MHT CET 2026 18th April Morning Shift

If the line $x + By + C = 0$ is the normal to the curve given by $x = a\sin^3 t$, $y = b\cos^3 t$, (where $a, b \neq 0$) at a point $t = \dfrac{\pi}{2}$, then $B - C = $

If the tangent to the curve $xy + ax + by = 0$ at $(1,1)$ makes an angle of $\tan^{-1}2$ with positive direction of the $x$-axis, then the value of $\dfrac{ab}{a+b}$ is...

If the side of an equilateral triangle increases at the rate of $\sqrt{3}\ \text{cm/sec}$, then the rate of change of increase of its area when the side is $12\ \text{cm}$ is ____

The area (in square units) bounded by the line $y = x$, the X-axis and the lines $x = -2$ and $x = 4$ is...

The area of the region (in sq. unit) bounded by x-axis, the tangent and normal to the circle $x^2 + y^2 = 4$, drawn at a point $(1, \sqrt{3})$ is

The tangent to the circle $x^2 + y^2 = 10$ at the point $(3,1)$ touches the circle $x^2 + y^2 - 2\sqrt{10}\,x - 20y + k = 0$, then the value of $k$ is...

The value of $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^{18} + \left(\dfrac{-1-i\sqrt{3}}{2}\right)^{18}$ is

The value of the definite integral $\int_0^{\log_e 5}\dfrac{e^x\sqrt{e^x-1}}{e^x+3}\,dx$ is equal to...

If $[\cdot]$ denotes the greatest integer function, then $\int_0^{\pi/3}[\tan x]\,dx = $

The equation of the curve whose slope is $\dfrac{y-1}{x^2+x}$ and which passes through the point $(1, 0)$ is

A body cools from $100^\circ\text{C}$ to $60^\circ\text{C}$ in 20 minutes, the temperature of the surroundings being $20^\circ\text{C}$. The total time taken (in minutes) for the body to cool down to $40^\circ\text{C}$ is ...

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

Let $t \in (0, 1)$ and $\alpha \in \left(0, \dfrac{\pi}{4}\right)$. If $x = \text{cosec}^{-1}\left(\dfrac{1+t^2}{2t}\right)$, $y = \cot^{-1}\left(\dfrac{\sqrt{1-t^2}}{t}\right)$ and $\dfrac{dy}{dx} = f(t)$, then the value of $f(\tan\alpha)$...

If $x\,e^{xy} = y + \sin^2 x$, then the value of $\dfrac{dy}{dx}$ at $x = 0$ is equal to

If $g(x) = (x^2 + 2x + 1)\cdot f(x)$ such that $f(0) = 5$ and $\lim\limits_{x \to 0}\dfrac{f(x)-5}{x} = 4$ then $g'(0) = $

If the eccentricity of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b)$ is $\dfrac{2}{3}$ and its focal chord is $3x + 2y - 6 = 0$, then the value of $a^2 + b^2$ is...

Let $f(x) = ax + b$ and $g(x) = cx + d$. The condition $f(g(x)) = g(f(x))$ holds for all $x$ if and only if ...

If $0 \leq x \leq 1$, $I_1 = \int\sin^{-1}\sqrt{1-x^2}\,dx$ and $I_2 = \int\sin^{-1}x\,dx$, then which of the following is true?

$\int\sin(\log x)\,dx = $

The value of $\int\dfrac{n\sqrt{\text{cosec}^2 x^n - 1}}{x^{(1-n)}}\,dx$ is ...

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

If $y = \tan^{-1}\left(\dfrac{\log\left(\dfrac{e}{x^3}\right)}{\log ex^3}\right) + \tan^{-1}\left(\dfrac{\log(e^4x^3)}{\log\left(\dfrac{e}{x^{12}}\right)}\right)$, $x \in \left(e^{-\frac{1}{3}}, e^{\frac{1}{12}}\right)$ then $\dfrac{dy}{dx}...

If $0 \leq x \leq 1$ and $(\sin^{-1}x)^3 + (\cos^{-1}x)^3 = a\pi^3$ then

If $\sum\limits_{n=1}^{2026}\tan^{-1}\left(\dfrac{1}{n^2+n+1}\right) = \tan^{-1}\left(1 - \dfrac{1}{x}\right)$, where $x \neq 0$, then $x = $

If $\lim\limits_{x \to \infty}\left[\dfrac{x^2+x+1}{x+1} - ax - b\right] = 3$, then $a - b = $...

The number of point / points where the function $f(x) = \dfrac{1}{x^2-5|x|+6}$ is discontinuous is......

