A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ white balls, $$3$$ red balls and $$4$$ black balls. A third box $${B_3}$$ contains $$3$$ white balls, $$4$$ red balls and $$5$$ black balls.

If $$2$$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $$2$$ balls are drawn from box $${B_2}$$ is

Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

If the function $${e^{ - x}}f\left( x \right)$$ assumes its minimum in the interval $$\left[ {0,1} \right]$$ at $$x = {1 \over 4}$$, which of the following is true?

Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

Which of the following is true for $$0 < x < 1?$$

Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

List $$I$$
$$P.$$$$\,\,\,\,\,$$ $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }^{ - 1}}y} \right)} \over {\cot \left( {{{\sin }^{ - 1}}y} \right) + \tan \left( {{{\sin }^{ - 1}}y} \right)}}} \right)}^2} + {y^4}} \right)^{1/2}}$$ takes value

$$Q.$$ $$\,\,\,\,$$ If $$\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$$ then
possible value of $$\cos {{x - y} \over 2}$$ is

$$R.$$ $$\,\,\,\,\,$$ If $$\cos \left( {{\pi \over 4} - x} \right)\cos 2x + \sin x\sin 2\sec x = \cos x\sin 2x\sec x + $$
$$\cos \left( {{\pi \over 4} + x} \right)\cos 2x$$ then possible value of $$\sec x$$ is

$$S.$$ $$\,\,\,\,\,$$ If $$\cot \left( {{{\sin }^{ - 1}}\sqrt {1 - {x^2}} } \right) = \sin \left( {{{\tan }^{ - 1}}\left( {x\sqrt 6 } \right)} \right),\,\,x \ne 0,$$
Then possible value of $$x$$ is

List $$II$$
$$1.$$ $$\,\,\,\,\,$$ $${1 \over 2}\sqrt {{5 \over 3}} $$

$$2.$$ $$\,\,\,\,\,$$ $$\sqrt 2 $$

$$3.$$ $$\,\,\,\,\,$$ $${1 \over 2}$$

$$1.$$ $$\,\,\,\,$$ $$1$$