MHT CET 2026 11th April Evening Shift

If the function $f(x) = ax^3 + bx^2 + 11x - 6$, defined on $[1, 3]$, satisfies all the conditions of Rolle's theorem for $c = 2 + \dfrac{1}{\sqrt{3}}$, then

The tangent to the curve $y^2 - xy + 9 = 0$ is vertical when

The value of $k$ such that the function $f(x) = \sin x - \cos x - kx + b$ is strictly decreasing for all real $x$ is

The area bounded by the curve $y = x\left|\sin\left(\dfrac{x}{2}\right)\right|$ and the X-axis between the lines $x = 0$ and $x = 4\pi$ (in square units), is....

If the area enclosed between the curves $y^2 = 4kx$ and $y = kx, k > 0$ is $\dfrac{2k}{3}$ sq. units then $k = $

If the equation $3x^2 + (3 - p)xy + qy^2 - 2px = 8pq$ represents a circle, then the area (in sq. units) of this circle is

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + x + 1 = 0$ then $\alpha^{2026} + \beta^{2026} = $

If $\int_{2}^{e}\left[\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2}\right] dx = a + \dfrac{b}{\log 2}$, then values of $a$ and $b$ are

If $\int_{0}^{k} \dfrac{1}{1 + 4^x} dx = \log_2\left(\dfrac{4}{3}\right)$, then $k = $ _____

The solution of the differential equation $2x\dfrac{dy}{dx} - y = 3$ represents a family of

The solution of the differential equation $x\dfrac{dy}{dx} + y = x^3 y^6$, is...

The derivative of $\tan^{-1}\left(\dfrac{\sqrt{1+x^2} - 1}{x}\right)$ with respect to $\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)$ is...

If the derivative of $\tan^{-1}(a + bx)$ with respect to x at $x = 0$ is $1$, then $a^6 - b^3 = $

If $x^3 + y^3 = 6$ and $\dfrac{d^2y}{dx^2} \cdot \dfrac{d^2x}{dy^2} = \dfrac{m}{(xy)^n}$, where $m, n \in R$ then $\dfrac{m}{n} = $

If $x = \sin(t), y = \cos(pt)$, then $(1 - x^2)\dfrac{d^2 y}{dx^2} - x\dfrac{dy}{dx} = $

The ellipse $4x^2 + 9y^2 = 1$ intersects positive Y-axis at point A. If S and S' are its foci, then the area (in sq. units) of $\Delta SAS'$ is _____

If $g(x) = x^2 + x - 2$ and $\dfrac{1}{2}(g \circ f)(x) = 2x^2 - 5x + 2$ then $f(x)$ is equal to

$\int \dfrac{\sin x}{\sin 4x} dx = $

$\int \sqrt{4^x(4^x + 4)}\, dx = \cdots$

The value of $\int \dfrac{1}{x^2 - 5x + 8} dx$ is equal to...

$\int \dfrac{2x}{4 - 3x - x^2} dx = \cdots$

If $\cot(\cos^{-1} x) = \sec\left(\tan^{-1} \dfrac{a}{\sqrt{b^2 - a^2}}\right)$, then the value of $x$ is

$\lim_{x \to 1} \left[\dfrac{x-2}{x^2-x} - \dfrac{1}{x^3 - 3x^2 + 2x}\right] = $

Let the function f be defined by $f(x) = \dfrac{x - |x|}{x}$, for $x \neq 0$ and $f(0) = 2$ then f is

The corner points of the feasible region determined by a system of linear constraints are $(0, 3), (1, 1)$ and $(3, 0)$. If the objective function is $z = px + qy$, where $p, q > 0$, then the condition on p and q such that the minimum of z ...

If $A = \{1, 2, 3, 4, 5, 6\}$ then which of the following is not true?

Consider the following statements$r$: If $p \rightarrow q$ is false then $p \vee q$ is false.$s$: If $p \leftrightarrow q$ is false then $p \vee q$ is false.The truth values of $r \rightarrow s$ and $s \rightarrow r$ are respectively ____

If $A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$ then which of the following is true ?

