List of top Mathematics Questions

If \(\sin x = p\), then prove that : (i) \(\cot x = \frac{\sqrt{1 - p^2}}{p}\) (ii) \(\frac{1 + \tan^2 x}{1 + \cot^2 x} = \frac{p^2}{1 - p^2}\)

Find the unit vector perpendicular to each of the vectors (\( \vec{a} + \vec{b} \)) and (\( \vec{a} - \vec{b} \)) where \[\vec{a} = \hat{i} + \hat{j} + \hat{k}, \, \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}.\]

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).

If three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) satisfying the condition \(\vec{a} + \vec{b} + \vec{c} = 0\). If \(|\vec{a}| = 3\), \[|\vec{b}| = 4 \text{ and } |\vec{c}| = 2, \text{ then find the value of } \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}.\] 5

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

If \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) be functions defined by \( f(x) = \cos x \) and \( g(x) = 3x^2 \) respectively, then prove that \( g \circ f \neq f \circ g \).

Find the angle between the vectors \(-2\hat{i} + \hat{j} + 3\hat{k}\) and \(3\hat{i} - 2\hat{j} + \hat{k}\).

If \( P(A) = 0.12, P(B) = 0.15 \) and \( P(B/A) = 0.18 \), then find the value of \( P(A \cap B) \).

Find the degree of the differential equation \[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\]

Prove that the function \( f(x) = |x| \), is continuous at \( x = 0 \).

A relation \( R = \{(a, b) : a = b - 1, b \geq 3\ \) is defined on set \( N \), then}

The modulus function \( f : \mathbb{R} \to \mathbb{R}^+ \) given by \( f(x) = |x| \) is

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is

The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{\frac{1}{3}} \text{ is} \]

The value of \( \displaystyle \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

Find the value of $k$ if $(x - 3)$ is a factor of the polynomial $x^3 - kx^2 + 15x - 6$.

Find the derivative of \( (\sin x)^x + \sin^{-1}(\sqrt{x}) \) with respect to \(x\).

Find the principal value of \( \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) \).

Evaluate the integral \[ \int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx . \]

Find the general solution of the differential equation \[ \frac{dy}{dx}+\frac{y}{x}=x^2 . \]

Show that the relation \(R\) in the set \(A = \{1,2,3,4,5\}\) given by \[ R=\{(a,b): |a-b| \text{ is even}\} \] is an equivalence relation.

If \[ A=\begin{bmatrix}2 & 3\\ 1 & -4\end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix}1 & -2\\ -1 & 3\end{bmatrix}, \] verify that \( (AB)^{-1} = B^{-1}A^{-1} \).

Solve the following system of equations using the matrix method: \[ 2x+3y+3z=5,\qquad x-2y+z=-4,\qquad 3x-y-2z=3. \]

The number of real solutions of the equation: $x|x+3| + |x-1| - 2 = 0$ is