List of practice Questions

Solve the differential equation: \[ \frac{dy}{dx} - y = \cos x \]
Find the area included between the circle $x^{2} + y^{2} = 8x$, parabola $y^{2} = 4x$ and upper part of $x$-axis.
Evaluate $\displaystyle \int \left[\log(\log x) + \frac{1}{(\log x)^{2}}\right] dx$.
Prove that the matrix \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \] satisfies the equation \[ A^{2} - 4A + I_{2} = 0, \] where $I_{2}$ is the $2 \times 2$ identity matrix and $0$ is the $2 \times 2$ zero matrix. Also, find $A^{-1}$ with the help of this.
Solve the system of equations by matrix method: \[ 3x - 2y + 3z = 8, 2x + y - z = 1, 4x - 3y + 2z = 4 \]
Maximize $Z = 8000x + 12000y$ under the constraints: \[ 9x + 12y \leq 180, 3x + 4y \leq 60, x + 3y \leq 30, x \geq 0, \; y \geq 0 \]
Evaluate \[ \int_0^{\pi/2} \frac{\sin^4x}{\sin^4x + \cos^4x} \, dx \]
Evaluate \[ \int_0^{\pi/2} \ln(\sin x) \, dx \]
If $y = \sin^{-1}x$, then prove that $(1 - x^{2}) \dfrac{d^{2}y}{dx^{2}} - x \dfrac{dy}{dx} = 0$.
Prove that the relation $R = \{(a,b) \in \mathbb{Z} \times \mathbb{Z} \;|\; (a-b)$ is divisible by $2\}$ is an equivalence relation.
If $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$, then prove that $(AB)^{-1} = B^{-1}A^{-1}$.
Differentiate $y = (\cos x)^{\tan x} + x^{x}$.
Find the interval in which the function \[ f(x) = 4x^3 - 6x^2 - 72x + 30 \] is (i) increasing and (ii) decreasing.
Solve the differential equation $(\tan^{-1}y - x)dy = (1+y^{2})dx$.
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \] \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \]
If a die is thrown three times, then find the probability of getting at least one appearing number odd.
If $y = A\cos\theta + B\sin\theta$, then prove that $\dfrac{d^{2}y}{d\theta^{2}} = -y$.
Solve the inequality $3x + 4y \leq 12$, $4x + 3y \leq 12$, $x \geq 0$, $y \geq 0$ by graphical method.
Prove that \[ \begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3 \]
If $A$ and $B$ are two non-singular square matrices of order $n$, then prove that \[ (AB)^{-1} = B^{-1}A^{-1} \]
Find the equation of tangent at $t = \dfrac{\pi}{2}$ on the curve $x = a \sin^{3}t$, $y = b \cos^{3}t$.
One die is thrown two times. If the sum of the appeared numbers on their faces is 6, find the conditional probability of appearing number 4 at least one time in that.
Find the vector equation of the line $\dfrac{x+3}{2} = \dfrac{y-5}{4} = \dfrac{z+6}{2}$.
Solve the differential equation $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$.
The modulus function $f: \mathbb{R} \to \mathbb{R}^{+}$ given by $f(x) = |x|$ will be: