MHT CET 2026 20th April Morning Shift

A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\ \text{cm}^3/\text{min}$. When the thickness of ice is $5$ cm, the rate at which the thickness of ice decreases is...

If the line $ax + by + 5 = 0$ is a normal to the curve $xy = 1$ then .....

On the interval $[0, 1]$, the function $f(x) = x^{25}(1 - x)^{75}$ attains its maximum value at the point $x = $....

The area of the region lying in the first quadrant and bounded by the curve $y = 4x^2$, the Y-axis, and the lines $y = 2$, $y = 4$ is...

The area bounded by the curves $y = |x| - 1$ and $y = -|x| + 1$ is

The equation of the circle which passes through the points $(2, 3)$ and $(4, 5)$ and whose centre lies on a straight line $4x - y - 3 = 0$, is

If $w$ is a complex cube root of unity , then the value of $w^{10} - w^7 + w^5 - w^2 + 1$ is.

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int_0^2 x^2[x]\, dx$ is equal to

If $\int_0^1 \dfrac{dx}{\sqrt{x + 1} - \sqrt{x}} = \sqrt{2}k$, then the value of $k$ is...

If $y = f(x)$ is a monotonically increasing function such that $\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx}$ and $y(0) = 5$, then $y(3) = \cdots$

If $\tan x$ is an integrating factor of the differential equation $\dfrac{dy}{dx} + Py = Q$, then $P$ can be

The order and degree of the differential equation $\left(\dfrac{d^3 y}{dx^3}\right)^{\frac{2}{3}} - 3\dfrac{d^2 y}{dx^2} + 5\dfrac{dy}{dx} + 4 = 0$ are respectively

If $\sec^{-1}\left(\dfrac{x^2 + y^2}{x^2 - y^2}\right) = 2a$ such that $y\dfrac{dy}{dx} = x \cdot f(a)$ then the value of $f\left(\dfrac{2\pi}{3}\right)$ is

If $\sqrt{y + x} + \sqrt{y - x} = c$ and $\dfrac{dy}{dx} = K - \sqrt{\dfrac{y^2}{x^2} - 1}$ , then the value of $K$ is

If $y = (x^2 + 1)^{\sin x}$ for $x > 0$ such that $\dfrac{dy}{dx} = y\left[\dfrac{2x \sin x}{g(x)} + \cos x \cdot \log[g(x)]\right]$, then the function $\dfrac{1}{g(x)}$ is...

The derivative of $f(\sin x)$ with respect to $g(\sec x)$ at $x = \dfrac{\pi}{4}$, given that $f'\left(\dfrac{1}{\sqrt{2}}\right) = 3$ and $g'(\sqrt{2}) = 1$ is.....

If $\theta$ is the eccentric angle of a point on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ such that the distance of the point from the center is $5$, then $\theta = $.......

The domain of the function $f(x) = \sqrt{x - 1} + \sqrt{6 - x}$ is...

The domain of the function $f(x) = {}^{(10-x)}C_{(x-6)}$ is...

If $f(x) = \int \dfrac{x}{(x^2 + 4)(x^2 + 9)}\, dx$ and $f(0) = \dfrac{1}{5}\log\left(\dfrac{2}{3}\right)$, then $f(1) = $

$\int \dfrac{x^4(x^{10} - 1)}{x^{20} + 3x^{10} + 1}\, dx = \cdots$

$\int \left(e^{\log(\sin x)} + \cos x\right) x\, dx = $

The value of $\int \sin 4x \cos 3x\, dx$ is

If $a_1, a_2, a_3, \ldots, a_n$ are in arithmetic progression with common difference d, then $\tan\left[\tan^{-1}\left(\dfrac{d}{1 + a_1 a_2}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_2 a_3}\right) + \ldots + \tan^{-1}\left(\dfrac{d}{1 + a_{n...

$\lim\limits_{x \to 0}\left[\dfrac{x \cdot \log(1 + 4x)}{\left(e^{4x} - 1\right)^2}\right] = \cdots$

Let $[x]$ denotes the greatest integer less than or equal to x and $f(x) = [\tan^2 x]$, then which of the following is true ?

The difference between the maximum value and the minimum value of the objective function $z = 3x + y$ subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, $y \geq 0$ is....

Consider the following statements.$p$: If $3^4 > 4^3$, then $3^3 > 4^4$$q$: The roots of the equation $x^2 - 2x + 2 = 0$ are real if and only if Mumbai is in Maharashtra.$r$: Statement $p$ is true or statement $q$ is false.Which of the foll...

Which of the following is/are not true ?I) If 1 is not a prime number, then 2 is not a prime number.II) e is a vowel and $12 \times 3 = 36$III) It is not true that 14 is a composite number and 3 is even number.IV) $\sqrt{5}$ is an irrationa...

