \( |\vec{b}| = 24 \), \( |\vec{c}| = 25 \) and
\( \vec{a} + \vec{b} + \vec{c} = \vec{0} \),
find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).
If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]
Evaluate: \[ I = \int_{0}^{\pi/2} \frac{ x \sin x \cos x }{ \sin^4 x + \cos^4 x } \, dx. \]
It is given that \[ X \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}. \text{ Then matrix X is:} \]
If $x = a \cos \theta + b \sin \theta$, $y = a \sin \theta - b \cos \theta$, then which one is true?
The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is:
corner points of the bounded
Assertion (A): The principal value of $\cot^{-1}(\sqrt{3})$ is $\frac{\pi}{6}$.
Reason (R): Domain of $\cot^{-1} x$ is $\mathbb{R} - \{-1, 1\}$.
The difference of the order and the degree of the differential equation \[ \Big( \frac{d^2 y}{dx^2} \Big)^2 + \Big( \frac{dy}{dx} \Big)^3 + x^4 = 0 \text{ is:} \]
The point which lies in the half-plane $2x + y - 4 \le 0$ is:
If $(\cos x)^y = (\cos y)^x$, then $\frac{dy}{dx}$ is:
If $\vec{a}$, $\vec{b}$ and $(\vec{a} + \vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is:
The projection of vector $\hat{i}$ on the vector $\hat{i} + \hat{j} + 2\hat{k}$ is:
A family has 2 children and the elder child is a girl. The probability that both children are girls is: