List of top Questions asked in MHT CET

A plane \( E_1 \) makes intercepts \( 1, -3, 4 \) on the coordinate axes. The equation of a plane parallel to \( E_1 \) and passing through \( (2,6,-8) \) is

The p.d.f. of a continuous random variable \( X \) is given by \[ f(x) = \frac{x + 2}{18}, \quad \text{if} \, -2<x<4, \quad f(x) = 0, \, \text{otherwise}. \] Then \( P[ |x|<1 ] = \)

If the p.m.f. of a random variable \( X \) is \[ \begin{array}{|c|c|c|c|c|} X & 1 & 2 & 3 & 4 & 5
P(X = x) & k & \frac{k}{3} & \frac{k}{4} & \frac{k}{2} & \end{array} \] then \( k = \)

Evaluate \( \int \frac{x^2}{(x+1)^2(x+2)^2} \, dx \)

Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \)

The equation of a circle passing through the origin and making x-intercept 3 and y-intercept -5 is:

The angle between the lines \[ \frac{x - 1}{4} = \frac{y - 3}{8} = \frac{z - 2}{2} \quad \text{and} \quad \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 4}{2} \] is

If the equation \( ax^2 + hy^2 + cx + cy = 0, c \neq 0 \) represents a pair of lines, then

The verbal statement of the same meaning, of the statement 'If the grass is green then it rains in July' is

Evaluate the integral \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \log |9e^{2x} - 4| + c \), then (where \( c \) is the constant of integration)

If \[ p \rightarrow \left( \sim p \vee q \right) \text{ is false, then the truth values of } p \text{ and } q \text{ are respectively} \]

Evaluate the integral \[ \int \frac{\sec x}{\sqrt{\log(\sec x + \tan x)}}\, dx \]

If $f'(x)=k(\cos x-\sin x)$, $f'(0)=3$, $f\!\left(\dfrac{\pi}{2}\right)=15$, then $f(x)=$

If \( f(x) = e^{x}g(x) \), \( g(0) = 4 \), and \( g'(0) = 2 \), then \( f'(0) = \)

If L.M.V.T. is applicable for the function $f(x) = x + \dfrac{1}{x}$ on $[1,3]$, then $c =$

If the vectors \( (2\mathbf{i}-q\mathbf{j}+3\mathbf{k}) \) and \( (4\mathbf{i}-5\mathbf{j}+6\mathbf{k}) \) are collinear, then the value of \( q \) is

If \( AX = B \), where \( A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4 \end{bmatrix} \), \( B = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), then \( x + y + z = \)

If \( | \vec{a} \, \vec{b} \, \vec{c} | = 3 \), then the volume of the parallelepiped with \( 2\vec{a} + \vec{b} \), \( 2\vec{b} + \vec{c} \), \( 2\vec{c} + \vec{a} \) as coterminous edges is:

The minimum value of \( Z = 5x + 8y \) subject to \[ x + y \geq 5, \quad 0 \leq x \leq 4, \quad y \geq 2, \quad x \geq 0, \quad y \geq 0 \] is

The equation of the curve which passes through the point \( (1,0) \) and has tangent with slope \[ 1+\frac{y}{x}+\left(\frac{y}{x}\right)^2 \] is

The measure of the acute angle between the lines given by the equation \( 3x^2 - 4\sqrt{3}xy + 3y^2 = 0 \) is

If \( f(x) = \log(\sec x + \tan x) \), then \[ f'\left( \frac{\pi}{4} \right) = \]

The acute angle included between the lines \[ x\sin\theta - y\cos\theta = 5 \] and \[ x\sin\alpha - y\cos\alpha + 11 = 0 \] is

The line cuts the \(X\)-axis and \(Y\)-axis at the points \(A\) and \(B\) respectively. The point \((5,6)\) divides the line segment \(AB\) internally in the ratio \(3:1\). Then the equation of the line is

Evaluate \[ \left[ \sin \left( \tan^{-1} \frac{3}{4} \right) \right]^2 + \left[ \sin \left( \tan^{-1} \frac{4}{3} \right) \right]^2 \]