List of top Mathematics Questions

Evaluate: \[ \int \sqrt{x^2 - 8x + 7}\, dx \]

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.

Find the point on the curve \[ y = x^3 - 11x + 5 \] at which the tangent is \[ y = x - 11. \]

If \[ A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, \] verify that $A'A = I$.

Prove that \[ \begin{vmatrix} x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]

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 $\dfrac{dy}{dx}$.

Evaluate: \[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \]

Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]

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 $x=0$.

Find the distance of the point $(2,5,-3)$ from the plane \[ \vec{r}\cdot(6\hat{i}-3\hat{j}+2\hat{k})=4. \]

Find the value of $\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b})$.

If $y = e^{a \cos^{-1}x}, \; -1 \leq x \leq 1$, then prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2 y = 0. \]

On the vector $\hat{i} + \hat{j}$ the projection of vector $\hat{i} - \hat{j}$ will be:

Find the value of $\cos(\sec^{-1}x)$.

If $A$ is a square matrix such that $A^{2} = A$, then $(I + A)^{3} - 7A$ will be:

The order of the differential equation $\left(\dfrac{d^{3}y}{dx^{3}}\right)^{2} + x^{2}\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3} + 7y = \sin x$ will be:

In the set $\{1, 2, 3, 4, 5, 6\}$, the relation $R$ defined by $R = \{(a, b) : b = a+1\}$ will be:

Solve the differential equation: \[ \frac{dy}{dx} - y = \cos x \]