MHT CET 2026 13th April Morning Shift

A point on the parabola $y^2 = \dfrac{36}{5}x$ at which the ordinate increases at thrice the rate of the abscissa is .....

The equation of the normal to the curve $xy + 7 = 0$ is $Ax + By + C = 0$, then

A wire 40 metre in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. The lengths of the two pieces so that the combined area of the square and the circle is minimum, are respectively

If a spherical balloon has a variable diameter $3x + \dfrac{9}{2}$ units, then the rate of change of its volume with respect to $x$ is

The area of the region (in sq.units) bounded by the curve $y = 2\sqrt{1 - x^2}$ and the X-axis is _____

If a circle with center $(-1, 1)$ touches the line $x + 2y + 4 = 0$, then the co-ordinates of the point of contact are

The polar form of the complex number $z = \dfrac{1}{1 + i}$, (where $i = \sqrt{-1}$) is

If $\displaystyle\int_0^a \dfrac{dx}{1 + 4x^2} = \dfrac{\pi}{8}$, then $a =$

The value of the integral $\displaystyle\int_{-\sqrt{3}}^{\sqrt{3}} \dfrac{(2x^9 + 3x^8 - 5x^7 + 9x^6 + 4x^3 - x + 3)}{x^2 + 3} \, dx$ is

If $\dfrac{dy}{dx} = y + 5$ and $y(0) = 4$ then $y(\log 2)$ is equal to

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x+y}{x-y}$ is

A spherical balloon expands at a rate proportional to its surface area. Initially its radius is 2 cm and 5 minutes later it increases upto 7 cm, then surface area of spherical balloon after 12 minutes will be

If $y = \tan^{-1}\left\{\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right\}$, where $|x| < 1$, then $\dfrac{dy}{dx}$ is equal to

If $y = x + e^x$, then $\dfrac{d^2 x}{dy^2}$ is equal to

Let $f(x) = x - 5$.If $g(x) = [f(4h(x) + 3)]^2$ and $h(1) = 4, h'(1) = -2$, then $g'(1) =$ .....

The domain of the function $f(x) = e^{\sqrt{5x - 3 - 2x^2}}$ is

$\displaystyle\int \dfrac{30x^{14} + 15^x \log 225}{x^{15} + 15^x} \, dx =$

If $\displaystyle\int \dfrac{\sin 2x}{\sin^4 x + \cos^4 x} \, dx = f(x) + c$ where c is constant of integration, then the value of $\tan(f(x))$ is

$\displaystyle\int e^{2x} \dfrac{2(\sin 2x \cos 2x - 1)}{2 \sin^2 2x} \, dx = A \, e^{2x} \cot 2x + c$(Where c is the constant of integration.), then $A^3 =$

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

$\displaystyle\lim_{x \to 0}\left[\dfrac{\log|2 + x| - \log|2 - x|}{\tan x}\right] = $ .......

If $f(x) = \begin{cases} \dfrac{(8 - 2x)^{\frac{1}{3}} - 2}{3 - (243 + 5x)^{\frac{1}{5}}}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$, then $k =$

In L.P.P., the corner points of the feasible region for the constraints $3x - y \geq 6, x \leq 3, y \leq 2, y \geq 0, x \geq 0$ are .....

Consider the statement patternsA. $(q \rightarrow p) \vee (p \rightarrow q)$B. $(\sim p \vee \sim q) \leftrightarrow \sim (p \wedge q)$C. $[(p \vee q) \wedge \sim p] \wedge \sim q$D. $(p \wedge q) \wedge (\sim p \vee \sim q)$then statement ...

The statement pattern $(p \wedge q) \rightarrow (r \vee \sim s)$ is false. Then the truth values of $p, q, r$ and $s$ are respectively

Let $A = \begin{bmatrix} -5 & -3 \\ 2 & 1 \end{bmatrix}$. The Row transformation $R_1 \rightarrow R_1 + 3R_2$ will transform matrix A into

If $A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}_{3 \times 3}$, and $A^{-1} = \begin{bmatrix} \gamma & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2 \end{bmatrix}_{3 \times 3}$, then $|\alpha \cdot \beta \cdot \gamma...

