MHT CET 2026 18th April Evening Shift

Let $g(x) = f(x) + f(1-x)$ and $f''(x) < 0, 0 \leq x \leq 1$, then $\ldots$

If the function $f(x) = ax^2 + bx + \sin x$ satisfies all the conditions of Rolle's theorem on $[0, \pi]$ and the slope of the tangent to the curve $y = f(x)$ at $x = \dfrac{\pi}{4}$ is zero, then $a - b = $

If a particle moves such that the displacement (s) is proportional to the square of the velocity (v), then its acceleration (a) is

If the area bounded by $y = x^3 + ax$ (where $a > 0$), the $x$-axis and the lines $x = -2$ and $x = 1$ is $\dfrac{37}{4}$ square units, then $\ldots$

The number of circles passing through the origin and touching the lines $x + y = 1$ and $x - y = 1$ is $\ldots$

If $\int_0^2 x(2 - x)^b\,dx = \dfrac{32}{7}$, where $b \in N$ then $b = $

$\int_0^2 |4x - 5|\,dx = \ldots$

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k$ $(k>0)$ is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of ...

The differential equation of the family of all parabolas whose axis is the $y$-axis is $\ldots$

The integrating factor of the differential equation $x\dfrac{dy}{dx} + 2y = x^2 \log x$ is

A spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius $r$ with time '$t$' is $\ldots$ (where $k$ is a positive constant)

The order and degree of the differential equation $\sqrt{1 + \dfrac{1}{\left(\dfrac{dy}{dx}\right)^2}} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac{3}{2}}$ are $\ldots$

For $x > 0$ and $(x \log x) < 1$, if $y = \cot^{-1}\left(\dfrac{x - \log x^{x^2}}{\log e^{x^2} + \log x^x}\right)$, then $\dfrac{dy}{dx} = \ldots$

Let $y = \sqrt[p]{x^3 y}$. If $\dfrac{dy}{dx} = \dfrac{3}{2}$ when $y = 1$, then the value of $p$ is equal to $\ldots$

If $f'(x) = \sin^2 x$ and $y = f\left(\dfrac{2x-1}{x^2+1}\right)$, then $\dfrac{dy}{dx}$ at $x = 1$ is

If $f : R \to R$ is an even function then

The domain of the function $f(x) = \sqrt{\dfrac{x}{1+x}}$ is $\ldots$

If $\int f'(x) \cdot e^{x^2}\,dx = (x - 1) \cdot e^{x^2} + k$, where $k$ is constant of integration, then $f(x) = \ldots$

If $\int e^{x + \tan^{-1}x}\left(\dfrac{x^2 + 2}{\sec^2(\tan^{-1}x)}\right)dx = e^{f(x)} + c$, then $\ldots$

Let $f(x) = x$, $f_1(x) = f(\log x)$, $f_2(x) = f_1(\log x)$, $f_3(x) = f_2(\log x)$, $\ldots$ and so on. Then $\int \dfrac{1}{f(x)\,f_1(x)\,f_2(x)\,\ldots f_{2026}(x)}\,dx = \ldots$

For $x > 0$, if $\sin(\cos^{-1}x + \tan^{-1}x) - \cos(\sin^{-1}x + \tan^{-1}x) = \sin(\cot^{-1}2)$ then $x = $

If $\tan^{-1}ax + \tan^{-1}3x = \dfrac{\pi}{4}$, where $3ax^2 < 1$, then value of $a$ for $x = \dfrac{1}{6}$ is $\ldots$

$\sec^2(\tan^{-1}3) - \tan^2(\sec^{-1}3) = $

If $\lim\limits_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2} = \lim\limits_{x \to 0} \dfrac{1 - \cos(2x)}{x \sin x}$, then the value of $k$ is $\ldots$

If $f(x) = \dfrac{3^x + 3^{-x} - 2}{\tan x \cdot \log(1+x)}$ for $x \neq 0$, is continuous at $x = 0$, then the value of $f(0)$ is equal to $\ldots$

The maximum value of $z = 4x + y$ subject to the constraints $x + y \leq 5, 2x + y \leq 7, 3x + 2y \leq 11, x \geq 0, y \geq 0$ is $\ldots$

In a switching circuit, if the combination $(S_1 \wedge S_2)$ is connected in parallel to the combination $(S_1' \wedge S_2')$, then the room is lit only when $\ldots$

If the truth value of the statement pattern $(\sim p \wedge q) \vee (\sim p \wedge \sim q) \vee (p \wedge \sim q)$ is $F$, then the truth values of $(p \vee \sim q)$ and $(p \to q)$ are $\ldots$ respectively.

