MHT CET 2026 16th April Evening Shift

A cylindrical tank without a top lid is being manufactured to hold a fixed volume of $125\pi$ cubic cm. The minimum surface area required to construct this tank is ............square cm

A spherical snow ball is melting so that its volume is decreasing at the rate of 8 c.c./sec then the rate of change of radius when the radius is 2 cm, is :

The equation of tangent to the curves $x = 1 - 3t^2$ and $y = t - 3t^3$ at the point $(-2, 2)$ is...

The area of the region enclosed by the lines $y = 2x, 2y = x$ and $x = 2$ (in sq.units) is...

The area of the region common to the parabolas $4y^2 = 9x$ and $3x^2 = 16y$ is...

Line $l : x + y = 4$ intersects the circle $x^2 + y^2 - 2x - 2y = 2$ at points $A$ and $B$. If C is the center of the circle, then the area of $\triangle ABC$ is...

If $n$ is a positive integer, then $(1 + i\sqrt{3})^{2n} + (1 - i\sqrt{3})^{2n}$ is equal to .........

The value of integral $\int_0^1\cot^{-1}(1 + x^2 - x)\,dx$ is...

The value of $\int_{-\pi}^{\pi}\dfrac{2x(1 + \sin x)}{1 + \cos^2 x}dx$ is...

For the differential equation $\dfrac{d^3y}{dx^3} + \cos\left(\dfrac{d^2y}{dx^2}\right) = 0$, which of the following is true ?

If water at $100^\circ\text{C}$ cools in 10 minutes to $80^\circ\text{C}$ and to $65^\circ\text{C}$ in the next 10 minutes, then the room temperature will be ...

The equation of the curve passing through the point (0, 1) and having a slope of the tangent at point $P(x, y)$ is equal to $\dfrac{y}{x + y}$, is

If $f(x)$ is a polynomial such that $f(x) = [f'(x)]^2$ and $f(2) = 0$, then $f(-2) = \ldots$

Let f(x) be a twice differentiable function such that $f''(x) = -f(x)$, $f'(x) = g(x)$ and $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. If $h(5) = 11$, then $h(10)$ is equal to ...

If $\cot[f(x)] = \dfrac{3x - x^3}{1 - 3x^2}$ and $\sin[g(x)] = \dfrac{1 - x^2}{1 + x^2}$, then $\lim\limits_{x \to t}\dfrac{f(x) - f(t)}{g(x) - g(t)} = \ldots$

If $y = \cos^{-1}\left(\dfrac{1 - 4^x}{1 + 4^x}\right)$, then $\dfrac{dy}{dx}$ at $x = 1$ is

If $e^x + e^{f(x)} = e$, then the domain of $f(x)$ is

If $f\left(\dfrac{x-1}{x+1}\right) = x + 1$, then $\int f(x)\,dx = \cdots$

If $f(x) = \cos x$, then the value of $\int\dfrac{e^{f(x)}(x\sin^3 x + f(x))}{1 - (f(x))^2}dx = $

$\int\dfrac{x^4 + 1}{x^6 + 1}dx = $

If $a > 0, b > 0$ and $\int\dfrac{1}{ax^2 + b}dx = \dfrac{1}{\sqrt{6}}\tan^{-1}\left(\dfrac{\sqrt{2}x}{\sqrt{3}}\right) + c$, then $\int\dfrac{1}{bx^2 + a}dx = \ldots$

The value of $\cot^{-1}\left[\dfrac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}\right]$, where $x \in \left(0, \dfrac{\pi}{2}\right)$ is...

If $(\tan^{-1}x)^2 + (\cot^{-1}x)^2 = \dfrac{5\pi^2}{8}$, then the value of $x$ is equal to...

If $\lim\limits_{x \to 0}\dfrac{45^x - 9^x - 5^x + 1}{(k^x - 1)(3^x - 1)} = 2$, then the value of $k$ is ...

If the function f is continuous at $x = \pi$, where $f(x) = \dfrac{1 - \cos[7(x - \pi)]}{5(x - \pi)^2}$, for $x \neq \pi$, then $f(\pi) = $

The minimum value of $z = 3x + 5y$, subject to constraints $x \leq 80$, $y \geq 60$, $x + y \leq 200$ & $x, y \geq 0$ occurs at the point...

