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JEE Mathematics Solved Question Paper 2014

Multiple Choice Questions

The number of solution(s) of the equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4 x - 1}$ is/are

The value of $|z^{2}| + {|z - 3|}^{2} + {|z - i|}^{2}$ is minimum when z equals

$2 - \frac{2}{3} i$
45 + 3i
$1 + \frac{i}{3}$
$1 - \frac{i}{3}$

If $\lim_{x \to 0} \frac{2 asin (x) - \sin (2 x)}{\tan^{3} (x)}$ exists and is equal to 1, then the value of $α$ is

The solution of the equation $\log_{101} (\log_{7} (\sqrt{x + 7} + \sqrt{x})) = 0$ is

The number of digits in 20³⁰¹ $(given, \log_{10} (2) = 0.3010)$ is

If R be the set of all real numbers and f : R ➔ R is given by f(x) = 3x² + 1. Then, the set f^-1([1, 6]) is

$\{- \sqrt{\frac{5}{3}}, 0, \sqrt{\frac{5}{3}}\}$
$[- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}]$
$[- \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}]$
$(- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}})$

7.
The value of $\tan (\frac{π}{2}) + 2 \tan (\frac{2 π}{5}) + 4 \cot (\frac{4 π}{5})$ is

$\cot (\frac{π}{5})$

$\cot (\frac{2 π}{5})$

$\cot (\frac{4 π}{5})$

$\cot (\frac{3 π}{5})$

$\cot (\frac{π}{5})$

$\tan (\frac{π}{5}) + 2 \tan (\frac{2 π}{5}) + 4 \cot (\frac{4 π}{5}) = \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + 2 \frac{\sin (\frac{2 π}{5})}{\cos (\frac{2 π}{5})} + 4 \frac{\cos (\frac{4 π}{5})}{\sin (\frac{4 π}{5})} = \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + 2 \frac{\sin (\frac{2 π}{5})}{\cos (\frac{2 π}{5})} + \frac{4 (\cos^{2} (\frac{2 π}{5}) - \sin^{2} (\frac{2 π}{5}))}{2 \sin (\frac{2 π}{5}) \cos (\frac{2 π}{5})} = \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + 2 \frac{\sin (\frac{2 π}{5})}{\cos (\frac{2 π}{5})} + \frac{2 (\cos (\frac{2 π}{5}) - \sin^{2} (\frac{2 π}{5}))}{\sin (\frac{2 π}{5}) \cos (\frac{2 π}{5})}$

$= \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + 2 [\frac{\sin^{2} (\frac{2 π}{5}) + \cos^{2} (\frac{2 π}{5}) - \sin^{2} (\frac{2 π}{5})}{\sin (\frac{2 π}{5}) \cos (\frac{2 π}{5})}] = \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + \frac{2 \cos (\frac{2 π}{5})}{\sin (\frac{2 π}{5})} = \frac{\sin (\frac{π}{5})}{\cos (\frac{π}{5})} + \frac{2 (\cos^{2} (\frac{π}{5}) - \sin^{2} (\frac{π}{5}))}{2 \sin (\frac{π}{5}) \cos (\frac{π}{5})} [∵ \sin (2 θ) = 2 \sin (θ) \cos (θ) and \cos (2 θ) = (\cos^{2} (θ) - \sin^{2} (θ))] = \frac{\sin^{2} (\frac{π}{5}) + \cos^{2} (\frac{π}{5}) - \sin^{2} (\frac{π}{5})}{\sin (\frac{π}{5}) \cos (\frac{π}{5})} = \frac{\cos^{2} (\frac{π}{5})}{\sin (\frac{π}{5}) \cos (\frac{π}{5})} = \cot (\frac{π}{5})$