List of top Mathematics Questions

Degree of the differential equation \[ \frac{dy}{dx} e^x + \left( \frac{dy}{dx} \right)^3 = x \]

The derivative of $\sin^{-1}\!\left(\dfrac{\sqrt{1+x}+\sqrt{1-x}}{2}\right)$ with respect to $\cos^{-1}x$ is

A body is heated to $110^\circ$C and placed in air at $10^\circ$C. After $1$ hour its temperature is $60^\circ$C. The additional time required for it to cool to $30^\circ$C is

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

Evaluate the integral \[ \int \frac{dx}{\cos 2x - \cos^2 x} \]

If $\displaystyle \int_{1}^{k} (3x^2 + 2x + 1)\,dx = 11$, then $k =$

Evaluate \[ \int \frac{\log x - 1}{1 + (\log x)^2} \, dx \]

The joint equation of bisectors of the angle between the lines represented by $3x^2+2xy-y^2=0$ is

The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is

If the elements of matrix \( A \) are the reciprocals of the elements of the matrix \( \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix} \), where \( \omega \) is a complex cube root of unity, then

Find the value of the following expression:
\[ 5^2 + 6^2 + 7^2 + \cdots + 20^2 = \]

If \[ \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \quad \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k}, \quad \mathbf{c} = x\hat{i} + \hat{j} + (x - 1)\hat{k} \] If the vector \( \mathbf{c} \) lies in the plane of \( \mathbf{a} \) and \( \mathbf{b} \), then \( x = \)

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

Evaluate the integral \[ \int \frac{\sin 2x}{\sin^2 x \cos^2 x}\, dx \]

If \( x = 3 \sin \theta \), \( y = 3 \cos \theta \cos \phi \), \( z = 3 \cos \theta \sin \phi \), then \( x^2 + y^2 + z^2 = \)

If the lines \[ \frac{1-x}{2} = \frac{y-8}{\lambda} = \frac{z-5}{2} \quad \text{and} \quad \frac{x-11}{5} = \frac{y-3}{3} = \frac{z-1}{1} \] are perpendicular, then \(\lambda =\)

If $3\sin^2x-8\sin x+4=0$, $x\in\left(\dfrac{\pi}{2},\pi\right)$, then $\tan x=$

If \( p, q \) are true statements and \( r \) is a false statement, then which of the following statements is a true statement?

Evaluate \( \int \frac{dx}{\cos 2x + \sin^2 x} \)

In a single throw of three dice, the probability of getting a sum of at least 5 is

If the probability density function of a continuous random variable is \[ f(x) = \frac{x^3}{3} \quad \text{if} \quad -1<x<2, \quad f(x) = 0 \quad \text{otherwise}, \] then the cumulative distribution function of \( X \) is

Solution of the differential equation \[ \frac{dy}{dx} + 2y = e^{-x} \text{ is:} \]

If \[ \tan^{-1} x + \tan^{-1} y = c \text{ is the general solution of the differential equation} \]

The solution of the differential equation \[ x^2 \frac{dy}{dx} = y^2 + xy \]