The negation of the inverse of the statement $\sim p \vee \sim q$ is...

If $A = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}$, $B = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ such that $XA = B^T$ and $A^{-1}Y = B$, then $XY = $

Let $A = \begin{bmatrix} a & 1 \\ 1 & b \end{bmatrix}$, where $a$ and $b$ are the roots of the equation $x^2 - 4x + 2 = 0$. If $A + A^{-1} = kI_2$, then the value of $k$ is ____

A random variable $X \sim B(n, p)$ follows a binomial distribution with $n = 6$. If $9P(X = 4) = P(X = 2)$, then the probability of success $p$ is..

Let X be a continuous random variable with the probability density function(p.d.f.) given by$f(x) = \begin{cases} kx, & 0 \leq x < 1 \\ k, & 1 \leq x < 2 \\ -kx + 3k, & 2 \leq x < 3 \\ 0, & \text{otherwise} \end{cases}$$P(2 < X \leq 3) = \c...

In a certain city, the ratio of men to women is $5:4$. It is found that $80\%$ of men and $90\%$ of women are literate. If a person selected at random is found to be illiterate, then the probability that the person is a man is....

In a right-angled $\triangle ABC$, the measures of the angles are in an Arithmetic Progression (A.P.) If its smallest side is $4$ units, then the area of $\triangle ABC$ is.....

The area of a triangle formed by a line with the coordinate axes is $49$ sq. units. If the perpendicular drawn from the origin to this line makes an angle of $45^\circ$ with the positive X-axis, then the equation of line is....

If the equation $16x^2 - 24xy + 9y^2 - 8x + 6y - 35 = 0$ represents a pair of straight lines, then the equation of the locus of points equidistant from these two lines is..........

The acute angle between the line $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$ and the plane $\vec{r} \cdot (2\hat{i} + p\hat{j} + \hat{k}) = 8$ is $\sin^{-1}\left(\dfrac{\sqrt{2}}{3}\right)$, then the va...

The shortest distance between the lines $\dfrac{x - 3}{3} = \dfrac{y - 8}{-1} = \dfrac{z - 3}{1}$ and $\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - 6}{4}$ is

A plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(1, r, r^2)$, then the equation of the plane is,

If M denotes the midpoint of the line joining A$(4, 5, -10)$ and B$(-1, 2, 1)$, then the equation of the plane through M and perpendicular to AB is:

The general solution of $\cos x + \sin x = \cos 2x + \sin 2x$ is $x = np\pi$ or $x = \dfrac{nq\pi}{3} + \dfrac{\pi}{6}$ for $n \in Z$ then $p : q = $

The number of common solutions of the pair of equations $2\sin^2\theta - 2\cos^2\theta + 1 = 0$ and $2\sin^2\theta + 3\sin\theta - 2 = 0$ in the interval $[0, 2\pi]$, is

The value of $\cos 15^\circ - \sin 15^\circ$ is ......

Let $\bar{a} = \hat{i} + \hat{j} + \hat{k}$, $\bar{b} = \hat{i} - 3\hat{j} + 2\hat{k}$ and $\bar{c} = 3\hat{i} - 2\hat{k}$. If a vector $\bar{p}$ satisfies the conditions $\bar{p} \cdot \bar{c} = 0$ and $\bar{p} \times \bar{a} = \bar{b} \ti...

The volume of the tetrahedron whose vertices are A$(-1, 2, 3)$, B$(3, -2, 1)$, C$(p, 1, 3)$, D$(-1, -2, 4)$ is $\dfrac{16}{3}$ cubic units then the value of p is

Let $\overline{OD} = \hat{i} + 2\hat{j} + 6\hat{k}$, $\overline{CB} = -3\hat{i} - 2\hat{k}$ be the diagonals of the parallelogram OBDC and $\overline{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ be another vector. Then the volume of a parallelopipe...

If $\bar{a}$ and $\bar{b}$ are unit vectors perpendicular to each other, then $\left[\bar{a} + (\bar{a} \times \bar{b})\quad \bar{b} + (\bar{a} \times \bar{b})\quad (\bar{a} \times \bar{b})\right] = \cdots$

If $\bar{a}$, $\bar{b}$ and $\bar{c}$ are three vectors such that $|\bar{a} + \bar{b} + \bar{c}| = 1$, $\bar{c} = \lambda(\bar{a} \times \bar{b})$ and $|\bar{a}| = \dfrac{1}{\sqrt{3}}$, $|\bar{b}| = \dfrac{1}{\sqrt{2}}$, $|\bar{c}| = \dfrac...