MHT CET 2026 19th April Evening Shift

Let PA and PB be the tangent segments drawn from point P$(6, 8)$ to the circle with the centre at origin O. The radius of circle for which the area of quadrilateral PAOB is maximum, is...

Rolle's theorem holds for monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem in the interval $[\beta, 3]$ where $g(3) = 0$, such that the...

If the rate of increase of surface area of a spherical balloon is $5\,\text{cm}^2/\text{sec}$ and rate of increase of volume of a spherical balloon is $10\,\text{cm}^3/\text{sec}$, then the radius of the balloon at that time is...

The area (in sq. units) of the region bounded by the curve $y = 2x - x^2$ and the X-axis is...

The area (in square units) of the region bounded by the circle $x^2 + y^2 = 9$ and the parabola $y^2 \leq 8x$ is...

A circle passes through the point $(0,1)$ and touches the parabola $y = x^2$ at the point $(1,1)$. The centre of the circle is...

If $z = \sum_{n=0}^{2026} i^n$, where $i = \sqrt{-1}$, then one of the values of $\sqrt{z}$ is...

Point A$(5, 12)$ rotated about the origin O in the XY-plane through an angle of $30^\circ$ in the anticlockwise direction to a new position B. The ordinate of point B is...

If $f(x) = |5x - 3|$ is defined on interval $[0, 1]$, then the value of $\int_0^1 f(x)\,dx$ is...

The value of integral $\int_0^\infty \dfrac{1}{1 + e^x}\,dx$ is...

If $f(x)$ is an antiderivative of $\dfrac{x + 1}{\sqrt{x - 1}}$ with respect to $x$, then value of $f(5) - f(2)$ is...

The differential equation of all lines where the length of the normal from the origin is p and the inclination of the normal is $\alpha$ is... (where p and $\alpha$ are arbitrary constants)

If the solution of the differential equation $(1 + x^3)\dfrac{dy}{dx} + 6x^2y = 1 + x^2$ is $y = \dfrac{1}{(1 + x^3)^s}\left[x + \dfrac{x^p}{p} + \dfrac{x^q}{q} + \dfrac{x^r}{r} + c\right]$, then the LCM of $p, q, r$ and $s$ is...

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

If $x^m + y^m = k, (m \neq 1)$ and $y'' = \dfrac{ax^b}{y^c}$ such that $a + b + c = 0$, then the value of k is...

If $y = \sin^{-1}\left(\dfrac{25 - x^2}{25 + x^2}\right)$, then $y'(1)$ is...

If $f(x)$ and $g(x)$ are inverse functions of each other and $f(x) = x + e^x$, then $g'(x) = $...

If $3y^2 - 2xy - x = 0$, then the value of $\dfrac{dy}{dx}$ at $y = 2$ is...

The eccentricity of the ellipse represented by the equation $7x^2 + 16y^2 - 14x + 64y - 377 = 0$ is...

If $g(x) = 1 - \sqrt{x}$ and $f(g(x)) = 5 + 4\sqrt{x} + x$, then the value of $f(6)$ is...

Let $f(x) = 1 - \dfrac{1}{x}, g_2(x) = f(f(x)), g_3(x) = f(f(f(x)))$ and so on. If $\int x \cdot g_{2026}(x)\,dx = \int g_{2025}(x)\,dx + h(x) + c$, then $h(x) = $...

The value of integral $\int x^3 \cos x\,dx$ is...

If $u$ and $v$ are functions of $x$, then $\int \dfrac{1}{v^3}\left(uv\dfrac{du}{dx} - u^2\dfrac{dv}{dx}\right)dx = $

The value of $\sin^{-1}\left(\sin\dfrac{5\pi}{6}\right) + \cos^{-1}\left(\cos\dfrac{7\pi}{6}\right) + \tan^{-1}\left(\tan\dfrac{2\pi}{3}\right)$ is equal to...

The lengths of the shadows of a tree of height p, thrown by sun's rays at three different moments are p, 2p and 3p. The sum of the angles of elevation of the sun's rays at these three moments is equal to...

If $\lim_{x \to 1} \dfrac{x^3 + ax^2 + bx + c}{x^2 - 2x + 1} = 2026$ then the value of $a - c$ is...

If the function $f(x) = \left(\dfrac{5x - 8}{8 - 3x}\right)^{\frac{3}{2x - 4}}$, for $x \neq 2$ is continuous at $x = 2$, then the value of $f(2)$ is...

