MHT CET 2026 16th April Morning Shift

A ball is thrown in the air. Its height at any time $t$ is given by $h = 3 + 14t - 5t^2$, then the maximum height it can reach

The function $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in the interval.....

The minimum value of $\dfrac{\log x}{x}$ in the interval $(2, \infty)$ is

The equations of the tangent to the curve $x^2 + y^2 = 10$, where the tangent is parallel to the line $2x + y - 1 = 0$, are

If $(1 + i) \cdot (1 + 2i) \ldots\ldots\ldots (1 + ni) = x + iy$ (Where $i = \sqrt{-1}$ ), then the value of $(2) \cdot (5) \cdot (10)\ldots\ldots\ldots(1 + n^2)$

$\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{\sin x - \cos x}{1 + \sin x\cos x}\,dx =$

$\displaystyle\int_0^{\log 5} \dfrac{e^x\sqrt{e^x - 1}}{e^x + 3}\,dx =$

The integrating factor of the differential equation $(1 + t^2) + \left(x - e^{\tan^{-1}t}\right)\dfrac{dt}{dx} = 0$ is

The degree of the differential equation obtained from the equation $(y - a)^2 = 4(x - b)$ [where $a$ and b are arbitrary constants] is

The solution of the differential equation $\left(x + 2y^3\right)\dfrac{dy}{dx} - y = 0$ is

The rate of reduction is proportional to the square root of a persons existing assets at that moment. If his assets at the beginning are 10000 and they dwindle down to 5625 in 2 years, then the person will be bankrupt in

If $y = \tan^{-1}\left[\dfrac{x - \sqrt{1 - x^2}}{x + \sqrt{1 - x^2}}\right]$ , then $\dfrac{dy}{dx} =$

The derivative of $\log_8(\log_5 x)$ w. r. t. $x$ is

If $y^{\frac{1}{m}} + y^{\frac{-1}{m}} = 2x$ , then $(x^2 - 1)y_1^{\ 2} =$

If $f(x) = \dfrac{4x + 3}{6x - 4}$, $x \neq \dfrac{2}{3}$ and $(\text{fof})(x) = g(x)$ where $g: \mathbb{R} - \left\{\dfrac{2}{3}\right\} \rightarrow \mathbb{R} - \left\{\dfrac{2}{3}\right\}$, then $(g\,o\,g\,o\,g\,o\,g\,o\,g)\,(3) =$

If the line $y = 2x + \lambda$ is a tangent to the hyperbola $36x^2 - 25y^2 = 3600$, then $\lambda =$

$\displaystyle\int \dfrac{(x + 1)(x + \log x)^2}{x}\,dx =$

$\displaystyle\int \cot^4 x\,dx$ is equal to

$\displaystyle\int \dfrac{\cos^3 x}{\sin^2 x + \sin x}\,dx =$

$\cos\left(\cos^{-1}\left(-\dfrac{1}{2}\right) + \dfrac{\pi}{3}\right) =$

$\sin^{-1}\left(\dfrac{12}{13}\right) + \cos^{-1}\left(\dfrac{4}{5}\right) + \tan^{-1}\left(\dfrac{63}{16}\right) =$

$\displaystyle\lim_{x \to 0}\dfrac{(5^x - 1)^4\,\text{cosec}\,(x\log 5)}{\tan(x\log 5) \cdot \log(1 + x^2\log 25)} = \ldots\ldots$

If the derivative of the function $f(x) = \begin{cases} ax^2 + b & \text{if } x < -1 \\ bx^2 + ax + 4 & \text{if } x \geq -1 \end{cases}$ is continuous everywhere then

The feasible region represented by the constraints $y - 2x \leq 4, x + y \geq 5, x \leq 4, y \geq 2, x, y \geq 0$ is ...........

