MHT CET 2026 15th April Morning Shift

If the function $f(x) = ax^3 - bx^2 - 8x - 4$ satisfies Roll's theorem in $[1,3]$, if $f'(2) = 0$ then $a - b$ is equal to...

The co-ordinates of the point on the curve $y = x\log x$ at which the normal is parallel to the line $2x - 2y = 3$ are...

The area of the smaller region bounded by the circle $x^2 + y^2 = 4$ and the line $x = 1$ is...

If the area bounded by the curve $x^2 = by$ and the lines $y = 1, y = 4$ in the first quadrant is $28$ sq. units, then the value of b is...

The centre and radius of the circle $(a+1)x^2 + 3y^2 - 6x + 9y + a + 4 = 0$ are respectively ...

If $\omega$ is a complex cube root of unity, then the value of the expression $2\left(1+\dfrac{1}{\omega}\right)\left(1+\dfrac{1}{\omega^2}\right) + 3\left(2+\dfrac{1}{\omega}\right)\left(2+\dfrac{1}{\omega^2}\right) + \ldots + (n+1)\left(n...

$\int_0^{\log 10}\left[\dfrac{e^x \sqrt{e^x - 1}}{e^x + 8}\right]dx = $

The value of the integral $\int_{-\pi/2}^{\pi/2}\left(\dfrac{x^2 \cos x}{1+e^x}\right)dx$ is equal to $\left(\dfrac{\pi^2}{A}\right) - B$. Then $\left(\dfrac{A}{B}\right) = $

The solution of the differential equation $\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}$ is...

If the differential equation $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$ for all $x$ and $f(0) = 1, f'(0) = 2$, then...

The solution of the differential equation $x^2 \dfrac{dy}{dx} - xy = 1$ is...

If $y = \log x^x$, then the value of $\left(\dfrac{dy}{dx}\right)^2$ at $x = e^2$ is

If $y = \cos^{-1}(\sin x)$, where $\dfrac{\pi}{2} < x < \pi$, then $\dfrac{dy}{dx} = ...$

If $y = \left(\dfrac{ax+b}{cx+d}\right)$, then $2\dfrac{dy}{dx} \cdot \dfrac{d^3 y}{dx^3}$ is equal to

Let $f(x) = 3x^3 + 4e^{\frac{x}{2}}$ and $g(x) = f^{-1}(x)$. Then the value of $g'(4)$ is equal to...

If $f(x-y) + f(x+y) = 2f(x)f(y)$ for all $x, y \in \mathbb{R}$, then $f(x)$ is .....

If the line $4x + 3y = 7$ touches the hyperbola $x^2 - y^2 = 7$, then the sum of the co-ordinates of the point of contact is...

If $\int \dfrac{1}{\cos^3 x \sqrt{\sin 2x}}\, dx = p(\tan^2 x + q)\sqrt{\tan x} + c$, then the values of $p$ and $q$ respectively are

If $\int \sqrt{2}\sqrt{1+\sin x}\, dx = -4\cos(ax+b) + c$, then the value of $a, b$ respectively are...

$\int \dfrac{\sqrt{x-2}}{x}dx = $

If $f(x)$ and $g(x)$ are integrable functions then $\left[\int f(x)dx\right]\left[\int g(x)dx\right] = $

If an antiderivative of $f(x)$ is $e^x$ and an antiderivative of $g(x)$ is $\cos x$, then $\int f(x)\cos x\, dx + \int g(x)e^x\, dx = $

If $\tan^{-1}\left[\dfrac{\sqrt{5-2\sqrt{6}}}{1+\sqrt{6}}\right] = \dfrac{\pi}{3} - \tan^{-1}(k)$, then $\sec^{-1}(k) = ...$

If $\lim\limits_{x \to 0} \dfrac{(4^x - 1)^3}{\tan\left(\dfrac{x}{4}\right)\log\left(1 + \dfrac{x^2}{3}\right)} = 96(\log a)^b$, then $(a + b) = $

If the function f is continuous at $x = 1$, where $f(x) = \dfrac{1 + \cos(\pi x)}{\pi(1-x)^2}$, for $x \neq 1$, then the value of $f(1)$ is....

The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6, x + y \geq 1, x \geq 0, y \geq 0$ is....

The dual of the converse of the inverse of the logical statement $p \to (q \to r)$ is equivalent to...

The statements p, q and r have truth values True, False and False respectively. The truth values of a logical statement $[\sim(p \wedge \sim q) \vee (q \vee \sim r)]$ and its dual are, respectively.....

