JEE (Advanced) 2013 Paper-1

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

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

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

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

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)

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

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 =

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 $$\frac{Probability of occurrence of E_1}{Probability of occurrence of E_3} = $$

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

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