List of top Questions asked in MHT CET

Two fair dice are tossed simultaneously. Let \(A\) be the event that the sum of the numbers is prime, and \(B\) be the event that at least one die shows a number 5. The odds against the occurrence of event \(B\) given that event \(A\) has already occurred is:

The Incenter of the triangle whose vertices are \(A(0,3,0)\), \(B(0,0,4)\), and \(C(0,3,4)\) is;

The probability mass function (p.m.f.) of a random variable X is as follows:
\(P(X=0)=3k^{3},\ P(X=1)=4k-10k^{2},\ P(X=2)=5k-1,\ P(X=x)=0\) for any value of \(x\), then the value of \(k\) is:

If \(X\sim B(10,0.4)\), then the probability that \(X\) takes a value at least 2 is:

The L.P.P. Maximize \(Z=0.75x+0.5y\) subject to \(3x+2y\leq 6\), \(x+4y\leq 4\), \(x\geq 0\), \(y\geq 0\) has

If the angle between the line \(x=\frac{y-5}{2}=\frac{z+8}{k}\) and the plane \(2x-3y+3z=9\) is \(\cos ^{-1}\left(\sqrt{\frac{13}{22}}\right)\), then \(k=\ldots\)

The coordinates of the foot of the perpendicular drawn from the point \(P(-1,3,2)\) to the line formed by the intersection of the planes \(x+y+z=1\) and \(2x-y+z=2\) are:

Consider the lines
\(L_{1}:\frac{x}{2}=\frac{y}{2}=-z,\)
\(L_{2}:\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-3}{2}.\)
The unit vector perpendicular to both \(L_{1}\) and \(L_{2}\) is:

If \(|\bar{a}|=24\), \(|\bar{b}|=7\), \(\bar{a}\cdot \bar{b}=0\), \(\bar{a}+\bar{b}+\bar{c}=\bar{0}\), then \(|\bar{c}|=\ldots\)

If \(\bar{a}=\hat{i}+2\hat{j}+2\hat{k}\), \(\bar{b}=2\hat{i}-2\hat{j}+\hat{k}\) and \(\bar{c}\) is a vector projection of \(\bar{a}\) on the line of \(\bar{b}\), then the value of \([2\bar{a}\ 3\bar{b}\ \sqrt{3}\bar{c}]\) is

If the lines \(\frac{1-x}{3}=\frac{3y-1}{2\lambda +1}=4-z\) and \(\frac{x+2}{2\mu }=\frac{1-2y}{6}=\frac{5-z}{7}\) are perpendicular, then \(3\mu +\lambda =\ldots\)

The area of the region bounded by the line \(4x+3y=12\), the X-axis, and the lines \(x=-1\) and \(x=2\) (in sq units) is:

The order and degree of the differential equation \(e^{\frac{dy}{dx}}+\frac{dy}{dx}=x\) is

If the scalar products of a vector \(\bar{a}=x\hat{i}+y\hat{j}+z\hat{k}\) with \(2\hat{i}+\hat{j}+\hat{k},\ \hat{i}-\hat{j}+\hat{k},\ \hat{i}+2\hat{j}+3\hat{k}\) are \(5,3,\) and \(5\) respectively, then \(x+y+z=\ldots\)

An ice ball melts. The rate at which it melts is proportional to the amount of the ice at that time. If half quantity of ice melts in \(20\) minutes, then the time when \((1/8)^{th}\) part of original amount of the ice is left is:

Let \(\bar{a}=2\hat{i}+\hat{k},\ \bar{b}=\hat{i}+\hat{j}+\hat{k},\) and \(\bar{c}=4\hat{i}-3\hat{j}+7\hat{k}\). If \(\bar{r}\) is a vector such that \(\bar{r}\times \bar{b}=\bar{c}\times \bar{b}\) and \(\bar{r}\cdot \bar{a}=0\), then the value of the scalar product \(\bar{r}\cdot \hat{i}\) is:

The solution of the differential equation \(\frac{dy}{dx}+1=e^{x+y}\) is

\(\int e^{x}\left(\log (\sin x)+\sqrt{\frac{1+\cos 2x}{1-\cos 2x}}\right)dx=\ldots\)

\(\int \frac{x}{x^{5/2}+4x^{3/2}}\ dx=\ldots\)

If \(\int _{-1}^{4}f(x)dx=9\) and \(\int _{2}^{4}(7-f(x))dx=10\), then \(\int _{2}^{-1}f(x)dx=\)

\(\int _{\pi /6}^{\pi /3}\sin 2x\log (\tan x)\ dx=\ldots\)

The integral \(\int \frac{\csc x}{\cos ^{2}[1+\log (\tan \frac{x}{2})]}\ dx\) is equal to:

The derivative of \(f(\cos x)\) with respect to \(g(\tan x)\) at \(x=\frac{\pi }{6}\), given that \(f'\left(\frac{\sqrt{3}}{2}\right)=4\) and \(g'\left(\frac{1}{\sqrt{3}}\right)=2\) is:

The approximate surface area of a sphere of radius \(2.01\) cm is

\(\int \frac{2x}{4-3x-x^{2}}\ dx=\ldots\)