JEE (Advanced) 2014 Paper-2
Let a, r, s, t be nonzero real numbers. Let $$P(at^2 , 2at), Q, R(ar^2 , 2ar)$$ and $$S(as^2 , 2as)$$ be distinct points on the parabola $$y^2 = 4ax$$. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0).
If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is
Given that for each $$a \in (0, 1)$$
$$\lim_{h \rightarrow 0^+} \int_{h}^{1 - h}t^{-a}(1 - t)^{a - 1}dt $$
exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $$x _i$$ be the number on the card drawn from the $$i^{th}$$ box, i = 1,2,3.
For the following questions answer them individually
Let $$Z_k = \cos \left(\frac{2k\pi}{10}\right) + i \sin \left(\frac{2k\pi}{10}\right); k = 1, 2 , ..., 9.$$
Let $$f_1 : R \rightarrow R, f_2 :[0, \infty) \rightarrow R, f_3 : R \rightarrow R$$ and $$f_4 : R \rightarrow [0, \infty)$$ be defined by
$$f_1 (x) = \begin{cases}|x| & if x < 0,\\e^x & if x \geq 0;\end{cases}$$
$$f_2 (x) = x^2;$$
$$f_3 (x) = \begin{cases}\sin x & if x < 0,\\x & if x \geq 0;\end{cases}$$
and
$$f_4 (x) = \begin{cases}f_2(f_1(x)) & if x < 0,\\f_2(f_1(x)) - 1 & if x \geq 0.\end{cases}$$
