MHT CET 2026 13th April Evening Shift

If Rolle's theorem is applicable for the function $f(x) = \log\left(\dfrac{x^2+a}{x}\right)$ on $[3, 4]$ with $c \in (3, 4)$ such that $f'(c) = 0$, then the value of $f''(c)$ is

If the tangent to the curve $2y^3 = x^3 + ax^2$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes such that $\alpha^2 + \beta^2 = 61$, then the value of $a$ is

Let $f(x)$ be the differentiable function for all $x$ such that $f'(x) \leq 5$ and $f(1) = 4$. The maximum value of $f(5)$ is...

The area of the region bounded by the curve $y = 2^{kx}$ and $x = 0, x = 2$, in first quadrant is $\dfrac{2}{\log 2}$, then the value of $k$ is

The area of the region bounded by the curve $y = x^3$ and the lines $y = 8$ and $x = 0$ is ... square units.

The equation of the circle concentric with circle $x^2 + y^2 - 6x + 7 = 0$ and which touches the line $x + y + 3 = 0$ is ....

If $x = \sqrt{-1-\sqrt{-1-\sqrt{-1-\ldots\infty}}}$, where $\omega$ is a non-real complex cube root of unity, then the value of $x$ is...

If $\int_0^a (x^2 - 4x + 1)dx = 6$, then the real value of $a$ is

If $\int_{\pi/6}^{\pi/3} \dfrac{1}{1+\sin x + \cos x}dx = \log 2$, then the value of $\int_{\pi/6}^{\pi/3} \dfrac{\cos x}{1+\sin x + \cos x}dx = $ ............

The differential equation of $3y = \sqrt[3]{x+c}$ is

The general solution of the differential equation $(x+y)\dfrac{dy}{dx} = 1$ is

A body cools according to Newton's law of cooling from $100^\circ$C to $60^\circ$C in $20$ minutes. The temperature of the surroundings being $20^\circ$C, then the total time required for the body to cool down to $30^\circ$C is

If $y = \tan^{-1}\left(\dfrac{1}{x^2+x+1}\right) + \tan^{-1}\left(\dfrac{1}{x^2+3x+3}\right) + \tan^{-1}\left(\dfrac{1}{x^2+5x+7}\right) + \ldots$ upto $n$ terms, then $y'(0) =$

$\dfrac{d}{dx}\left[\sin^2\left\{\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right\}\right] =$ ...

If $f(x) = \sqrt{x^2+1}, g(x) = \dfrac{x+1}{x^2+1}, h(x) = 2x - 3$, then $f'\left(h'(g'(x))\right) =$

If $\sqrt{y+x} + \sqrt{y-x} = c$ such that $\dfrac{dy}{dx} = f(x) - \sqrt{[f(x)]^2 - 1}$, then $f(x) =$ ____

If $g(x)$ is the inverse function of $f(x)$, where $f(x) = \dfrac{5x+3}{4x-1}$, then $g(1) =$

If $\int x^2 \cdot e^x dx = e^x f(x) + c$, then the minimum value of $f(x)$ is ...

$\int \dfrac{(x+1)}{x(1+xe^x)^2}dx =$

If $\int \dfrac{x+1}{x^2+1}dx = \tan^{-1}x + g(x) + c$, where $c$ is constant of integration, then the function $g(x)$ is monotonically increasing in the interval

$\int \sqrt{1+\sin x}\,dx$ is equal to

Let $f(x) = (\sin^{-1} x)^2 + (\cos^{-1} x)^2$ be a real-valued function defined on its domain. Then the sum of the greatest and the least values of $f(x)$ is

The value of $\dfrac{\tan^{-1}(1) + \cos^{-1}\left(\dfrac{1}{2}\right) + \sin^{-1}\left(\dfrac{1}{2}\right)}{\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)}$ is equal to

The quadratic polynomial $p(x)$ has roots $1$ and $\alpha$, while quadratic polynomial $q(x)$ has roots $1$ and $\beta$. Let $\alpha$ and $\beta$ be the roots of $r(x) = p(x) + q(x)$. Then $\lim_{x\to\infty}\left[\sqrt{p(x)} - \sqrt{q(x)}\r...