The maximum value of $Z = 4x + 5y$, subject to the constraints $3x + y \leq 15, 3x + 4y \leq 24, x \geq 0, y \geq 0$ is

The correct logical equivalence from the following is /are ___(I) $p \to (q \to r) \equiv (p \wedge q) \to r$(II) $(p \to q) \to r \equiv p \to (q \vee r)$(III) $(p \to q) \to r \equiv (p \to r) \wedge (\sim q \to r)$(IV) $p \to (q \to r) \...

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

The negation of the converse of $p \vee q$ is

Let $A = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$. If $B = \text{adj}\,A$, then the matrix $B^{-1}$ is equal to...

Let $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & -13 \\ 5 & -10 \end{bmatrix}$ be two matrices. If the variables $x$ and $y$ satisfy the matrix equation $((A^{-1})^2 + B)\begin{bmatrix} x \\ y \end{bmatr...

11 players of the Indian cricket team are sitting at a circular table. The number of ways they can sit so that two players, the wicket keeper and the captain never sit together is ___

A fair coin is tossed 9 times. On each toss, a man predicts that the outcome will be heads. The probability that the number of successful predictions is strictly greater than the number of unsuccessful predictions is...

Let $F(x)$ be the cumulative distribution function (c.d.f.) of a continuous random variable $X$. If $F(b) = 0.7$ and $P(X > a) = 0.4$, then the value of $P(a < X < b)$ is ...

If the probabilities of a student succeeding in the entrance tests for institutes A, B and C are $0.6$, $0.5$ and $0.4$ respectively, while the probability of succeeding in both A and B is $0.3$, in both B and C is $0.2$, in both A and C is...

In $\triangle ABC$, if $\angle C = \dfrac{\pi}{3}$, then the value of $\cos^2 A + \cos^2 B + \cos A\cos B$ is...

In triangle ABC, with usual notations, if $(a+b+c)(a+b-c) = ab$, then the measure of angle C is...

In $\triangle ABC$, with usual notation, if $a = 13, b = 14, c = 15$, then the sum of the values of $\sin\left(\dfrac{A}{2}\right)$ and $\sin A$ is....

The line $L_1$ given by $\dfrac{x}{p} + \dfrac{y}{2} = 1$ passes through the point $(5,0)$. The line $L_2$ given by $\dfrac{x}{10} + \dfrac{y}{q} = 1$ is parallel to $L_1$. Then the distance between the lines $L_1$ and $L_2$ is...

Two lines are given by $x^2 - 4xy + 4y^2 + kx - 2ky = 0$, then the value of k so that the distance between them is 3 is

The line $\ell$ passes through the point $(2, 1, 1)$ and is parallel to the plane $x + y + 2z = 18$. If line $\ell$ intersects the line $\dfrac{x+2}{3} = \dfrac{y+1}{-1} = \dfrac{z-2}{1}$, then equation of the line $\ell$ is...

The acute angle $\theta$ between the $xy$-plane and the plane passing through the point $(1, 2, 4)$ and parallel to the vectors with direction ratios $3, 2, -1$ and $1, -2, -2$ is...

If the plane $\bar{r} = (\lambda + \mu)\hat{i} + (2 + \mu)\hat{j} + (3\lambda + 2\mu)\hat{k}$, where $\lambda$ and $\mu$ are parameters, intersects coordinate axes at points $(a, 0, 0)$, $(0, b, 0)$ and $(0, 0, c)$ then $a + b + c = $...

The coordinates of the point of intersection of the lines $\dfrac{x-3}{1} = \dfrac{y-5}{2} = \dfrac{z-1}{-1}$ and $\dfrac{x-4}{2} = \dfrac{y-2}{-1} = \dfrac{z-4}{2}$ are...

If the plane $2x + 3y + z = 6$ cuts coordinate axes at A, B and C, then the volume of tetrahedron OABC (where O is the origin) is ......cubic units.

If $\bar{a} = \hat{i} - \hat{k}$, $\bar{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}$ and $\bar{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}$ then $[\bar{a}\ \bar{b}\ \bar{c}]$ depends on

Let $\bar{u}, \bar{v}, \bar{w}$ be three vectors such that $|\bar{u}| = 1, |\bar{v}| = 2, |\bar{w}| = 3$. If the projection of $\bar{v}$ along $\bar{u}$ is equal to the projection of $\bar{w}$ along $\bar{u}$ and $\bar{v}, \bar{w}$ are perp...

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors of equal magnitude such that the angle between $\bar{a}$ and $\bar{b}$ is $\alpha$, $\bar{b}$ and $\bar{c}$ is $\beta$, $\bar{c}$ and $\bar{a}$ is $\gamma$.Then the minimum value of $\cos\alp...

A vector which is orthogonal to the vector $\bar{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and coplanar with the vectors $\bar{b} = 3\hat{i} + 2\hat{j}$ and $\bar{c} = 2\hat{i} + \hat{j} + 3\hat{k}$ is