Let $A = \begin{bmatrix} 2k-1 & 1 & 1 \\ 0 & 2k-1 & 1 \\ 0 & 0 & 2k-1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & 1 \\ 1-2k & 0 & k \\ -1 & -k & 0 \end{bmatrix}$ where k is a real number. If $\det(\text{adj} A) + \det(\text{adj} B) =...

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then the matrix $A^{-3}$ when $\theta = \dfrac{\pi}{6}$ is equal to...

The number of five-digit numbers formed using the digits $2, 3, 5, 7, 9$ without repetition and which are greater than $24000$ are

The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is $0.8$. If $8$ people are given this vaccine, then the probability that exactly $4$ wi...

The Variance of the following probability distribution of X isX$0\(1\)2\(3$P(X)$q^3\)3q^2 p\(3qp^2\)p^3$where $0 < p < 1, q = 1 - p$

If A and B are two events such that $P(A) = \dfrac{2}{3}$, $P(B) = \dfrac{1}{2}$ and $P(A \mid B) = \dfrac{2}{3}$, then $P(A' \cup B) + P(A \cup B') = $

In $\triangle ABC$, with the usual notations, $\angle C = 90^\circ$, then $\sin(A - B)$ is equal to....

In $\triangle ABC$ with usual notations, if $1 + \tan\left(\dfrac{A}{2}\right)\tan\left(\dfrac{B}{2}\right) = \dfrac{k}{s}$ (where $s$ is the semi-perimeter), then the value of $k$ is...

If a triangle $ABC$ has vertices $A(1,-6), B(2,-3)$ and $C(3,-2)$, then the coordinates of its orthocenter are....

If $\theta$ is the acute angle between the lines represented by the equation $x^2 - 3xy + 2y^2 = 0$, then $\dfrac{3\sin\theta + 2\cos\theta}{3\sin\theta - 2\cos\theta} = $

The triangle formed by the lines $2x^2 - 3xy - 2y^2 = 0$ and $3x - y = 7$ is...

If the lines $\dfrac{x-5}{5m+2} = \dfrac{2-y}{5} = \dfrac{1-z}{-1}$ and $x = \dfrac{2y+1}{4m} = \dfrac{1-z}{-3}$ are perpendicular to each other, then the value of m is ...

For the line $\dfrac{x+1}{1} = \dfrac{y-2}{2} = \dfrac{z+3}{3}$, identify the incorrect statement among the following.

The direction cosines of a line which is perpendicular to the lines $\dfrac{x-7}{2} = \dfrac{y+17}{-3} = \dfrac{z-6}{1}$ and $\dfrac{x+5}{1} = \dfrac{y+3}{2} = \dfrac{z-6}{-2}$ are...

The equation of the plane containing the lines $\dfrac{x-1}{2} = \dfrac{y+1}{\lambda} = \dfrac{z}{2}$ and $\dfrac{x+1}{5} = \dfrac{y+1}{2} = \dfrac{z}{\lambda}$ is

If $\cos(p\theta) + \cos(q\theta) = 0$ and $p \neq q$, then the general value of $\theta$ (where n is any integer) is...

If $P = \tan 20^\circ$, then the value of $\dfrac{\tan 160^\circ - \tan 110^\circ}{1 + \tan 160^\circ \tan 110^\circ}$ in terms of $P$, is...

If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ ($0 < \theta < \pi$) is the angle between them, then the value of $\dfrac{|\bar{a} + \bar{b}|}{|\bar{a} - \bar{b}|}$ is equal to...

If ABCDEF is a regular hexagon and $\overline{AB} + \overline{AC} + \overline{AD} + \overline{AE} + \overline{AF} = p\overline{AD} = q\overline{AO}$, where O is the center of the hexagon, then the values of $p$ and $q$ respectively are

The vector $\bar{r}$ whose magnitude is $3\sqrt{2}$ units and which makes angles of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ with the positive y- and z-axes respectively is....

Let $(\bar{p} \wedge \bar{q})$ denote the angle between $\bar{p}$ and $\bar{q}$. If $\bar{a} + \bar{b} + \bar{c} = \bar{0}, |\bar{a}| = 7, |\bar{b}| = 5$ and $|\bar{c}| = 3$ then (take $\pi = \dfrac{22}{7}$)

If a unit vector makes angles $\dfrac{\pi}{4}$ with $\hat{i}$, $\dfrac{\pi}{3}$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$, then a value of $\theta$ is equal to...