Let $$f:R \to R$$ be a function defined by
$$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.$$ where $$\left[ x \right]$$ is the greatest integer less than or equal to $$x$$, if $$I = \int\limits_{ - 1}^2 {{{xf\left( {{x^2}} \right)} \over {2 + f\left( {x + 1} \right)}}dx,} $$ then the value of $$(4I-1)$$ is

Let $$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$ for all $$x \in R$$ and $$f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]$$ be a continuous function. For $$a \in \left[ {0,{1 \over 2}} \right],\,$$ $$F'(a)+2$$ is the area of the region bounded by $$x=0, y=0, y=f(x)$$ and $$x=a,$$ then $$f(0)$$ is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $$0.96,$$ is

Match the following :

Column I	Column II
(A) In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector $ \alpha \hat{i} + \beta \hat{j} $ on $ \sqrt{3}\hat{i} + \hat{j} $ is $ \sqrt{3} $ and if $ \alpha = 2 + \sqrt{3}\beta $, then possible value(s) of $ |\alpha| $ is (are)	$(P)\ 1$
(B) Let $ \alpha $ and $ b $ be real numbers such that the function $ f(x)= \begin{cases} -3\alpha x^2-2, & x<1 \\[4pt] bx+\alpha^2, & x\ge 1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $. Then possible value(s) of $ \alpha $ is (are)	$(Q)\ 2$
(C) Let $ \omega \ne 1 $ be a complex cube root of unity. If $ (3-3\omega+2\omega^2)^{4n+3} +(2+3\omega-3\omega^2)^{4n+3} +(-3+2\omega+3\omega^2)^{4n+3}=0, $ then possible value(s) of $ n $ is (are)	$(R)\ 3$
(D) Let the harmonic mean of two positive real numbers $ a $ and $ b $ be $ 4 $. If $ q $ is a positive real number such that $ a,\ 5,\ q,\ b $ is an arithmetic progression, then the value(s) of $ |q-a| $ is (are)	$(S)\ 4$
	$(T)\ 5$