JEE (Advanced) 2015 Paper-1

Instructions

For the following questions answer them individually

Question 41

The number of distinct solutions of the equation
$$\frac{5}{4} \cos^2 2x + \cos^4 x + \sin^4x + \cos^6 x + \sin^6 x = 2$$
in the interval $$[0, 2\pi]$$ is

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Question 42

Let the curve C be the mirror image of the parabola $$y^2 = 4x$$ with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y = -5, then the distance between A and B is

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Question 43

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

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Question 44

Let n be the numberof ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $$\frac{m}{n}$$ is

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Question 45

If the normals of the parabola $$y^2 = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $$(x - 3)^2 + (y + 2)^2 = r^2$$, then the value of $$r^2$$ is

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Question 46

Let $$f : R \rightarrow R$$ be a function defined by $$f(x) = \begin{cases}[x], & x \leq 2\\0 & x > 2\end{cases}$$,
where [x] is the greatest integer less than or equal to x. If $$I = \int_{-1}^{2} \frac {xf(x^2)}{2 + f(x + 1)} dx,$$ then the value of (4I - 1) is

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Question 47

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V mm^3$$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $$\frac{V}{250 \pi}$$ is

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Question 48

Let $$F(x) = \int_{x}^{x^2 + \frac{\pi}{6}} 2 \cos^2t dt$$ for all $$x \in R$$ and $$f : \left[0, \frac{1}{2}\right] \rightarrow [0, \infty)$$ be a continuous function. For $$a \in \left[0, \frac{1}{2}\right]$$, if $$F' (a) + 2 $$ is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is

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Question 49

Let X and Y be two arbitrary, $$3 \times 3$$, non-zero, skew-symmetric matrices and Z be an arbitrary $$3 \times 3$$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

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Question 50

Which of the following values of $$\alpha$$ satisfy the equation

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