JEE (Advanced) 2014 Paper-2


For the following questions answer them individually

Question 41

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

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Question 42

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

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Question 43

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

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Question 44

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

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Question 45

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

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Question 46

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

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Question 47

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

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Question 48

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

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Question 49

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

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Question 50

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

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