List of top Mathematics Questions asked in MHT CET

The equation of the plane passing through the point \( (1, 1, 1) \) and perpendicular to the planes \( 2x + y - 2z = 5 \) and \( 3x - 6y - 2z = 7 \) is:
The equation \( (\cos p - 1)x^2 + (\cos p)x + \sin p = 0 \), where \( x \) is a variable with real roots. Then the interval of \( p \) may be any one of the following:
Maximize \( z = x + y \) subject to: \[ x + y \leq 10, \quad 3y - 2x \leq 15, \quad x \leq 6, \quad x, y \geq 0. \] Find the maximum value.
A wire of length 20 units is divided into two parts such that the product of one part and the cube of the other part is maximum. The product of these parts is:
The value of \( \sin(\cot^{-1}(x)) \) is:
If \( x^y = e^{x-y} \), at \( x = 1 \), find \( \frac{dy}{dx} \)?

Evaluate: $$ \lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2x)^3} $$ 

Evaluate: $$ \int \frac{e^{\tan^{-1} x}}{1 + x^2} \left[ \left(\sec^{-1}\sqrt{1 + x^2}\right)^2 + \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right) \right] dx, \quad x > 0 $$ 

If $x^{2}+y^{2}=t+\frac{1}{t}$ and $x^{4}+y^{4}=t^{2}+\frac{1}{t^{2}}$, then $\frac{dy}{dx}$ is equal to
The value of c of Lagrange's mean value theorem for $f(x)=\sqrt{25-x^{2}}$ on $[1,5]$ is

Given \[ 0 \le x \le \frac{1}{2}, \] find the value of \[ \tan \left( \sin^{-1}\left( \frac{x}{\sqrt{2}} + \frac{\sqrt{1 - x^2}}{\sqrt{2}} \right) - \sin^{-1} x \right). \] 

If $\int \frac{x^2}{\sqrt{1-x }}dx = p\sqrt{1-x} (3x^2 + 4x + 8) + c$ where $c$ is a constant of integration, then the value of $p$ is
The value of \[ \int e^{x} \left(\frac{x^{2} + 4x + 4}{(x+4)^{2}}\right)\, dx \] is:
If $\sum_{r=1}^{50} \tan^{-1}\frac{1}{2r^{2 = p}}$ then $\tan p$ is
If \[ x = \sqrt{e^{\sin^{-1} t}} \quad \text{and} \quad y = \sqrt{e^{\cos^{-1} t}}, \] then find \[ \frac{d^2 y}{dx^2}. \]
The volume of parallelopiped, whose coterminous edges are given by $\overline{u}=\hat{i}+\hat{j}+\lambda\hat{k}, \vec{v}=\hat{i}+\hat{j}+3\hat{k}, \overline{w}=2\hat{i}+\hat{j}+\hat{k}$ is 1 cu. units. If $\theta$ is the angle between $\overline{u}$ and $\overline{w}$, then the value of $\cos\theta$ is
If the Cartesian equation of a line is $6x-2=3y+1=2z-2$, then the vector equation of the line is
The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of the vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $\lambda\hat{i}+2\hat{j}+3\hat{k}$ is equal to 1, then value of $\lambda$ is
The derivative of $f(\sec x)$ with respect to $g(\tan x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(\sqrt{2})=4$ and $g^{\prime}(1)=2$, is
If $[(\overline{a}+2\overline{b}+3\overline{c})\times(\overline{b}+2\overline{c}+3\overline{a})]\cdot(\overline{c}+2\overline{a}+3\overline{b})=54$ then the value of $[\overline{a}\overline{b}\overline{c}]$ is
A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is 3 mm and 1 hour later it reduces to 2 mm, then the expression for the radius R of the raindrop at any time t is
$\int\frac{1}{\sin(x-a)\sin x}dx=$
The number of discontinuities of the greatest integer function $f(x)=[x]$, $x\in(-\frac{7}{2},100)$
If $\cos^{-1}x-\cos^{-1}\frac{y}{3}=\alpha$, where $-1\le x\le1$, $-3\le y\le3$, $x\le\frac{y}{3}$, then for all $x, y$, $9x^{2}-6xy\cos\alpha+y^{2}$ is equal to
A vector $\vec{n}$ is inclined to X-axis at 45°, Y-axis at 60° and at an acute angle to Z-axis. If $\vec{n}$ is normal to a plane passing through the point $(-\sqrt{2},1,1)$ then equation of the plane is