List of top Questions asked in MHT CET- 2020

In a single throw of three dice, the probability of getting a sum of at least 5 is

If $3\sin^2x-8\sin x+4=0$, $x\in\left(\dfrac{\pi}{2},\pi\right)$, then $\tan x=$

If \[ \cos x + \cos y = -\cos \alpha, \quad \sin x + \sin y = -\sin \alpha, \quad \text{then} \quad \cot \left( \frac{x + y}{2} \right) = \]

Evaluate \( \int \frac{dx}{\cos 2x + \sin^2 x} \)

The length of the latus rectum of the parabola whose focus is at \( (1,-2) \) and directrix is the line \( x + y + 3 = 0 \) is

With usual notations, in \( \triangle ABC \), if \( b\cos^2\frac{C}{2} + c\cos^2\frac{B}{2} = \frac{3a}{2} \), then

If the lines \[ \frac{1-x}{2} = \frac{y-8}{\lambda} = \frac{z-5}{2} \quad \text{and} \quad \frac{x-11}{5} = \frac{y-3}{3} = \frac{z-1}{1} \] are perpendicular, then \(\lambda =\)

If \( f : \mathbb{R} \rightarrow \mathbb{R} \), \( g : \mathbb{R} \rightarrow \mathbb{R} \) are two functions defined by \( f(x) = 2x - 3 \), \( g(x) = x^3 + 5 \), then find \( (f \circ g)^{-1(x) \).}

Evaluate the integral \[ \int \frac{\sin 2x}{\sin^2 x \cos^2 x}\, dx \]

The differential equation of the family of lines having \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is

If the angle between the lines given by the equation \[ x^2 - 3xy + Ay^2 + 3x - 5y + 2 = 0, \quad \lambda \geq 0, \quad \text{is} \, \tan^{-1} \left( \frac{1}{3} \right), \, \text{then} \, \lambda = \]

If \(3\cos x \ne 2\sin x\), then the general solution of \[ \sin^2 x-\cos 2x=2-\sin 2x \] is

The points of discontinuity of the function \[ f(x) = \begin{cases} \frac{1}{x - 1} & \text{if } 0 \leq x \leq 2 \\ \frac{x + 5}{x + 3} & \text{if } 2<x \leq 4 \end{cases} \] in its domain are

Simplify \( \sqrt{2} + \sqrt{2} + 2 \cos 4\theta = \)

If two angles of \( \triangle ABC \) are \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \), then the ratio of the smallest and greatest side is

The radius of the circle passing through the points \[ (5, 7), \quad (2, -2), \quad (-2, 0) \] is

The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290K and the metal temperature drops from 370K to 330K in 10 minutes, then the time required to drop the temperature up to 295K is

The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{5} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{8} \right) \] is

If the vectors \( i + j + k \), \( i - j + k \) and \( 2i + 3j + mk \) are coplanar, then \( m = \)

The integral \[ \int \cot x \cdot \log \left[ \log (\sin x) \right] \, dx = \]

The shortest distance between the lines \[ \vec{r} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k} \] and \[ \vec{r} = (p + 1)\hat{i} + (2p - 1)\hat{j} + (2p + 1)\hat{k} \] is

If \[ f(x) = \left[ \tan \left( \frac{\pi}{4} + x \right) \right]^{\frac{1}{x}} \quad \text{if} \quad x \neq 0 \] \[ f(x) = k \quad \text{if} \quad x = 0 \] is continuous at \( x = 0 \), then \( k = \)

Which of the following statement patterns is a tautology? \[ S_1 \equiv \neg p \rightarrow (q \leftrightarrow p) \] \[ S_2 \equiv \neg p \vee q \] \[ S_3 \equiv (p \rightarrow q) \land (q \rightarrow p) \] \[ S_4 \equiv (q \rightarrow p) \vee (\neg p \leftrightarrow q) \]

A body cools according to Newton’s law from \(100^\circ\text{C}\) to \(60^\circ\text{C}\) in 20 minutes. The temperature of the surroundings being \(20^\circ\text{C}\), the temperature of the body after one hour is

If \( f(x) = \begin{cases \dfrac{81^x - 9^x}{k^x - 1}, & x \neq 0
2, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( k \) is}