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

The negation of the contrapositive of the statement $(p \vee \sim q) \to (p \wedge \sim q)$ is

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$, $B^{-1} = \begin{bmatrix} \dfrac{1}{5} & \dfrac{2}{5} \\ \dfrac{2}{5} & -\dfrac{1}{5} \end{bmatrix}$, then $(AB)^{-1} = $

If $A = [a_{ij}]_{3 \times 3}$ is a matrix such that $a_{ij} = |2i - 5j|$, where $|.|$ denotes the modulus function, then the element in the $2^{\text{nd}}$ row and $3^{\text{rd}}$ column of $A^{-1}$ is ...

If $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$ and $A^2 + xA + yI_2 = O_2$, where $I_2$ and $O_2$ are the identity matrix and null matrix of order 2 respectively, then:

The number of 4-letter words formed from the English alphabet such that there are exactly 2 vowels and 2 consonants and no vowel is repeated, but consonants may be repeated is:

If a fair coin is tossed 8 times, then the probability of getting at most 2 heads is...

If a random variable $X$ has a probability mass function $P(x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, 4 \\ 0, & \text{otherwise} \end{cases}$, then the mean of $X$ is ...

If $E_1$ and $E_2$ are equally likely, mutually exclusive and exhaustive events and $P(A|E_1) = 0.2$, $P(A|E_2) = 0.3$, then $P(E_1|A)$ equal to...

In triangle ABC, with usual notations, if $a = 4, b = 5$ and $c = 6$, then angle C is equal to...

The line $(2 + k)x + (1 + k)y = 5 + 7k$ passes through the fixed point for different values of k. If 'd' is the distance of a fixed point from the origin, then $d^2 = \ldots$

If $\theta$ is the acute angle between the lines $2x^2 + 7xy + 3y^2 = 0$, then the value of $\dfrac{2\cos\theta - 3\sin\theta}{4\sin\theta + 5\cos\theta} = \ldots$

The combined equation of the pair of lines passing through the origin and making an angle $\dfrac{\pi}{4}$ with the line $3x + y - 6 = 0$ is...

The equation of a line in cartesian form passing through (0, 0, 0) and (4, 3, c) and parallel to $\vec{a} \times \vec{b}$ where $\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 4\hat{j}$ is

If the lines $2x = ky = -z$ and $6x = -y = -4z$ are perpendicular to each other then the value of $k$ is ...

The vector equation of plane in parametric form, passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line $\dfrac{x-1}{1} = \dfrac{2y+1}{2} = \dfrac{z+1}{-1}$ is

If $d$ is the distance of point (2, 5, 10) from the plane containing the lines $\bar{r} = (4\hat{j} - \hat{k}) + \lambda(\hat{i} + 2\hat{j} - 2\hat{k})$ and $\bar{r} = (2\hat{i} + \hat{j}) + \mu(\hat{i} + 2\hat{j} - 2\hat{k})$, then $d^2 = ...

The values of p and q so that the line joining the points (7, p, 2) and (q, -2, 5) may be parallel to the line joining the points (2, -3, 5) and (-6, -15, 11) are

The general solution of the equation $\cot\theta \cdot \cot 2\theta = 1$ is...

The value of $\cos(60^\circ - A)\cdot\cos A\cdot\cos(60^\circ + A)$ is

If $\bar{a} = 4\hat{i} + \hat{j} + \hat{k}$, $\bar{b} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\bar{c} = 3\hat{i} + 4\hat{j} + 5\hat{k}$, then $(\bar{a} + \bar{b}) \cdot (\bar{b} + \bar{c}) = $

If $\bar{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\bar{b} = \hat{i} + 3\hat{k}$ and $\bar{c}$ is a unit vector, then the maximum value of the scalar triple product $[\bar{a}\ \bar{b}\ \bar{c}]$ is

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three coplanar unit vectors. A unit vector $\bar{d}$ is perpendicular to them. If $(\bar{a} \times \bar{b}) \times (\bar{c} \times \bar{d}) = \dfrac{3}{26}\hat{i} - \dfrac{2}{13}\hat{j} + \dfrac{6}{13...

The volume of a parallelopiped with coterminous edges $\bar{a}, \bar{b}, \bar{c}$ is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges $(\bar{a} \times \bar{b}), (\bar{a} \times 2\bar{c}), (\bar{b} \times 2\...