If $\begin{bmatrix} 2 & 1 & -1 \\ -1 & 2 & 1 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ -2 \end{bmatrix}$ then the distance of point $P(a, b, c)$ from the plane $2x + y + 2z = 10$ is

The number of ways that 8 beads of different colours can be strung to form a necklace is

If $X \sim B(n, p)$, then the value of $\dfrac{P(X = k)}{P(X = k - 1)}$ is

A player tosses two fair coins, he wins Rs. $5$ if two heads appear, Rs. $3$ if one head appears and Rs. $2$ if no head appears, then the variance of the winning amount is

The probability mass function of a random variable X is given by$P(X = x) = \dfrac{{}^{5}C_x}{2^5}$, for $x = 0, 1, 2, 3, 4, 5$ and $= 0$, otherwise.Then which of the following is correct?

If events A and B are such that $P(A) = \dfrac{1}{4}$, $P(A \cap B) = \dfrac{1}{10}$ and $P(A/B) = 2P(B/A)$ then $P(B) = $...........

In $\Delta ABC$, if $\angle A = 90^\circ$, then $\sin(B - C) = $

In triangle ABC, with usual notations, if the angles are in Arithmetic Progression and $3c^2 = 2b^2$, then the measure of angle A is

The line $2x+y=4$ intersects X and Y axes at points A and B respectively. Let $C(p, q)$ be the point such that the lines AC and BC are perpendicular. The distance of point C from the midpoint of segment AB is...

If the distance between the lines represented by $(x - 2y)^2 + k(x - 2y) = 0$ is $3$ units and $k > 0$ then the equations of the lines are

If the slope of one of the two lines given by $\dfrac{x^2}{a} + \dfrac{2xy}{h} + \dfrac{y^2}{b} = 0$ is twice that of the other, then $h^2 : ab = $ ...

The vector equation of the plane passing through the point $A(-2, 7, 5)$ and parallel to the vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ is ...

The point of intersection of the two lines $\dfrac{x-3}{3} = \dfrac{y-3}{-1}, z - 1 = 0$ and $\dfrac{x-6}{2} = \dfrac{z-1}{3}, y - 2 = 0$ is....

If the centroid of a tetrahedron $OABC$ is $(1, 2, -1)$, where O is the origin, $A(a, 2, 3), B(1, b, 2), C(2, 1, c)$ are the other vertices, then the distance of the point $P(a, b, c)$ from the origin is...

If the lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-3}{1} = \dfrac{y-c}{2} = \dfrac{z}{1}$ intersect, then the radius of circle $x^2 + y^2 - 4x + 10y + c = 0$ is

The general solution of $\cos 4\theta = \cos 3\theta$ is $\theta = $.....

The value of $\cos^2 10^\circ - \cos 10^\circ \cos 50^\circ + \cos^2 50^\circ$ is equal to....

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\bar{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k}$ are three vectors in which $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$, then $x = \cdots$

If $|\bar{a}| = 4, |\bar{b}| = 3$ and $\bar{a} \cdot \bar{b} = 8$ then $[\bar{a}\ \ \bar{a} + \bar{b}\ \ \bar{a} \times \bar{b}] = $

Let $\bar{a}, \bar{b}, \bar{c}$ be the unit vectors such that $\bar{a}$ is perpendicular to $\bar{b}$ and the angle between $\bar{b}$ and $\bar{c}$ is $120^\circ$. If $\bar{a} + \bar{c}$ is perpendicular to $\bar{b} + \bar{c}$ then

Let $x_0$ be the point of local maxima of $f(x) = \bar{a} \cdot (\bar{b} \times \bar{c})$ where $\bar{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, \bar{b} = -2\hat{i} + x\hat{j} - \hat{k}$ and $\bar{c} = 7\hat{i} - 2\hat{j} + x\hat{k}$ then the val...

The parallelopiped is determined by vectors $\bar{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\bar{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, $\bar{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of parallelopiped on the parallelogram base determi...