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JEE Mathematics Solved Question Paper 2016

Multiple Choice Questions

The line y = x + $λ$ is tangent to the ellipse 2x² + 3y² = 1. Then, $λ$ is

- 2
1
$\sqrt{\frac{5}{6}}$
$\sqrt{\frac{2}{3}}$

Let S be the set of points, whose abscissae and ordinates are natural numbers. Let p e S, such that the sum of the distance of P from (8, 0) and (0, 12) is minimum among all elemants in S. Then, the number of such points P in S is

The cosine of the angle between any two diagonals of a cube is

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

4.
If x is a positive real number different from 1 such that $\log_{a} (x), \log_{b} (x), \log_{c} (x)$ are in AP, then

$b = \frac{a + c}{2}$

$\sqrt{ac}$

$c^{2} = {(ac)}^{\log_{a} (b)}$

None of these

$c^{2} = {(ac)}^{\log_{a} (b)}$

Given, $\log_{a} (x), \log_{b} (x), \log_{c} (x)$ are in AP.

$∴ 2 \log_{b} (x) = \log_{a} (x) + \log_{c} (x)$

$\Rightarrow 2 {og}_{b} (x) = \frac{1}{\log_{x} (a)} + \frac{1}{\log_{x} (c)}$

$\Rightarrow \frac{2}{\log_{x} (b)} = \frac{\log_{x} (ac)}{\log_{x} (a) \log_{x} (c)}$

$\Rightarrow 2 \log_{x} (c) = \frac{\log_{x} (b)}{\log_{x} (a)} (\log_{x} (ac))$

$\Rightarrow 2 \log_{x} (c) = \log_{a} (b) . \log_{x} (ac)$

$\Rightarrow \log_{x} (c^{2}) = \log_{x} ({ac}^{\log_{a} (b)})$

$\Rightarrow c^{2} = {(ac)}^{\log_{a} (b)}$

If a, x are real numbers and $|a|$ < 1, $|x|$ < 1, then 1 + (1+ a) x + (1+ a + a²)x² + ... $\infty$ is equal to

$\frac{1}{(1 - a) (1 - ax)}$
$\frac{1}{(1 - a) (1 - x)}$
$\frac{1}{(1 - x) (1 - ax)}$
$\frac{1}{(1 + ax) (1 - a)}$

If $\log_{0.3} (x - 1) < \log_{0.09} (x - 1)$ , then x lies in the interval

$(2, \infty)$
(1, 2)
(- 2, - 1)
None of these

The value of $\sum_{n = 1}^{13} (i^{n} + i^{n + 1}), i = \sqrt{- 1}$ is

i
i - 1
1
0

If $z_{1}, z_{2}, z_{3}$ are imaginary numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}|$ = 1, then $|z_{1} + z_{2} + z_{3}|$ is