Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

Two integers $$x$$ and $$y$$ are chosen with replacement from the set $$\{0,1,2,3,\dots ,10\}$$. Then the probability that $$|x-y|>5$$, is :

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $$i^{\text{th }}$$ roll than the number obtained in the $$(i-1{)}^{\text{th }}$$ roll, $$i=2,3$$, is equal to

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn $$\mathrm{A}$$ is :

If the mean of the following probability distribution of a radam variable $$\mathrm{X}$$ : .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-baqh{text-align:center;vertical-align:top} $$\mathrm{X}$$ 0 2 4 6 8 $$\mathrm{P}(\mathrm{X})$$ $$a$$ $$2a$$ $$a+b$$ $$2b$$ $$3b$$ is $$\frac{46}{9}$$, then the variance of the distribution is

There are three bags $$X,Y$$ and $$Z$$. Bag $$X$$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $$Y$$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $$Z$$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag $$\mathrm{Y}$$, is :

Let the sum of two positive integers be 24 . If the probability, that their product is not less than $$\frac{3}{4}$$ times their greatest possible product, is $$\frac{m}{n}$$, where $$\gcd (m,n)=1$$, then $$n$$-$$m$$ equals

The coefficients $$a,b,c$$ in the quadratic equation $$a{x}^2+bx+c=0$$ are chosen from the set $$\{1,2,3,4,5,6,7,8\}$$. The probability of this equation having repeated roots is :

The coefficients $$\mathrm{a},\mathrm{b},\mathrm{c}$$ in the quadratic equation $$\mathrm{a}{x}^2+\mathrm{bx}+\mathrm{c}=0$$ are from the set $$\{1,2,3,4,5,6\}$$. If the probability of this equation having one real root bigger than the other is p, then 216p equals :

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is :

A company has two plants $$A$$ and $$B$$ to manufacture motorcycles. $$60\%$$ motorcycles are manufactured at plant $$A$$ and the remaining are manufactured at plant $$B.80\%$$ of the motorcycles manufactured at plant $$A$$ are rated of the standard quality, while $$90\%$$ of the motorcycles manufactured at plant $$B$$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $$p$$ is the probability that it was manufactured at plant $$B$$, then $$126p$$ is

Suppose $$f(x)=\frac{\left(2^x+2^{-x}\right)\tan x\sqrt{{\tan }^{-1}\left(x^2-x+1\right)}}{{\left(7x^2+3x+1\right)}^3}$$. Then the value of $$f^{\prime }(0)$$ is equal to

$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1

Let $$f:\mathbb{R}-\{0\}\to \mathbb{R}$$ be a function satisfying $$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$ for all $$x,y,f(y)\ne 0$$. If $$f^{\prime }(1)=2024$$, then

Let $$g:\mathbf{R}\to \mathbf{R}$$ be a non constant twice differentiable function such that $${\mathrm{g}}^{\prime }\left(\frac{1}{2}\right)={\mathrm{g}}^{\prime }\left(\frac{3}{2}\right)$$. If a real valued function $$f$$ is defined as $$f(x)=\frac{1}{2}[g(x)+g(2-x)]$$, then

If $$f(x)=\left|\begin{array}{ccc}2{\cos }^{4}x& 2{\sin }^{4}x& 3+{\sin }^{2}2x\\ 3+2{\cos }^{4}x& 2{\sin }^{4}x& {\sin }^{2}2x\\ 2{\cos }^{4}x& 3+2{\sin }^{4}x& {\sin }^{2}2x\end{array}\right|,$$ then $$\frac{1}{5}{f}^{\prime }(0)=$$ is equal to :

If $${\log }_{e}y=3{\sin }^{-1}x$$, then $$(1-x^2)y^{\prime \prime }-x{y}^{\prime }$$ at $$x=\frac{1}{2}$$ is equal to

Let $$f(x)=a{x}^3+b{x}^2+cx+41$$ be such that $$f(1)=40,f^{\prime }(1)=2$$ and $$f^{\prime \prime }(1)=4$$. Then $$a^2+b^2+c^2$$ is equal to:

Let $$f(x)=x^5+2{\mathrm{e}}^{x/4}$$ for all $$x\in \mathbf{R}$$. Consider a function $$g(x)$$ such that $$(g\circ f)(x)=x$$ for all $$x\in \mathbf{R}$$. Then the value of $$8g^{\prime }(2)$$ is :

If $$y(\theta )=\frac{2\cos \theta +\cos 2\theta }{\cos 3\theta +4\cos 2\theta +5\cos \theta +2}$$, then at $$\theta =\frac{\pi }{2},y^{\prime \prime }+y^{\prime }+y$$ is equal to :