JEE (Advanced) 2015 Paper-2

Let $$E_1$$ and $$E_2$$ be two ellipses whose centers are at the origin. The major axes of $$E_1$$ and $$E_2$$ lie along the x-axis and the y-axis, respectively. Let S be the circle $$x^2 + (y - 1)^2 = 2.$$ The straight line x + y = 3 touches the curves S, $$E_1$$ and $$E_2$$ at P, Q and R, respectively. Suppose that $$PQ = PR = \frac{2 \sqrt 2}{3}.$$ If $$e_1$$ and $$e_2$$ are the eccentricities of $$E_1$$ and $$E_2$$, respectively, then the correct expression(s) is(are)

Consider the hyperbola $$H : x^2 - y^2 = 1$$ and a circle S with center $$N(x_2, 0).$$ Suppose that H andS touch each other at a point $$P(x_1, y_1)$$ with $$x_1 > 1$$ and $$y_1 > 0.$$ The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle $$\triangle PMN$$, then the correct expression(s) is(are)

The option(s) with the values of a and L that satisfy the following equation is(are)
$$\frac{\int_{0}^{4\pi}e^t (\sin^6 at + \cos^4 at)dt}{\int_{0}^{\pi}e^t (\sin^6 at + \cos^4 at)dt} = L$$ ?

Let $$f,g:[-1, 2] \rightarrow R$$ be continuous functions which are twice differentiable on the interval (-1, 2). Let the valuesof f and g at the points —1, 0 and 2 be as given in the following table:

In each of the intervals (-1,0) and (0, 2) the function $$(f - 3g)"$$ never vanishes. Then the correct statement(s) is(are)

Let $$f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$$ for all $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2} \right).$$ Then the correct expression(s) is(are)

Let $$f'(x) = \frac{192x^3}{2 + \sin^4 \pi x}$$ for all $$x \in R$$ with $$f\left(\frac{1}{2}\right) = 0$$. If $$m \leq \int_{\frac{1}{2}}^{1} f(x) dx \leq M,$$ then the possible values of m and M are

One of the two boxes, box I and box II, was selected at random anda ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is $$\frac{1}{3},$$ then the correct option(s) with the possible values of $$n_1, n_2, n_3$$ and $$n_4$$ is(are)

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $$\frac{1}{3},$$ then the correct option(s) with the possible values of $$n_1 and n_2$$ is(are)

The correct statement(s) is(are)

If $$\int_{1}^{3}x^2 F'(x)dx = -12$$ and $$\int_{1}^{3}x^3 F"(x)dx = 40,$$ then the correct expression(s) is(are)