A board has 16 squares as shown in the figure : Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

$A$ and $B$ alternately throw a pair of dice. A wins if he throws a sum of 5 before $B$ throws a sum of 8 , and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5 . The probability, that A wins if A makes the first throw, is

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is :

Two number ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of ${\mathrm{i}}^{{\mathrm{k}}_1}+{\mathrm{i}}^{{\mathrm{k}}_2},(\mathrm{i}=\sqrt{-1})$ is non-zero, equals

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is

If A and B are two events such that $P(A) = 0.7$, $P(B) = 0.4$ and $P(A\cap \overline{B})=0.5$, where $\overline{B}$ denotes the complement of B, then $P\left(B\mid (A\cup \overline{B})\right)$ is equal to

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n}$, $\gcd (m,n)=1$, then $n^2-m^2$ is equal to :

Let a random variable X take values 0, 1, 2, 3 with P(X=0)=P(X=1)=p, P(X=2)=P(X=3) and E(X2)=2E(X). Then the value of 8p−1 is :

$$\text{ Given three indentical bags each containing }10\text{ balls, whose colours are as follows : }$$ $$\begin{array}{lccc}& \text{ Red }& \text{ Blue }& \text{ Green }\\ \text{ Bag I }& 3& 2& 5\\ \text{ Bag II }& 4& 3& 3\\ \text{ Bag III }& 5& 1& 4\end{array}$$ A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left(\frac{1}{p}+\frac{1}{q}\right)$ is:

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is

If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x)=k(x+1)3^{-x},x=0,1,2,3\dots$, where $k$ is a constant, then $P(X\ge 3)$ is equal to

The product of all solutions of the equation ${\mathrm{e}}^{5{\left({\log }_{\mathrm{e}}x\right)}^2+3}=x^8,x>0$, is :

Let $f:(0,\infty )\to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2{f}^{\prime }(x)=2xf(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to :

Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that $(\sin x\cos y)(f(2x+2y)-f(2x-2y))=(\cos x\sin y)(f(2x+2y)+f(2x-2y))$, for all $x,y\in \mathbf{R}$. If $f^{\prime }(0)=\frac{1}{2}$, then the value of $24f^{\prime \prime }\left(\frac{5\pi }{3}\right)$ is :

Let y = y(x) be the solution of the differential equation : $\cos x{\left({\log }_e(\cos x)\right)}^{2}dy+\left(\sin x-3y\sin x{\log }_e(\cos x)\right)dx=0$, x ∈ (0, $\frac{\pi }{2}$ ). If $y(\frac{\pi }{4})$ = $-\frac{1}{{\log }_{e}2}$, then $y(\frac{\pi }{6})$ is equal to :

If for the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+(\tan x)y=\frac{2+\sec x}{(1+2\sec x{)}^2}$, $x\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right),f\left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi }{4}\right)$ is equal to:

Let $x=x(y)$ be the solution of the differential equation $y^2\mathrm{d}x+\left(x-\frac{1}{y}\right)\mathrm{d}y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f^{\prime }(y)+f^{\prime }(x)f(y)$ for all $x,y\in \mathbf{R}$. Then \sum_\limits{n=1}^{100} \log _e f(n) is equal to :