JEE (Advanced) 2015 Paper-2

Instructions

For the following questions answer them individually

Question 51

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)

Video Solution
Question 52

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)

Video Solution
Question 53

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

Video Solution
Question 54

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)

Video Solution
Question 55

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)

Video Solution
Question 56

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

Video Solution
Instructions

Let $$n_1$$ and $$n_2$$ be the number of red and black balls, respectively, in box I. Let $$n_3$$ and $$n_4$$ be the number of red and black balls, respectively, in box II.

Question 57

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)

Video Solution
Question 58

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)

Video Solution
Instructions

Let $$F : R \rightarrow R$$ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = -4 and $$F'(x) < 0$$ for all x \in $$(\frac{1}{2} , 3)$$. Let $$f(x) = xF(x)$$ for all $$x \in R$$.

Question 59

The correct statement(s) is(are)

Video Solution
Question 60

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)

Video Solution

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