List of practice Questions

Find the intervals in which the function $f(x)=\sin x + \cos x,\; 0\leq x \leq 2\pi$ is increasing or decreasing.
Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]
Find the equation of the curve passing through $(-2,3)$ on which the gradient of the tangent at any point $(x,y)$ is \[ \frac{dy}{dx} = \frac{2x}{y^2}. \]
The probabilities of solving a particular problem by $A$ and $B$ independently are respectively $\tfrac{1}{2}$ and $\tfrac{1}{3}$. If both try to solve the problem independently, find the probability that (i) the problem is solved, (ii) only one of them solves the problem.
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}). \]
Prove that cone of the maximum volume inscribed in a sphere of radius $R$ is $\tfrac{8}{27}$ of the volume of the sphere.
Solve by matrix method the system of equations: \[\begin{aligned} 2x - 3y + 5z &= 11 \\ 3x + 2y - 4z &= -5 \\ x + y - 2z &= -3 \end{aligned}\]
Find the vector and Cartesian equation of the plane which passes through the point $(1,0,-2)$ and on which the vector $\hat{i} + \hat{j} - \hat{k}$ is perpendicular.
Eliminating arbitrary constants $a$ and $b$, find the differential equation represented by the family of curves: \[ y^2 = a(b^2 - x^2) \]
If \[ x = a\left(\cos t + \log \tan \frac{t}{2}\right), y = a \sin t, \] find the value of $\dfrac{dy}{dx}$.
If \[ A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, \] verify that $A'A = I$.
Prove that \[ \begin{vmatrix} x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]
Prove that in the set $\mathbb{Z}$ of integers the relation $R$ given by $R = \{(a,b): 5 \text{ divides } (a-b)\}$ is an equivalence relation.
Find the point on the curve \[ y = x^3 - 11x + 5 \] at which the tangent is \[ y = x - 11. \]
Evaluate: \[ \int \sqrt{x^2 - 8x + 7}\, dx \]
If \[ A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}, B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix} \] find the matrix $AB$.
Test whether the function $f : \mathbb{R} \to \mathbb{R}$ defined by \[f(x) = \begin{cases} x^3 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}\] is continuous at $x=0$.
Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]
If $A$ and $B$ are two events such that $P(A) = 0.6$, $P(B) = 0.3$ and $P(A \cap B) = 0.15$, find $P(A|B)$.
If $y = e^{a \cos^{-1}x}, \; -1 \leq x \leq 1$, then prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2 y = 0. \]
Evaluate: \[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \]
Find the value of $\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b})$.
Find the distance of the point $(2,5,-3)$ from the plane \[ \vec{r}\cdot(6\hat{i}-3\hat{j}+2\hat{k})=4. \]
The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^{2} + 5$ will be:
If $A$ is a square matrix such that $A^{2} = A$, then $(I + A)^{3} - 7A$ will be: