MHT CET 2026 19th April Morning Shift

A tank with a rectangular base and rectangular sides, open at the top is made. Depth of the tank is $4$ m and its volume is $36$ cubic meters. For making a tank cost of base material used is Rs. $100$ per sq. meter and that of sides is Rs. ...

A particle is fired straight up from the ground. Its height in feet after $t$ second is given by $s(t) = 128t - 16t^2$. The velocity of the particle when it hits the ground is...

The value of $c$ satisfied by the Rolle's theorem for the function $f(x) = x^2(1 - x)^2$, $x \in [0, 1]$ is...

The value of $x$ so that the volume of the parallelopiped formed by the vectors $\hat{i} + x\hat{j} + \hat{k}$, $\hat{j} + x\hat{k}$ and $x\hat{i} + \hat{k}$ is minimum, is

If the area of the region bounded by the parabola $y^2 = 4kx$ and the line $x = k$, (where $k > 0$) is $\dfrac{128}{3}$ sq. units, then the value of $\sin^{-1}\left(\dfrac{2}{k}\right)$ is equal to...

The area (in sq. units) of the region enclosed by $\{(x, y) \mid y \leq x^2, xy \leq 8, y \geq 1\}$ is...

A line making equal intercepts on coordinate axes and is tangent to the circle $x^2 + y^2 = 4$. The length of each intercept made by line on the coordinate axes is ...

The smallest positive integer $n$ for which $\dfrac{(1 + i)^n}{(1 - i)^{n-2}}$ is a real number, is ...

If $[x]$ is the greatest integer function not greater than $x$, then the value of $\displaystyle\int_0^2 x[x^2]\,dx$ is...

The value of $\displaystyle\int_2^4 (\{x\} + [x])\,dx =$...(where $\{x\}$ and $[x]$ are the fractional part function and the greatest integer function, respectively)

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x - y}{x + y}$, when $x = 0$ and $y = 0$ represents ....

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

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

Let $x = at^2 - 1$, where $a > 0$ and $y = t^3 + 1$. If at $t = 1$, $\dfrac{d^2y}{dx^2} = \dfrac{3}{16}$, then the value of $a$ is...

If $y = \cot^{-1}\left(\dfrac{1 + \sin 5x}{\cos 5x}\right)$, then the value of $\dfrac{dy}{dx}$ is

The derivative of $\log_{10} x$ with respect to $\log_x 10$ is

A tangent having slope $-\dfrac{1}{2}$ to the ellipse $3x^2 + 4y^2 = 12$ intersects the X-axis and Y-axis at the points A and B respectively. if O is the origin, then the area of $\triangle AOB$ is ...

If $f(x) = x^2$ and $g(x) = [x^2]$ where $[\cdot]$ represents the greatest integer function then, $(f \circ g)\left(\dfrac{3}{2}\right) + (g \circ f)\left(\dfrac{3}{2}\right)$ is equal to ...

The value of integral $\displaystyle\int \dfrac{dx}{\sin^2 x + \tan^2 x}$ is...

The value of integral $\displaystyle\int \dfrac{\sqrt{x^2 + 1}\,[\log(x^2 + 1) - 2\log x]}{x^4}\,dx$ is equal to...

If $f(x) = \dfrac{1}{\log x}$ and $g(x) = \dfrac{1}{(\log x)^2}$, then the value of $\displaystyle\int [f(x) - g(x)]\,dx$ is...

If $\displaystyle\int \dfrac{dx}{x^{7/2}(x^4 + 1)^{3/8}} = m\left(\dfrac{x^4 + 1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $\dfrac{n}{m}$ is...

$\sum\limits_{k=1}^{2026} \sin^{-1}\left(\cos\dfrac{k\pi}{4}\right) =$

If $\sin^{-1}\left(\tan\dfrac{\pi}{4}\right) - \sin^{-1}\left(\sqrt{\dfrac{3}{x}}\right) = \dfrac{\pi}{6}$ then $x$ is a root of the equation

If the function $f(x) = \dfrac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x}$ is continuous at $x = \dfrac{\pi}{4}$, then the value of $f\left(\dfrac{\pi}{4}\right)$ is ...

The value of $\lim\limits_{x \to 0}\left(\dfrac{8}{x^8}\right)\left[1 - \cos\dfrac{x^2}{2} - \cos\dfrac{x^2}{4} + \cos\dfrac{x^2}{2}\cdot\cos\dfrac{x^2}{4}\right]$ is equal to ...