The minimum number of switches in the simplified form of the following switching circuit is

If matrix A and its inverse $A^{-1}$ are given by $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \dfrac{1}{2} & -\dfrac{1}{2} & \dfrac{1}{2} \\ -4 & 3 & y \\ \dfrac{5}{2} & -\dfrac{3}{2...

Let $A = \begin{bmatrix} -3 & 2 \\ 1 & 4 \end{bmatrix}$ and if $A^2 - 2A + I = \begin{bmatrix} 18 & p \\ q & 11 \end{bmatrix}$, then $\ldots$

A parabola has its focus on the positive X-axis and the Y-axis as its directrix. If $P(\alpha , 4)$ is a point on this parabola such that the tangent to the parabola at point P passes through the origin, then the distance of P from origin i...

In a test, there are 7 questions of the type 'True or False'. No student got all the answers correct. If the sequence of answers for every student is unique, then the maximum number of students who appeared for the test is $\ldots$

A fair die is thrown 4 times. If getting a prime number on the die is considered as a success, then the probability of getting no success at all is $\ldots$

If the p. d. f. of a continuous random variable X is given by $f(x) = \begin{cases} k(9 + 8x - x^2), & \text{for } -1 \leq x \leq 4 \\ 0, & \text{otherwise} \end{cases}$ then the value of $k$ is $\ldots$

In an entrance test, there are multiple choice questions. There are four possible answers to each question, only one of which is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to...

In $\triangle ABC$, with usual notations, if $\Delta$ denotes the area of triangle $ABC$ then the value of $2s(b+c-a)\tan\left(\dfrac{A}{2}\right)$ is equal to $\ldots$

The difference between the roots of the equation $x^2 + 2x + 4 = 0$ is $\ldots$ (where $i = \sqrt{-1}$)

The coordinates of the orthocenter of the triangle whose sides are represented by the lines $4x - 7y + 10 = 0$, $x + y = 5$ and $7x + 4y = 15$ are $\ldots$

The combined equation of lines parallel to the coordinate axes and passing through the point of intersection of lines represented by $x^2 - 6xy + 5y^2 + 10x - 14y + 9 = 0$ is $\ldots$

The equation of the plane passing through the points having position vectors $(\bar{a} + \bar{b}), (\bar{b} + \bar{c})$ and $(\bar{a} + \bar{c})$ is $\ldots$

If the equation of a plane passing through $A(1, p, 2)$, $B(3, 2, 4)$ and parallel to the z axis, is $3x - 2y - q = 0$, then $\ldots$

The angle between the line $\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$ and the plane $\bar{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 8$ is $\ldots$

The acute angle between the lines $2x = 3y = -z$ and $6x = -y = -4z$ is $\ldots$

Let $f_k(x) = \dfrac{1}{k}(\cos^k x + \sin^k x)$ where $k \in N$, then $f_6(x) - f_4(x) = \ldots$

If the volume of the tetrahedron whose coterminous edges are given by the vectors $\bar{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}$, $\bar{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}$, $\bar{c} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ is 11 cubic u...

The value of $\theta \in \left(0, \dfrac{\pi}{2}\right)$ for which vectors $\bar{a} = (\sin\theta)\hat{i} + (\cos\theta)\hat{j}$ and $\bar{b} = \hat{i} - \sqrt{3}\hat{j} + 2\hat{k}$ are perpendicular is

If $|\bar{a}| = |\bar{b}| = 1, |\bar{c}| = 2$ and $\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}$, then the acute angle between $\bar{a}$ and $\bar{c}$ is $\ldots$

The value of $|\bar{a} \cdot \bar{b}|^2 + |\bar{a} \times \bar{b}| \cdot |\bar{a} \times \bar{b}|$ is $\ldots$

If $\bar{a} \cdot \bar{b} = \beta$ and $\bar{a} \times \bar{b} = \bar{c}$ then $\bar{a} = $