MHT CET 2026 17th April Evening Shift

The coordinates of the points on the curve $4y = x^2$ that are nearest to the point $(0,5)$ are ...

The equation of the tangent to the curve $y = \sqrt{9 - 3x^2}$ at the point where the ordinate and abscissa equal is...

If the line $y = 4x - 5$ is tangent to the curve $y^2 = ax^3 + b$ at the point $(2,3)$, then the value of $7a - 2b$ is...

The area of the region bounded by curves $y = \sin x, y = \cos x$ and the lines $x = 0$, $x = \dfrac{\pi}{4}$ is

If the circles $x^2 + y^2 = 16$ and $x^2 + y^2 + 2ax + 4y + 4 = 0$ touch each other internally then $a =$

$\dfrac{(\cos 2\theta + i\sin 2\theta)^7}{(\cos 4\theta + i\sin 4\theta)^3} =$

The function $f(x) = \int_0^x\dfrac{dt}{1 + \cos t}$ satisfies which of the following differential equations?

If $\int_a^b(x^2 - x)\, dx = 18, \int_a^b x^3\, dx = 0$, then $a + b$ is equal to

The value of the definite integral $\int_0^{\pi}\dfrac{1}{5 + 4\cos x}\, dx$ is equal to ...

Let $f(x) = x - [x]$ for every real number $x$, where $[x]$ is integral part of $x$. then $\int\limits_{-1}^{1} f(x)\, dx$ is

The general solution of the differential equation $\sec y + (x - e^{\sin y})\dfrac{dy}{dx} = 0$ is...

The general solution of the differential equations $\dfrac{dy}{dx} = (9x + y + 5)^2$ is...

If $f(x) = \cos x\,\cos 2x\,\cos 4x\,\cos 8x\,\cos 16x$, then $f'\left(\dfrac{\pi}{4}\right) =$

If $y = \dfrac{1}{3x + 5}$, then the value of $\dfrac{d^9 y}{dx^9}$ is..

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

If $y = \sqrt{\cos x^2 + \sqrt{\cos x^2 + \sqrt{\cos x^2 + \ldots \infty}}}$ and $\dfrac{dy}{dx} = \dfrac{f(x)}{2y - 1}$ then, $\int f(x)\, dx = \ldots$

If $f(x) = \cot^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$, then $\lim\limits_{x \to a}\dfrac{f(x) - f(a)}{g(x) - g(a)}$, $\left(0 < a < \dfrac{1}{2}\right)$ is

If the line $3x + 4y + k = 0$ touches the ellipse $9x^2 + 16y^2 = 144$, then the value of k is.

If $f(x) = \dfrac{1 - x}{1 + x}$, then $f(f(\cos x)) =$

If $\int\dfrac{\sin x}{\sin 4x}\, dx = \alpha\log\left|\dfrac{1 + \sin x}{1 - \sin x}\right| + \beta\log\left|\dfrac{1 + \sqrt{2}\sin x}{1 - \sqrt{2}\sin x}\right| + c$, then the value of $32(\alpha + \beta^2) =$

If $\int f(x)\, dx = g(x)$ then $\int x^3 f(x^2)\, dx$ is equal to

If $\int\dfrac{3x + 7}{x^2 - 3x + 2}\, dx = m\log\left(\dfrac{x - 2}{x - 1}\right) + n\log(x - 2) + c$, where $m, n \in R$ and $c$ is an integration constant, then $m + n =$

If $f(x) = \dfrac{\sin^{-1}x}{\sqrt{1 - x^2}}$ and $g(x) = e^{\sin^{-1}x}$, then the value of $\int f(x)g(x)\, dx = \ldots$

If $\sin^{-1}(x - 2) + \cos^{-1}(x) + \tan^{-1}(x + 2) + \cot^{-1}(x + 4) = \sec^{-1}(\sqrt{k}) - \dfrac{\pi}{2}$, then $\cos(2\,\text{cosec}^{-1}\sqrt{k - 1}) = \ldots$

$\sin\left(3\sin^{-1}\left(\dfrac{1}{5}\right)\right) =$

$\cos^{-1}\left(\cos\dfrac{4\pi}{3}\right) + \sin^{-1}\left(\sin\dfrac{4\pi}{3}\right) = \ldots$

If the function $f(x) = \dfrac{4\sqrt{2}(\sin 3x + \sin x)}{2\sin 2x\sin\dfrac{3x}{2} + \cos\dfrac{5x}{2} - \cos\dfrac{3x}{2}}$ for $x \neq \dfrac{\pi}{2}$ is continuous at $x = \dfrac{\pi}{2}$, then the value of $f\left(\dfrac{\pi}{2}\righ...

