JEE (Advanced) 2014 Paper-2

The function y = f(x) is the solution of the differential equation
$$\frac{\text{d}y}{\text{d}x} + \frac{xy}{x^2 - 1} = \frac{x^4 + 2x}{\sqrt {1 - x^2}}$$
in (−1, 1) satisfying f(0) = 0. Then
$$\int_{-{\frac{\sqrt 3}{2}}}^{\frac{\sqrt 3}{2}} f(x) dx $$ is

The following integral
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2 cosec x)^{17} dx $$
is equal to

Coefficient of $$x^{11}$$ in the expansion of $$(1 + x^2)^4 (1 + x^3)^7 (1 + x^4 )^{12}$$ is

Let $$f: [0, 2] \rightarrow R$$ be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let
$$F(x) = \int_{0}^{x^2} f(\sqrt t) dt $$
for $$x \in [0, 2].$$ If $$F^{'}(x) = f^{'} (x)$$ for all $$x \in (0, 2)$$ then F(2) equals

The common tangents to the circle $$x^2 + y^2 = 2$$ and the parabola $$y^2 = 8x$$ touch the circle at the points P, Q and the parabola at the points R, S. Then the area of the quadrilateral PQRS is

For $$x \in (0, \pi)$$, the equation $$\sin x + 2 \sin 2x - \sin 3x = 3$$ has

In a triangle the sum of two sides is x and the product of the same two sides is y. If $$x^2 - c^2 = y,$$ where c is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has