An airplane can carry a maximum of $250$ passengers. A profit of Rs $1500$ is made on each executive class ticket and a profit of Rs $900$ is made on each economy class ticket. The airline reserves at least $30$ seats for executive class. H...

If the truth value of $[(p \vee q) \wedge (q \to r) \wedge (\sim r)] \to (p \wedge q)$ is false, then which of the following is NOT true?

If p, q, r are simple propositions with truth values T, F, T respectively, then which of the following is not a true statement?

If $\sim p \to q$ is false and $q \leftrightarrow r$ is false, then the truth value of $(p, q, r)$ is...

If matrix $A = \begin{bmatrix} -1 & 2025 & 2026 \\ 0 & 2 & 2027 \\ 0 & 0 & -1 \end{bmatrix}$, then the sum of all elements in $\text{adj}(A^{-1})$ is equal to...

The inverse of the matrix $A = \begin{bmatrix} 2 & -1 & 4 \\ 4 & -3 & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is $B = \dfrac{1}{37}\begin{bmatrix} -5 & 9 & 11 \\ -3 & -2 & 14 \\ 11 & -5 & k \end{bmatrix}$, then the value of $k$ is...

If ${}^{n}C_4, {}^{n}C_5$ and ${}^{n}C_6$ are in arithmetic progression (A.P.), then the value of n is...

The probability that a bomb will hit the target is $0.8$. Out of 6 bombs dropped, probability that at least 1 will miss the target is...

Two cards are drawn at random from a box which contains 5 cards numbered 1, 1, 2, 2 & 3. If X denotes the sum of the numbers, then the expected value of X is...

Bag A contains 3 white and 5 black balls while bag B contains 4 white and 3 black balls. A ball is selected at random from bag A and put in bag B. If a ball is now selected at random from bag B then the probability that this ball is white b...

In $\triangle ABC$, with usual notations, $(a + b + c)(b + c - a)(c + a - b)(a + b - c) = 3b^2c^2$, then $\angle A = $

If ABC is a triangle of area $\Delta$ with $a = 2, b = \dfrac{7}{2}, c = \dfrac{5}{2}$, where $a, b, c$ are the lengths of the sides of the triangle opposite to angles A, B and C respectively, then $\dfrac{2\sin A - \sin 2A}{2\sin A + \sin ...

The family of straight lines $4ax + 3by + c = 0$ such that $a + b + c = 0$ (where a, b, c are real constants) are concurrent at the point...

If two lines represented by $x^2 - (1 + \sqrt{3})xy + \sqrt{3}y^2 = 0$ make angles $\alpha$ and $\beta$ with the X-axis, then $\tan(\alpha + \beta)$ is...

The perpendicular distance from the origin to the plane containing the points $(1, -2, 1), (2, -1, -3)$ and $(0, 1, 5)$ is...(in units)

If $\alpha, \beta, \gamma$ are the direction angles of the line $x = 4z + 3$ and $y = 2 - 3z$, then the value of $\cos\alpha + \cos\beta + \cos\gamma$ is...

The shortest distance between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(\hat{i} + 4\hat{j} - 5\hat{k})$ is...

Let $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 7\hat{k}) = 32$ is the equation of a plane and the line having direction ratios $(5, b, 3)$ is parallel to the plane, then the value of b is...

A line passing through the points $(1, -1, 2)$ and $(2, 0, 1)$ meets the XY plane and the YZ plane at points A and B respectively. The distance AB is equal to...

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $\vec{a} \perp (\vec{b} + \vec{c}), \vec{b} \perp (\vec{c} + \vec{a}),$ and $\vec{c} \perp (\vec{a} + \vec{b})$ and $|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 3$, then $|\vec{a} + \...

Two adjacent sides of a parallelogram ABCD are given by $\overline{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$ and $\overline{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$. The side $AD$ is rotated by an acute angle $\alpha$ in the plane of the paralle...

The maximum volume of a parallelopiped (in cubic units) with vectors $(2a\hat{i} + \hat{k}), (a\hat{j} - a\hat{k})$, and $(3\hat{i} + a\hat{j})$, where $a \in [0, 1]$, as its coterminous edges is...

Let O$(0, 0)$, A$(-1, 2)$ and B$(1, 3)$ be the vertices of $\triangle$OAB. The bisector of angle O intersects side AB at point D. The value of $\vec{OD} \cdot \vec{AB}$ is equal to...