In the following figure, the shaded region represents the system of constraints:

Which of the following statements is/are False?$S_1 : \exists\, n \in N$, such that $n^2 + n + 2$ is divisible by 4.$S_2 : \exists\, x \in N$, such that $x - 17 < 20$.$S_3 : \forall\, n \in N, \quad x^2 + 3x - 10 = 0$.$S_4 : \forall\, n \in...

The negation of $(p \wedge q) \rightarrow ((p \vee r) \rightarrow\, \sim q)$ is equivalent to ...

The statement pattern $[(p \wedge q) \rightarrow (\sim p \vee r)] \vee [(\sim p \vee r) \rightarrow (p \wedge q)]$ is

If $A = \begin{bmatrix} 1 & -\tan\dfrac{\theta}{2} \\ \tan\dfrac{\theta}{2} & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & \tan\dfrac{\theta}{2} \\ -\tan\dfrac{\theta}{2} & 1 \end{bmatrix}$ then $A^{-1}B$ is equal to

If $A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$; $B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ then, $(2A + 3B)^{-1} =$ _______

In a test, there are 5 questions of the 'true or false' type. No student has got all the answers correct and the sequence of answers for every student is unique. The maximum number of students who could have appeared for the test is....

On average, one out of 10 persons is busy. If six persons are selected at random, then the probability that at least 5 of them will be busy is...

If a discrete random variable $X$ takes the values $1, 2, 3, 4$ such that $2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4)$, then $P(X = 4) = \ldots$

Three numbers are chosen from 1 to 30. The probability that minimum is 10 and maximum is 26 is _______

In $\triangle ABC$, with usual notation, if cot A, cot B, cot C are in arithmetic progression, then

The line $y = 2x + c$ passes through a point that is equidistant from both the axes and lies in the first quadrant $(x > 0, y > 0)$. Then the value of c is...

If the equation $ax^2 + 4xy - 2y^2 + 4x + 8y + 1 = 0$ represents a pair of straight lines, then the coordinates of their point of intersection are.........

The equation of the perpendicular line from the point $(2,-3,1)$ to the line $\dfrac{x + 1}{2} = \dfrac{y - 3}{3} = \dfrac{z + 2}{-1}$ is

The angle between the line $x - 1 = 2 - y = \dfrac{2z - 6}{4}$ and the plane $\vec{r} \cdot (2\hat{i} + \hat{j} + \hat{k}) = 10$ is

The sum of the coordinates of one of the points on the line $\dfrac{x - 2}{1} = \dfrac{y + 3}{-2} = \dfrac{z + 5}{2}$ which is at a distance of 3 units from the point $(2, -3, -5)$ is ...

If A and B are the feet of the perpendiculars drawn from $(1, 2, 3)$ to planes $YZ$ and $ZX$, then the equation of the plane passing through the points A, B and the origin is

If the distance of point $B(2,1,-3)$ from the line passing through the point $A(4,-2,2)$, and parallel to the vector $\vec{c} = -4\hat{i} - 6\hat{j} - 2\hat{k}$ is $x$, then $x^4 + x^2 + 541 =$

If $\tan A$ and $\tan B$ are the roots of the equation $5x^2 - 4x + 1 = 0$, then the value of $A + B$ is...

Let $\vec{a}$ and $\vec{b}$ be linearly independent vectors such that$|\vec{a}| = \sqrt{3}, |\vec{b}| = 3$ and $|\vec{a} - \vec{b}| = 4$.If $\vec{a} \times (2\hat{i} + 2\hat{j} - \hat{k}) = (2\hat{i} + 2\hat{j} - \hat{k}) \times \vec{b}$ an...

The value of $b$ such that the scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with the unit vector parallel to the sum of the vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $b\hat{i} + 2\hat{j} + 3\hat{k}$ is one, is...

If $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, $\vec{a} \cdot \vec{b} < 0$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\sin\theta + \tan\theta$ is...

Let $\vec{a} = \lambda\hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 4\hat{j} + 4\hat{k}, \vec{c} = \hat{i} + \mu\hat{j} + \hat{k}$If $\vec{a}$ is parallel to $\vec{b}$ and $\vec{b}$ is perpendicular to $\vec{c}$ then $\lambda - \mu = \l...