The area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 48y$ to the ends of its latus rectum, is .....

The number of cubic equations that can be formed with the coefficients 0, 3, 2, 4, 5, 6 when repetition of coefficients is allowed, is....

The odds in favour of A winning a game of table tennis against B are 1:2. If 3 games are to be played, then the probability of A winning at least two games out of the three is.....

A random variable X has the following probability distribution$X = x\(1\)2\(3\)4\(5\)6\(7\)8\(P(X=x)\)0.15\(0.23\)0.10\(0.12\)0.20\(0.08\)0.07$$0.05$For the events $E = \{X \text{ is a prime number}\}$, $F = \{X < 4\}$, $P(E \cup F)$ is

A random variable X takes the values 0, 1, 2, 3 and its mean is $1.3$. If $P(X=3) = 2P(X=1)$ and $P(X=2) = 0.3$, then $P(X=0)$ is

Let $X \sim B\left(6, \dfrac{1}{2}\right)$. Then the maximum probability occurs at

In a triangle ABC, with usual notations if $\dfrac{s-a}{11} = \dfrac{s-b}{12} = \dfrac{s-c}{13}$ and $\lambda \tan^2 \dfrac{A}{2} = 455$, then $\lambda =$

With usual notations, in $\triangle ABC$, if $2a^2 = b^2 + c^2$, then $\dfrac{\cos 3A}{\cos A} + 2 =$

With usual notations, in $\triangle ABC$, $(b - c)^2 \cos^2 \dfrac{A}{2} + (b + c)^2 \sin^2 \dfrac{A}{2} =$

The area of the triangle formed by the co-ordinate axes and a line $px + qy = r$ (where p, q and r are positive real numbers) is $\dfrac{1}{54}$ sq.units, then ....

The joint equation of the pair of lines through the origin and making an angle of $\dfrac{\pi}{4}$ with the line $3x + y - 6 = 0$ is

The point having position vector $4\hat{i} - 11\hat{j} + 2\hat{k}$ lies on the line

The shortest distance between the lines, where the first line passes through $(0,0,0)$ and $(2,0,3)$ and the second line passes through $(2,5,0)$ and $(0,4,0)$ is

The co-ordinates of the foot of the perpendicular from the origin to the plane $2x - 3y - 6z = 49$ are...

Let a plane P pass through the point $(3, 7, -7)$ and contain the line $\dfrac{x - 2}{-3} = \dfrac{y - 3}{2} = \dfrac{z + 2}{1}$. If the distance of the plane P from the origin is d, then $d^2$ is

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

The number of solutions in $\left[0, \dfrac{\pi}{2}\right)$ of the equation $\cos 3x \cdot \tan 5x = \sin 7x$ is

$\tan 105^\circ = $ ......

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors having magnitudes 1, 1 and 2 respectively. If $\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}$, then the acute angle between $\bar{a}$ and $\bar{c}$ is

The volume of a tetrahedron with vertices $5\hat{i} - \hat{j} + \hat{k}, 7\hat{i} - 4\hat{j} + p\hat{k}, \hat{i} - 6\hat{j} + 10\hat{k}$ and $-\hat{i} - 3\hat{j} + 7\hat{k}$ is 11 cubic units, then one of the values of p is

Let $\bar{a}, \bar{b}, \bar{c}$ be unit vectors such that $\bar{a}$ is perpendicular to the plane of $\bar{b}$ and $\bar{c}$. If the angle between $\bar{b}$ and $\bar{c}$ is $\dfrac{\pi}{3}$, then $|\bar{a} + \bar{b} + \bar{c}| =$

If a parallelogram is constructed on the vectors $\bar{a} = 3\bar{p} - \bar{q}, \bar{b} = \bar{p} + 3\bar{q}$ and $|\bar{p}| = 3, |\bar{q}| = 2$ and angle between $\bar{p}$ and $\bar{q}$ is $\dfrac{\pi}{3}$, then the ratio of the lengths of...

A vector $\bar{r}$ of magnitude $3\sqrt{2}$ units which makes angles of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ respectively with Y and Z axes is