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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

11.

If $θ \in (\frac{π}{2}, \frac{3 π}{2})$ , then the value of $\sqrt{4 \cos^{4} (θ) + \sin^{2} (2 θ)} + 4 \cot (θ) \cos^{2} (\frac{π}{4} - \frac{θ}{2})$ is

$- 2 \cot (θ)$
$2 \cot (θ)$
$2 \cos (θ)$
$2 \sin (θ)$

12.

The number ofreal solutions of the equation $(\sin (x) - x) (\cos (x) - x^{2}) = 0$ is

13.

The value of ${(\frac{1 + \sqrt{3} i}{1 - \sqrt{3} i})}^{64} + {(\frac{1 - \sqrt{3} i}{1 + \sqrt{3} i})}^{64}$

14.

Find the maximum value of $|z|$ when $|z - \frac{3}{z}| = 2$ , where z being a complex number.

1 + $\sqrt{3}$
3
$1 + \sqrt{2}$
1

15.

Given that, x is a real number satisfying $\frac{5 x^{2} - 26 x + 5}{3 x^{2} - 10 x + 3} < 0$ , then

$x < \frac{1}{5}$
$\frac{1}{5} < x < 3$
x > 5
$\frac{1}{5} < x < \frac{1}{3} or 3 < x < 5$

16.

The value of $λ$ such that the system of equations 2x - y - 2z = 2; x - 2y + z = - 4; x + y + $λ$ z = 4, has no solution, is

17.
Let $X_{n} = {(1 - \frac{1}{3})}^{2} {(1 - \frac{1}{6})}^{2} {(1 - \frac{1}{10})}^{2} . . . (1 - \frac{1}{\frac{n (n + 1)}{2}}), n \geq 2$
Then, the value of $\lim_{n \to \infty} x^{n}$ is

1/3

1/9

1/81

0

1/9

We have,

$X_{n} = {(1 - \frac{1}{3})}^{2} {(1 - \frac{1}{6})}^{2} {(1 - \frac{1}{10})}^{2} . . . (1 - \frac{1}{\frac{n (n + 1)}{2}})$

$\Rightarrow x_{n} = {[\prod_{n = 2}^{n} (\frac{n^{2} + n - 2}{n (n + 1)})]}^{2} = {[\prod_{n = 2}^{n} (\frac{(n + 2) (n - 1)}{n (n + 1)})]}^{2} = {[\prod_{n = 2}^{n} (\frac{n + 2}{n + 1}) . \prod_{n = 2}^{n} (\frac{n - 1}{n})]}^{2} = {[\prod_{n = 2}^{n} (\frac{n + 2}{n + 1})]}^{2} {[\prod_{n = 2}^{n} (\frac{n - 1}{n})]}^{2}$

$\Rightarrow x_{n} = {(\frac{4}{3} . \frac{5}{4} . \frac{6}{5} . . . \frac{n + 2}{n + 1})}^{2} {(\frac{1}{2} . \frac{2}{3} . \frac{3}{4} . . . \frac{n - 1}{n})}^{2} \Rightarrow x_{n} = {(\frac{n + 2}{3})}^{2} {(\frac{1}{n})}^{2} \Rightarrow x_{n} = \frac{1}{9} {(\frac{n + 2}{n})}^{2} \Rightarrow x_{n} = \frac{1}{9} {(1 + \frac{2}{n})}^{2} \Rightarrow \lim_{n \to \infty} x_{n} = \frac{1}{9} {(1 + 0)}^{2} = \frac{1}{9}$

18.

The variance of first 20 natural numbers is

133/4
279/12
133/2
399/4

19.

If the letters of the word 'PROBABILITY' are written down at random in a row, then probability that two B's are together, is

$\frac{2}{11}$
$\frac{10}{11}$
$\frac{3}{11}$
$\frac{6}{11}$

20.

A fair coin is tossed-at a fixed number of times. If the probability of getting exactly 3 heads equals the probability of getting exactly 5 heads, then the probability of getting exactly one head is