The value of $\lim_{n\to\infty} \dfrac{(n+2)! + (n+1)!}{(n+2)! - (n+1)!}$ is ____

If $f(x)$ is continuous at $x = 0$, where $f(x) = \dfrac{8^x - 2^x}{k^x - 1}$, for $x \neq 0$ and $f(0) = 2$, then the value of $k$ is ...

The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ of a linear programming problem subject to constraints $5x + 10y \leq 50$, $x + y \geq 1$, $y \leq 4$ and $x \geq 0, y \geq 0$ is $3\lambda$....

If the truth value of the compound statement$[(p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)] \rightarrow (p \wedge q)$ is false, then the truth values of $p \rightarrow q$ and $q \rightarrow p$ are respectively...

Which of the following statements is logically equivalent to $\sim(p \leftrightarrow q)$ ?

If $A = \begin{bmatrix} \sec\theta & -\tan\theta \\ -\tan\theta & \sec\theta \end{bmatrix}$, $\theta \in \left(0, \dfrac{\pi}{2}\right)$ such that $A + adj\,A = 4I$, then $\theta =$

The element in 1st row and 2nd column of the inverse of the matrix $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}$ is ...

The distance of the focus of the parabola $y^2 = 16x$ from its directrix is...

In a multiple-choice examination, there are $10$ questions with one correct option out of $4$ options for each question. A student gets $4$ marks for each correct answer and $1$ mark is deducted for each incorrect answer. The probability th...

It is known that a box of $8$ batteries contains $3$ defective pieces and a person randomly selects $2$ batteries from this box. Then the probability distribution of the number of defective batteries is

A certain disease has a prevalence of $1\%$ in the population. A diagnostic test for the disease has a sensitivity of $98\%$ and a specificity of $95\%$. If a person from this population tests positive, then the probability that they actual...

The equation of the straight line passing through the point $(-5, 3)$ such that the portion of the line intercepted between the axes is divided by the point in the ratio $4:3$ (from the X-axis to the Y-axis) is...

If the equation $kx^2 - 5xy + 6y^2 + x - 3y = 0$ represents a pair of straight lines, then the co-ordinates of their point of intersection are

The pair of straight lines represented by the equation $\sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0$ are

The vector equation of the plane which is at a distance of $5$ units from the origin and normal to the vector $2\hat{i} + \hat{j} - 2\hat{k}$ is

The equation of a line passing through a point $(4, -2, 3)$ and perpendicular to the XZ-plane is....

The equation of the plane passing through the intersection of planes $2x - y + z = 3$, $4x - 3y + 5z = -9$ and parallel to the line $\dfrac{x+1}{2} = \dfrac{y+3}{4} = \dfrac{z-3}{5}$ is...

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the sum of values of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ and $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is ...

If $\sin 3\alpha = 4\sin\alpha \cdot \sin(x+\alpha) \cdot \sin(x-\alpha)$ where $\alpha \neq n\pi, n \in Z$, then all possible values of $x$ are given as

The general solution of $\cos\theta - \sin\theta = 1$ is

If $\cos(\theta-\alpha)=a$ and $\sin(\theta-\beta)=b$, then the value of $\cos^2(\alpha-\beta)+2ab\sin(\alpha-\beta)+\cos^2(\theta-\alpha)$ is...

If $\bar{a} = \hat{i} + \hat{j}$ and $\bar{b} = \hat{i} - \hat{k}$, then the point of intersection of the lines $\bar{r} \times \bar{a} = \bar{b} \times \bar{a}$ and $\bar{r} \times \bar{b} = \bar{a} \times \bar{b}$ is

If $\bar{a}$ and $\bar{b}$ have the same magnitude and angle between them is $60^\circ$ and their scalar product is $\dfrac{1}{2}$, then $|\bar{a}|$ is ____

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i}, \bar{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ with $c_1 = 1$, $c_2 = 2$, then value of $c_3$ such that $\bar{a}, \bar{b}, \bar{c}$ are coplanar is ____

If $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar unit vectors such that the angle between any two of them is $60^\circ$, and the vector $\bar{d} = x\bar{a} + y\bar{b} + z\bar{c}$ is perpendicular to both $\bar{a}$ and $\bar{b}$, then th...

If $\bar{a}, \bar{b}, \bar{c}$ are three non zero and non-coplanar vectors such that $\bar{a} \times (\bar{b} \times \bar{c}) = \dfrac{\bar{b}}{2}$, then the angle between $\bar{a}$ and $\bar{b}$ is ...