List of top Mathematics Questions asked in MHT CET

Evaluate the integral \[ \int x^3 e^{x^2} \, dx \]

If \( \dfrac{d^2y}{dx^2} = \sin x + e^x \), \( y(0) = 3 \) and \( \left.\dfrac{dy}{dx}\right|_{x=0} = 4 \), then the equation of the curve is

If \[ \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 4, \quad \text{then} \quad \frac{dy}{dx} = \]

The general solution of the differential equation \[ \sec^2 x \tan y\,dx + \sec^2 y \tan x\,dy = 0 \] is

Water at \(100^\circ\text{C}\) cools in 15 minutes to \(75^\circ\text{C}\) in a room temperature of \(25^\circ\text{C}\). Then the temperature of water after 30 minutes is

If \[ \sec x + \tan x = 3, \quad x \in \left( 0, \frac{\pi}{2} \right), \] then \( \sin x = \)

The coordinates of the point where the line $\dfrac{x-1}{2}=\dfrac{y-2}{-3}=\dfrac{z+3}{4}$ meets the plane $2x+4y-z=1$ are

Evaluate $\displaystyle \int_{0}^{1} x(1-x)^5 \, dx$

Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]

If the equation \[ kxy + 5x + 3y + 2 = 0 \text{ represents a pair of lines, then } k = \]

The general solution of \( \tan 3x = 1 \) is

If \( y = \tan^{-1} \left( \frac{x - \sqrt{1 - x^2}}{x + \sqrt{1 - x^2}} \right) \), then \( \frac{dy}{dx} \) is

If the lines \[ \frac{x - 1}{5} = \frac{y + 1}{3} = \frac{3 - z}{\lambda} \quad \text{and} \quad \frac{x + 1}{4} = \frac{1 - 3y}{15} = \frac{z + 1}{1} \] are perpendicular to each other, then \( \lambda = \)

If \[ A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix} \] then

If \[ A = \begin{pmatrix} 2 & 5 \\ 0 & 1 \\ 3 & 0 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} 3 & -1 \\ -6 & 6 \\ -5 & 2 \end{pmatrix} \] then the values of \( \alpha \) and \( \beta \) are, respectively

If \[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}, \] such that \[ A^2 - 4A + 3I = 0, \] then \( A^{-1} \) is

The area bounded by the parabola \( x^2 = 4y \), the lines \( y = 2 \), \( y = 4 \) and the Y-axis is

A fair coin is tossed 2 times. A person receives \( X^3 \) if he gets \( X \) number of heads. His expected gain is

Out of 100 people selected at random, 10 have common cold. If five persons are selected at random from the group, then the probability that at most one person will have common cold is

An urn contains 4 red and 5 white balls. Two balls are drawn one after the other without replacement. Find the probability that both the balls are red.

Evaluate the integral: \[ \int_{-5}^{5} \frac{e^x + e^{-x}}{e^x - e^{-x}} \, dx. \]

If $B$ is the end point of minor axis of the ellipse $b^2x^2+a^2y^2=a^2b^2\ (a>b)$ and $S$ and $S'$ are the foci of the ellipse such that $\triangle BSS'$ is an equilateral triangle, then the eccentricity $e$ is

If the line \( 6x - y - 4 = 0 \) touches the curve \( y^2 = ax^3 + b \) at the point (1, 2), then \( a + b = \)

The rational form of the number \( 1.41 \overline{41} \) is

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