JEE (Advanced) 2013 Paper-1

Instructions

For the following questions answer them individually

Question 51

Let $$S_n = \sum_{k = 1}^{4n}(-1)^{\frac {k (k + 1)}{2}} k^2.$$ Then $$S_n$$ can take value(s)

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Question 52

For $$3 \times 3$$ matrices M and N, whichof the following statement(s) is (are) NOT correct ?

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Question 53

Let $$f(x) = x \sin \pi x, x > 0$$ Then for all natural numbers n, $$f'(x)$$ vanishes at

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Question 54

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8: 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

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Question 55

A line l passing through the origin is perpendicular to the lines
$$l_1 : (3 + t)\widehat{i} + (-1 + 2t)\widehat{j} + (4 + 2t)\widehat{k}, -\infty < t < \infty$$
$$l_2 : (3 + 2s)\widehat{i} + (3 + 2s)\widehat{j} + (2 + s)\widehat{k}, -\infty < s < \infty$$
Then, the coordinate(s) of the point(s) on $$l_2$$ at a distance of $$\sqrt 17$$ from the point of intersection of $$l$$ and $$l_1$$ is (are)

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Question 56

The coefficients of three consecutive terms of $$(1 + x)^{n + 5}$$ are in the ratio $$5 : 10 : 14$$. Then n =

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Question 57

A pack contains n cards numberedfrom 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k - 20 =

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Question 58

Of the three independent events $$E_1, E_2$$ and $$E_3$$ the probability that only $$E_1$$ occurs is $$\alpha$$ only $$E_2$$ occurs is $$\beta$$ and only $$E_3$$ occurs is $$\gamma$$. Let the probability p that none of events $$E_1, E_2$$ or $$E_3$$ occurs satisfy the equations $$(\alpha - 2\beta) p = \alpha \beta$$ and $$(\beta - 3\gamma) p = 2 \beta \gamma$$. All the given probabilities are assumed tolie in the interval (0, 1).
Then $$\frac{Probability of occurrence of E_1}{Probability of occurrence of E_3} = $$

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Question 59

A vertical line passing through the point (h, 0) intersects the ellipse $$\frac{x^2}{4} + {y^2}{3} = 1$$ at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If $$\triangle (h)$$ = area of the triangle PQR,

, then $$\frac{8}{\sqrt 5} \triangle_1 - 8 \triangle_2 =$$

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Question 60

Consider the set of eight vectors $$V = \left\{{a\widehat{i} + b\widehat{j} + c \widehat{k} : a, b, c \in \left\{-1, 1\right\}} \right\}$$. Three non-coplanar vectors can be chosen from Vin $$2^p$$ ways. Then p is

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