The shaded region in the provided graph represents the solution set for which of the following systems of linear inequalities?

The simplified switching circuit for the following circuit is

The logical statement $(p \vee q) \wedge [(\sim p \wedge q) \vee (p \wedge \sim q)] \wedge \sim q$ is logically equivalent to ...

Which of the following logical statements is a tautology?

If $A = [a_{ij}]_{3\times3}$, where $a_{ij} = \begin{cases} 1, & \text{if } i+j \text{ is even} \\ 0, & \text{if } i+j \text{ is odd} \end{cases}$, then $\text{adj}(A) = \ldots$ ..

The inverse of matrix $\begin{bmatrix} 1+pq & p & 0 \\ q & 1+pq & p \\ 0 & q & 1 \end{bmatrix}$ is ...

Four men and three women are to be arranged at a round table. The number of arrangements in which no two women are together is ....

Let $a$ be an integer selected at random from the set $\{0, 1, 2, 3, \ldots, 9\}$. The probability that the equation $ax^2 - ax + 1 = 0$ has real roots is ...

If in $6$ trials, X is a binomial random variable which follows the relation $9P(x = 4) = P(x = 2)$, then the probability of failure is...

Given the probability density function (p.d.f.) of the random variable X, $f(x) = \dfrac{1}{2a}$, $0 < x < 2a$, $a > 0$$= 0$, otherwise, then which of the following is correct ?

Consider a game of tossing a six sided fair die. If the face that comes up is $6$, the player wins Rs. $36$ and he loses Rs. $k^2$, where $k$ is the face that comes up $k = \{1, 2, 3, 4, 5\}$, then the expected winning amount in this game i...

In $\triangle ABC$, with usual notations, if the sides $a$, $b$ and $c$ are in the ratio $18 : 17 : 7$, then $\cot\dfrac{A}{2} : \cot\dfrac{B}{2} : \cot\dfrac{C}{2} =$

A straight line L passes through the point of intersection of the lines $x - y + 1 = 0$ and $2x + y - 7 = 0$. If L intersects the positive x-axis at $A(a, 0)$ and the positive y-axis at $B(0, b)$, then the minimum area of the triangle $OAB$...

If the combined equation of angle bisectors of the lines $x^2 - 2pxy - y^2 = 0$ is $x^2 - 2qxy - y^2 = 0$, then which of the following is true?

The plane $\dfrac{x}{2} + \dfrac{y}{3} + \dfrac{z}{4} = 1$ cuts the axes at the points A, B, C then the area of triangle ABC is

From a point P $(a, b, c)$, perpendiculars PA and PB are drawn to XY plane and ZX plane respectively. If O is the origin, then the equation of plane OAB is

If $p$ is the shortest distance between the lines $\dfrac{x+1}{7} = \dfrac{y+1}{-6} = z+1$ and $\vec{r} = (3\hat{i} + 5\hat{j} + 7\hat{k}) + \mu(\hat{i} - 2\hat{j} + \hat{k})$ then $[p]$ is... ,(where $[\,.\,]$ denotes the greatest integer ...

The symmetric form of the equation of the line $x = ay + b$, $z = cy + d$ is

Let $x \in [0, 6\pi]$ satisfy the equation $\cos x - \sin x = -1$. If $x = k\left(\dfrac{\pi}{3}\right)$ where $k \in N$, then find the number of possible values of $k$ is .....

The value of $\cos\left(\dfrac{\pi}{5}\right)$ is ...

If $A(\vec{a})$, $B(\vec{b})$ and $C(\vec{c})$ are vertices of $\triangle ABC$. Point D divides segment BC internally in the ratio $2 : 1$. Point E divides segment AD internally in the ratio $1 : 2$, then the position vector of E is ____

If $|\vec{a}| = 3$, $|\vec{b}| = 4$, $|\vec{c}| = 5$ such that each vector is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda\hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda\hat{j} + 2\hat{k}$, $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda\hat{k}$ are coplanar is...

If $ABC$ is a right-angled triangle in which $BC$ is the longest side and the position vector of $B$ and $C$ are respectively $3\hat{i} - 2\hat{j} + \hat{k}$ and $5\hat{i} + \hat{j} - 3\hat{k}$, then the value of $\overline{AB} \cdot \overl...