List of top Questions asked in MHT CET- 2023

The value of $\tan^{-1} \left(\frac{1}{8}\right) + \tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{5}\right)$ is

If $f(a) = 2$, $f'(a) = 1$, $g(a) = -1$, $g'(a) = 2$, then as $x$ approaches $a$, $\frac{g(x)f(a)-g(a)f(x)}{(x-a)}$ approaches}

The differential equation representing the family of curves $y^2 = 2c(x + \sqrt{c})$, where $c$ is a positive parameter, is of}

If in a regular polygon, the number of diagonals are 54, then the number of sides of the polygon are}

Let $P = (-3, 0)$, $Q = (0,0)$ and $R = (3,3\sqrt{3})$ be three points. Then the equation of the bisector of the angle PQR is

If a body cools from $80^{\circ}\text{C}$ to $50^{\circ}\text{C}$ in the room temperature of $25^{\circ}\text{C}$ in 30 minutes, then the temperature of the body after 1 hour is

Let $x_0$ be the point of local minima of $f(x) = \vec{a} \cdot (\vec{b} \times \vec{c})$ where $\vec{a} = x\hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = -2\hat{i} + x\hat{j} - \hat{k}$, $\vec{c} = 7\hat{i} - 2\hat{j} + x\hat{k}$, then value of $\vec{a} \cdot \vec{b}$ at $x = x_0$ is

The diagonal of a square is changing at the rate of $0.5$ cm/sec. Then the rate of change of area when the area is $400$ cm$^{2}$ is equal to}

Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c} = \vec{b} \times \vec{a}$ and $\vec{c} \cdot \vec{a} = 0$, then $\vec{c} \cdot \vec{b}$ is

Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be the vectors such that $|\vec{u}| = 1$; $|\vec{v}| = 2$; $|\vec{w}| = 3$. If the projection of $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v}$, $\vec{w}$ are perpendicular to each other, then $|\vec{u} - \vec{v} + \vec{w}|$ is equal to}

If $y = 4x - 5$ is a tangent to the curve $y^{2} = px^{3} + q$ at $(2, 3)$, then $p-q$ is

The function $f(x) = x^3 - 6x^2 + 9x + 2$ has maximum value when $x$ is

If $I_n = \int_0^{\pi/4} \tan^n \theta d\theta$, then $I_{12} + I_{10}$ is equal to}

A fair die with numbers 1 to 6 on their faces is thrown. Let $X$ denote the number of factors of the number, on the uppermost face, then the probability distribution of $X$ is

The centre of the circle whose radius is 3 units and touching internally the circle $x^2 + y^2 - 4x - 6y - 12 = 0$ at the point $(-1, -1)$ is

If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}, B = \begin{bmatrix} 1 & 1 \\ 4 & -1 \end{bmatrix}$, then $(A+B)^{-1}$ is

If the slope of the tangent of the curve at any point is equal to $-y+e^{-x}$, then the equation of the curve passing through origin is

The function $f$ defined on $\left(-\frac{1}{3}, \frac{1}{3}\right)$ by $f(x) = \left\{ \begin{array}{ll} \frac{1}{x} \log\left(\frac{1+3x}{1-2x}\right) & , x \neq 0 \\> k & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then $k$ is

If $|\vec{a}| = 2$, $|\vec{b}| = 3$, $|\vec{c}| = 5$ and each of the angles between the vectors $\vec{a}$ and $\vec{b}$, $\vec{b}$ and $\vec{c}$, $\vec{c}$ and $\vec{a}$ is $60^\circ$, then the value of $|\vec{a} + \vec{b} + \vec{c}|$ is

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by}

The mirror image of $P(2, 4, -1)$ in the plane $x - y + 2z - 2 = 0$ is $(a, b, c)$, then the value of $a+b+c$ is

If the statement $p \leftrightarrow (q \rightarrow p)$ is false, then true statement/statement pattern is

Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is

If both mean and variance of 50 observations $x_1, x_2, \ldots, x_{50}$ are equal to 16 and 256 respectively, then mean of $(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2$ is

Let $z \in C$ with $\text{Im}(z) = 10$ and it satisfies $\frac{2z-n}{2z+n} = 2i - 1$, $i = \sqrt{-1}$ for some natural number $n$, then}