List of top Questions asked in MHT CET- 2025

In L.P.P., the maximum value of objective function $Z = 6x + 3y$ subject to $x + y \leq 5, x + 2y \geq 4, 4x + y \leq 12, x, y \geq 0$ is \dots

The order of the differential equation whose general solution is given by $y = (C_1 + C_2) \sin(x + C_3) - C_4 e^{x+C_5}$ is \dots}

If $(\tan^{-1} x)^2 + (\cot^{-1} x)^2 = 5\pi^2/8$, then $x^2 + 1 = \dots$

If the vectors $\vec{a} = c (\log_7 x) \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_7 x) \hat{i} + 3c (\log_7 x) \hat{j} - 4\hat{k}$ make obtuse angle for any x > 0, then c belongs to ______.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable function having $f(3) = 3, f'(3) = 1/27$ and $g(x) = \begin{cases} \int_3^{f(x)} \frac{3t^2}{x-3} dt, & x \neq 3 \\ K, & x = 3 \end{cases}$ is continuous at $x = 3$, then $K = \dots$

If the line $\frac{x-3}{2} = \frac{y+5}{1} = \frac{z+2}{2}$ lies in the plane $\alpha x + 3y - z + \beta = 0$, then values of $\alpha$ and $\beta$ respectively are \dots}

$\int_{\log(1/2)}^{\log 2} \sin \left( \frac{e^x - 1}{e^x + 1} \right) dx = \_\_\_\_\_\_.$

The area bounded by the curve $y = 4x - x^2$ and X-axis in square units, is \dots

The Cartesian equation of the plane $\vec{r} = (2\hat{i} - 3\hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})$ is \dots}

The altitude through vertex A of $\triangle ABC$ with position vectors of points A, B, C as $\vec{a}, \vec{b}, \vec{c}$ respectively is ______.

The lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ are \dots}

$\int_{\pi/4}^{\pi/2} 2\sin^{-4} x dx = \_\_\_\_\_\_.$
Note: The initial OCR showed "$23.4 \frac{/2}{/4}$". The "4" was a misread coefficient. The mathematical evaluation of the options indicates a coefficient of 2 is present in the intended question.

$\int \frac{dx}{x(x^3 + 1)} = \dots$

If $\vec{b}$ and $\vec{c}$ are unit vectors and $|\vec{a}| = 7$, $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) = \frac{1}{2} \vec{a}$, then angle between the vectors $\vec{a}$ and $\vec{c}$ and angle between the vectors $\vec{b}$ and $\vec{c}$ are respectively \dots
Note: The original question text displayed $\frac{1}{3}\vec{a}$, which is a known OCR/print typo in this standard exam question format. The correct standard value is $\frac{1}{2}\vec{a}$ to yield standard angular options.

$\int \frac{dx}{(x + a)^{9/7} (x - b)^{5/7}}$ = ______.

The length of the perpendicular drawn from the origin on the normal to the curve $x^2 + 2xy - 3y^2 = 0$ at the point $(2, 2)$ is ______.

If $\tan^{-1}(x + 1) + \tan^{-1} x + \tan^{-1}(x - 1) = \tan^{-1} 3$, then for $x < 0$ the value of $500x^4 + 270x^2 + 997 = \dots$

Let $f : \mathbb{R} - \{2\} \rightarrow \mathbb{R} - \{1\}$ defined by $f(x) = \frac{x-3}{x-2}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x) = 3x - 2$, then sum of all values of $x$ for which $f^{-1}(x) + g^{-1}(x) = 19/6$ is ______.

If $f(x) = \log(1 + x) - \frac{2x}{2 + x}$, then $f(x)$ is increasing in ______.

There are 11 points in a plane of which 5 points are collinear. Then the total number of distinct quadrilaterals with vertices at these points is ______.

$\int \frac{x^4 \cos(\tan^{-1} x^5)}{1 + x^{10}} dx$ equals ______.

Let $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 5\hat{i} - 3\hat{j} + 2\hat{k}$ and if $\vec{b} \times \vec{c} = \vec{a}$ then $|\vec{b}|$ = ______.

The angle $\theta$, at which the curves $y = 3^x$ and $y = 7^x$ intersect, is given by ______.

If $x = \sin t$ and $y = \sin pt$, then the value of $(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + p^2 y = \dots$

The circumradius of a triangle whose sides are 10 units, 8 units and 6 units is ______.