List of top Physics Questions on Random Variables

If \[ (4^{\sec^{2}\alpha})x^{2}+2x+\left(\beta^{2}-\beta+\frac{1}{2}\right)=0 \] has real roots, then the value/value(s) of \[ (\cos\alpha+\cos^{-1}\beta) \] is/are:
If \(A_{1},A_{2},A_{3},\dots,A_{1006}\) be independent events such that \[ P(A_{i})=\frac{1}{2i}\quad (i=1,2,\dots,1006) \] and the probability that none of the events occurs be \[ \frac{\alpha!}{2^{\alpha}(\beta!)^{2}}, \] then:
Four natural numbers selected at random are multiplied together, then the probability that the digit in the unit's place in the product be 1, 3, 7 or 9 is:
If \(f\) be a real valued function defined for all real numbers \(x\) such that for some fixed \(a>0\), it satisfies \(f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^{2}}\ \forall x\), then \(f(x)\) is periodic with period:
The quantities \(a_{1},a_{2},a_{3},\ldots\) form an infinite decreasing G.P. If \(a_{1}=1\), then the common ratio of the progression for which the expression \(6a_{5}-16a_{4}-3a_{3}+12a_{2}\) is at a maximum is:
Let \(1\) lie between the roots of the equation \(y^{2}-my+1=0\) and \([x]\) denotes the greatest integer function. Then the value of \(\left[\left(\frac{4|x|}{x^{2}+16}\right)^{m}\right]\) is:
Let \(A_{1},A_{2},\dots,A_{6}\) be six sets, each with four elements and \(B_{1},B_{2},\dots,B_{n}\) be \(n\) sets, each with two elements. Let \[ S=A_{1}\cup A_{2}\cup\dots\cup A_{6}=B_{1}\cup B_{2}\cup\dots\cup B_{n}. \] Given that each element of \(S\) belongs to exactly four of the \(A\)'s and to exactly three of the \(B\)'s, then \(n\) is:
Let \[ \vec{a}=(x,y,z) \] be the vector with \(|\vec{a}|=2\sqrt{3}\), which makes equal angles with the vectors \[ \vec{b}=(y,-2z,3x) \] and \[ \vec{c}=(2z,3x,-y), \] and is perpendicular to the vector \[ \vec{d}=(1,-1,2). \] If the angle between \(\vec{a}\) and the unit vector \(\hat{j}\) is obtuse, then \(\vec{a}\) is:
If \[ \int\frac{(1-x^{2})\,dx}{\sqrt{x}\sqrt{(1+x^{2})^{3}}} =\alpha\frac{x^{\beta}}{(1+x^{2})^{\gamma}}+C, \] \(\alpha,\beta,\gamma\in\mathbb{R}\) and \(C\) is constant of integration, then \(\alpha:\beta:\gamma\) will be:
If \(a\) is an integer lying in \([-5,30]\), then the probability that the graph of \(y=x^{2}+2(a+4)x-5a+64\) lies above the x-axis is:
Let \(f:(0,1)\rightarrow(0,1)\) be a bijective differentiable function such that \(f^{\prime}(x)\ne0 \ \forall x\in(0,1)\) and \(f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}\). Suppose for all \(x\), \[ \lim_{t\rightarrow x}\frac{\int_{0}^{t}\sqrt{1-(f(s))^{2}}\,ds-\int_{0}^{x}\sqrt{1-(f(s))^{2}}\,ds}{f(t)-f(x)}=f(x) \] Then the value of \(f\left(\frac{1}{4}\right)\) belongs to:
Let \[ \det A=\left|\begin{matrix}l&m&n\\ p&q&r\\ 1&1&1\end{matrix}\right|. \] If \[ (l-m)^{2}+(p-q)^{2}=9, \] \[ (m-n)^{2}+(q-r)^{2}=16, \] \[ (n-l)^{2}+(r-p)^{2}=25, \] then the value of \((\det A)^{2}\) is:
If for two real numbers \(a,b\) with \(|a|\le1\) and \(|b|\le1\), \[ \frac{1}{3}+\frac{\sin^{-1}a+\sin^{-1}b}{4}+\frac{(\sin^{-1}a+\sin^{-1}b)^{2}}{16}+\frac{(\sin^{-1}a+\sin^{-1}b)^{3}}{64}+\dots=\frac{2(8-3\pi)}{3(16+3\pi)}, \] then the value of \[ \sin^{-1}(a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}}) \] is:
Let us define the power of a matrix \(A\) as the maximum \(m\in\mathbb{Z}^{+}\) such that \(A^{m}=I\). For two matrices \(A\) and \(B\) if \(A^{5}=I\) and \(ABA^{-1}=B^{2}\), then the power of the matrix \(B\) is between:
Suppose \(A\) is denoted the set of all numbers between 1 and 700 which are divisible by 3 and let \(B\) is denoted the set of all numbers between 1 and 300 which are divisible by 7. If \(C=\{(a,b)\mid a\in A,b\in B, a\ne b \text{ and } a+b=\text{even number}\}\), then order of \(C\) is:
The number of 3-digit numbers of the form $xyz$ with $x<y$, $z<y$ and $x\ne0$ is:
Let domain and range of \(f(x)\) and \(g(x)\) is \([0,\infty)\). If \(f(x)\) is an increasing function, \(g(x)\) is a decreasing function, \(h(x)=f\{g(x)\}\), \(h(0)=0\) and \(p(x)=h(x^{3}-2x^{2}+2x)-h(4)\) then for all \(x\in(0,2)\):
Let 10 Bags \(B_{1},B_{2},\dots,B_{10}\) which contain 21, 22, \dots, 30 different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is:
Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in (0,90)$; (where $[\cdot]$ denotes the greatest integer function) is:
Let \(a_{1},a_{2},a_{3},\dots\) are in G.P. such that \(n>m\), \(a_{n}>a_{m}\) and \(a_{1}+a_{n}=66\), \(a_{2}\cdot a_{n-1}=128\). If \(\sum_{r=1}^{n}a_{r}=126\), then \(n\) is:
Consider the sequence of numbers \[ \{1,2,3,\dots,13\}. \] A person chooses three numbers at random from the sequence. The probability that the chosen three numbers form an A.P. is:

A force \(F =\left(5+3 y^2\right)\) acts on a particle in the \(y\)-direction, where \(F\) is in newton and \(y\) is in meter The work done by the force during a displacement from \(y=2 m\) to \(y=5 m\) is___ \(J\).