List of top Questions asked in MHT CET

The perpendiculars are drawn to lines $L_1$ and $L_2$ from the origin making an angle $\frac{\pi}{4}$ and $\frac{3\pi}{4}$ respectively with the positive direction of X-axis. If both the lines are at unit distance from the origin, then their joint equation is

If $\int \frac{x^3}{\sqrt{1 + x^2}} \, dx = a(1 + x^2)\sqrt{1 + x^2} + b\sqrt{1 + x^2} + c$ (where $c$ is a constant of integration), then the value of $3ab$ is

The range of the function $f(x) = \frac{x^2}{x^2 + 1}$ is

Let $B\equiv(0,3)$ and $C\equiv(4,0)$. The point A is moving on the line $y=2x$ at the rate of 2 units/second. The area of $\triangle ABC$ is increasing at the rate of

If $y = \left[(x+1)(2x+1)(3x+1)\cdots(nx+1)\right]^n$, then $\frac{dy}{dx}$ at $x=0$ is

If $y = \left[(x+1)(2x+1)(3x+1)\cdots (nx+1)\right]^n$, then $\left.\frac{dy}{dx}\right|_{x=0}$ is

If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$ respectively, then $\frac{1}{a^2} + \frac{1}{b^2}$ equals

If $\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = 3\pi$, then the value of $x^{2025} + x^{2026} + x^{2027}$ is

$\int \frac{\sin x + \sin^3 x}{\cos 2x} \, dx = A \cos x + B \log f(x) + c$ (where $c$ is a constant of integration). Then values of $A$, $B$ and $f(x)$ are

If $f(x) = \begin{cases} 3(1 - 2x^2), & 0<x<1 \\ 0, & \text{otherwise} \end{cases}$ is a probability density function of $X$, then $P\left(\frac{1}{4}<x<\frac{1}{3}\right)$ is

Let the curve be represented by $x=2(\cos t+t\sin t)$, $y=2(\sin t-t\cos t)$. Then normal at any point '$t$' of the curve is at a distance of _____ units from the origin.

Equation of the plane passing through $(1,-1,2)$ and perpendicular to the planes $x+2y-2z=4$ and $3x+2y+z=6$ is

The integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec^{\frac{2}{3}} x \, \csc^{\frac{4}{3}} x \, dx$ is equal to

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{c}$ and $\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$, then $|\vec{a} \times \vec{c}|$ is equal to

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three vectors with magnitudes $\sqrt{3}$, $1$, $2$ respectively, such that $\vec{a} \times (\vec{a} \times \vec{c}) + 3\vec{b} = \vec{0}$, if $\theta$ is the angle between $\vec{a}$ and $\vec{c}$, then $\sec^2 \theta$ is

The principal solutions of the equation $\sec x+\tan x=2\cos x$ are

If $a>0$ and $z = \frac{(1+i)^2}{a - i}$ ($i = \sqrt{-1}$) has magnitude $\frac{2}{\sqrt{5}}$, then $\overline{z}$ is

If $f'(x) = \tan^{-1}(\sec x + \tan x)$, $-\frac{\pi}{2}<x<\frac{\pi}{2}$ and $f(0) = 0$, then $f(1)$ is

$\int \frac{\sin 2x \left(1 - \frac{3}{2}\cos x\right)}{e^{\sin^2 x + \cos^3 x}} \, dx =$

If $\int \frac{\cos \theta}{5 + 7\sin \theta - 2\cos^2 \theta} \, d\theta = A \log_e |f(\theta)| + c$ (where $c$ is a constant of integration), then $\frac{f(\theta)}{A}$ can be

In $\triangle ABC$, with usual notations, $2ac \sin\left(\frac{A-B+C}{2}\right)$ is equal to

Let \( A=\begin{bmatrix}2 & -1 \\ 0 & 2\end{bmatrix} \). If \( B=I-{}^{3}C_{1}(\mathrm{adj}\,A)+{}^{3}C_{2}(\mathrm{adj}\,A)^{2}-{}^{3}C_{3}(\mathrm{adj}\,A)^{3} \), then the sum of all elements of the matrix \( B \) is

The angle between the lines, whose direction cosines \( l, m, n \) satisfy the equations \( l + m + n = 0 \) and \( 2l^{2} + 2m^{2} - n^{2} = 0 \), is

Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a function such that \( f(x)=x^{3} + x^{2}f^{\prime}(1) + x f^{\prime\prime}(2) + 6 \) for \( x \in \mathbb{R} \), then \( f(2) \) equals

If $y = (\sin^{-1}x)^2 + (\cos^{-1}x)^2$, then $(1 - x^2)\,y'' - x\,y' = $