Simplest form of the following switching circuit is

Inverse of $\begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$ is

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A\,(\text{adj }A) = AA^T$, Then $5a + b =$

The number of arrangements of the numbers strictly between 10 and 1000 formed with the digits 0, 1, 2, 3, 4, 5, 6 without repetition is

A fair die is thrown at random. Let A be the event that the number obtained on the die is a non-even prime number and B be the event that the number obtained on the die is an odd number.Let $p : P(A) = \dfrac{1}{3}$, $q$ : A and B are indep...

Three ships A, B, C sail from England to India. If the odds in favour of their safe arrival are 2:5, 3:7 and 6:11 respectively, then the probability that the exactly two ships arrive safely is

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

The probability distribution of a random variable X is given by$X = x\(1\)2\(3\)4\(P(X = x)\)k\(2k\)3k$$4k$Then the c.d.f. of X is given by

Two dice are thrown successively 4 times and getting a doublet is called a success. The probability of getting at least 1 success is

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

In a triangle ABC, with usual notations $a = \sqrt{3} + 1$, $b = \sqrt{3} - 1$ and $\angle C = 60^\circ$ then the values of $\angle A$ and $\angle B$ respectively are

The number of lines passing through A (3, 4) and the sum of whose non-zero intercepts is zero is (are)

If $4ab = 3h^2$, then the ratio of the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is...

A triangle has two fixed vertices A $(a, 0)$ and B$(0, b)$ . Let its third vertex C is moving along the line $x = y$. If $s$ is the area of triangle ABC, then $\dfrac{ds}{dx} =$

The lines $\dfrac{x - 2}{1} = \dfrac{y - 3}{1} = \dfrac{z - 4}{-k}$ and $\dfrac{x - 1}{k} = \dfrac{y - 4}{2} = \dfrac{z - 5}{1}$ are coplanar if

The direction ratios of the normal to the plane passing through (1, 0, 0), (0, 1, 0) which makes an angle of measure $45^\circ$ with the plane $2x + 3y = 7$ are....

The lines $\dfrac{x - 1}{-1} = \dfrac{y + 2}{1} = \dfrac{z - 3}{-2}$ and $\dfrac{x - 1}{1} = \dfrac{y + 2}{1} = \dfrac{z + 1}{-2}$ are

If the product of the distances of the point (1, 2, 3) from the origin and the plane $2x - 3y + z + k = 0$ is 7, then the value of k is

The vector equation of the line whose cartesian equations are $x = 2, 2y - 3z + 7 = 0$

If $0 \leq x \leq \dfrac{\pi}{2}$ , then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

The value of $\dfrac{\sin^2 3A}{\sin^2 A} - \dfrac{\cos^2 3A}{\cos^2 A}$ is

If a vector $3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are $a + 1, -3, 5$ . The possible values of $a$ is

Let A, B, C, D be the points in the plane with position vectors $-2\hat{i} - \hat{j}$, $4\hat{i}$, $3\hat{i} + 3\hat{j}$ and $-3\hat{i} + 2\hat{j}$ respectively, then $\square$ABCD is

Let $\bar{a} = \hat{i} + \hat{j}$, $\bar{c} = \hat{i} - \hat{j}$ and a vector $\bar{b}$ be such that $\bar{a} \times \bar{b} = \bar{c}$ and $\bar{a} \cdot \bar{b} = 3$ then $|\bar{b}| =$

The acute angle between the vector $2\hat{i} + \hat{j} - 3\hat{k}$ and the plane containing the vectors $2\hat{i} + 3\hat{j} - \hat{k}$ and $\hat{i} - \hat{j} + 2\hat{k}$ is

A parallelogram is constructed on $5\bar{a} + 2\bar{b}$ and $\bar{a} - 3\bar{b}$ as its adjacent sides, with $|\bar{a}| = 2\sqrt{2}, |\bar{b}| = 3$ . The angle between $\bar{a}$ and $\bar{b}$ is $\dfrac{\pi}{4}$ . Then the length of the dia...