List of top Questions asked in MHT CET

The probability distribution of a discrete random variable (r.v.) \(X\) is given below:

\(X=x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X=x)\)	\(k\)	\(2k\)	\(4k\)	\(2k\)	\(k\)

Then the value of \(P[X\leq E(X)]\) is:

It is known that \(10\%\) of certain articles manufactured are defective. What is the probability that in a random sample of \(12\) such articles, \(9\) are defective?

The linear programming problem:
Maximize : \(Z=x+y\)
Subject to :
\(x+2y\leq 200,x+y\leq 150,0\leq y\leq 60,x\geq 0\) has:

A husband and wife appeared for an interview for two vacancies in an office. The probability of the husband's selection is \(\frac{1}{5}\) and that of the wife's selection is \(\frac{1}{7}\), then the probability that exactly one of them is selected is...

The position vectors of vertices \(A\), \(B\) and \(C\) of triangle \(ABC\) are \(\hat{i}-\hat{j}-3\hat{k}\), \(2\hat{i}+\hat{j}-2\hat{k}\), and \(-5\hat{i}+2\hat{j}-6\hat{k}\) respectively. The length of the bisector \(AD\) of angle \(BAC\), where \(D\) is on the line segment \(BC\), is

The equation of the plane containing the line \(\bar{r}=(2\hat{i}-\hat{k})+\lambda (\hat{i}+\hat{j}+2\hat{k})\) and perpendicular to the plane \(\bar{r}\cdot (3\hat{i}-\hat{j}+\hat{k})=20\) is……

If \(l,m,n\) are direction cosines of a vector \(\bar{r}\) such that \(l+m+n=1\) and \(l^{2}+m^{2}+n^{2}=1\), then:
(where \(l,m,n\geq 0\))

If the line \(\frac{x+4}{a}=\frac{y-2}{b}=\frac{z+1}{c}\) passes through the point \((-3,1,-2)\) and its direction is chosen such that it makes an acute angle \(\gamma\) with the positive direction of Z- axis, then value of \(\cos \gamma\) is:

If the direction cosines of the line \(\frac{2x+5}{3}=\frac{2-3z}{-1};3y=-2\) are \(\frac{9}{\sqrt{85}},m,\frac{2}{\sqrt{85}}\) then the value of \(m\) is:

A unit vector \(\bar{u}\) makes an angle of \(45^{\circ }\) with the X- axis. If \(\bar{u}+\hat{j}+\hat{k}\) is also a unit vector, then the sum of the projections of \(\bar{u}\) on the coordinate axes is:

For a certain substance, the rate of change of vapor pressure \(P\) with respect to temperature \(T\) is directly proportional to the vapor pressure \(P\), and inversely proportional to the square of the temperature \(T\). The representation of this statement in the form of a differential equation is:
(where \(k\) is a constant of proportionality)

Let \(\bar{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\bar{b}=\hat{i}+2\hat{j}-2\hat{k}\). If a third vector \(\bar{c}=x\hat{i}+y\hat{j}+z\hat{k}\) is coplanar with \(\bar{a}\) and \(\bar{b}\), is perpendicular to \(\bar{a}\), and satisfies \(|\bar{c}|=\sqrt{45}\), then the sum of all possible values of \(y\) is:

If the general solution of the differential equation \(y\sec ^{2}\left(\frac{x}{y}\right)dx-\left(x\sec ^{2}\left(\frac{x}{y}\right)-y\right)dy=0\) is \(\tan \left(\frac{x}{y}\right)+g(y)=c\), then \(g(y)=\ldots\)

The general solution of the differential equation \((x+1)\frac{dy}{dx}-y=e^{x}(x+1)^{2}\) is given by:

If \(\bar{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\bar{b}=\hat{i}+3\hat{j}-\hat{k}\), \(\bar{c}=-2\hat{i}+\hat{j}-3\hat{k}\) and \(\bar{d}=3\hat{i}+2\hat{j}+5\hat{k}\), then the value of \(x^{2}+z^{2}-y\) is \(\cdots\), where \(\bar{d}=x\bar{a}+y\bar{b}+z\bar{c}\)

The area bounded by the parabola \(y=4x-x^{2}\) and the x-axis (in sq. units) is:

If \(\int x(a-x)^{9}dx=\frac{[f(x)]^{11}}{11}-a\frac{[f(x)]^{10}}{10}+c\), where \(c\) is a constant of integration, then \(f'(x)=\cdots\)

If \(\int (\sqrt{\tan x}+\sqrt{\cot x})dx=a\tan ^{-1}\left(\frac{\tan x-1}{\sqrt{b\tan x}}\right)+c\), for \(x\in \left(0,\frac{\pi }{2}\right)\), then \(a^{10}+b^{5}=\cdots\)

The value of \(\int \tan ^{-1}\left(\frac{6x}{1-9x^{2}}\right)dx\), where \(|x|\lt \frac{1}{3}\) is:

The value of \(I=\int _{1}^{3}e^{x}\frac{(x+2)^{2}}{(x+4)^{2}}dx\) is:

The value of the integral \(\int _{1/2}^{2}\frac{1}{x}\text{sin}^{101}\left(x-\frac{1}{x}\right)dx\) is equal to:

\(\int _{\pi /5}^{3\pi /10}\frac{1}{1+\sqrt{\tan x}}dx=\cdots\)

A right circular cone is melting such that its volume is decreasing at the rate of \(12\pi\) \(\text{cm}^{3}/\text{sec}\). The radius of the base is always equal to half of its height. Then the rate of change of the radius when the radius is \(2\) cm is:

If \(f(x)=\frac{x-1}{4}+\frac{(x-1)^{3}}{12}+\frac{(x-1)^{5}}{20}+\frac{(x-1)^{7}}{28}+\ldots\) where \(0\lt x\lt 2\), then \(f'(x)=\)

If the derivative of \(\tan ^{-1}(a+bx)\) at \(x=0\) is \(1\), then \(1-b^{3}+a^{6}\) is: