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JEE Mathematics Solved Question Paper 2016

Multiple Choice Questions

In how many ways can 5 keys be put in a ring ?

4!
5!
$\frac{4!}{2}$
$\frac{5!}{2}$

The value of ${(\sqrt{5} + 1)}^{5} - {(\sqrt{5} - 1)}^{5}$ is

The sum of the series $\frac{1}{2 . 3} + \frac{1}{4 . 5} + \frac{1}{6 . 7} + . . .$ is

$\log (\frac{e}{2})$
$\log (\frac{2}{e})$
$\frac{e}{2}$
$\frac{2}{e}$

The maximum value of $3 \cos (θ) + 4 \sin (θ)$ is

3
4
5
None of these

The period of $\sin (θ) - \sqrt{3} \cos (θ)$ is

$\frac{π}{4}$
$\frac{π}{2}$
$π$
$2 π$

If the tangent at the point $(2 \sec (θ), 3 \tan (θ))$ of the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$ is parallel to 3x - y + 4 = 0, then the value of $θ$ is

$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{6}$
$\frac{π}{2}$

7.
If $\cos (θ) = \frac{\cos^{2} (α) + \cos^{2} (β) + \cos^{2} (γ)}{\sin^{2} (α) + \sin^{2} (β) + \sin (γ)}$ , where $α, β, γ$ are the angles made by a line with the positive directions of the axes of reference, then the measure of $θ$ is

$60 °$

$90 °$

$30 °$

$45 °$

$60 °$

$Given, \cos (θ) = \frac{\cos^{2} (α) + \cos^{2} (β) + \cos^{2} (γ)}{\sin^{2} (α) + \sin^{2} (β) + \sin (γ)} . . . (i) where, α, β, γ are the angles made by a line with posItIve directIon of the axes of reference . We know that, l^{2} + m^{2} + n^{2} = 1 ∴ \cos^{2} (α) + \cos^{2} (β) + \cos^{2} (γ) = 1 . . . (ii) [∵ l = \cos (α), m = \cos (β) and n = \cos (γ)] \Rightarrow (1 - \sin^{2} (α)) + (1 - \sin^{2} (β)) + (1 - \sin^{2} (γ)) = 1 \Rightarrow 3 - (\sin^{2} (α) + \sin^{2} (β) + \sin^{2} (γ)) = 1 \Rightarrow \sin^{2} (α) + \sin^{2} (β) + \sin^{2} (γ) = 2 . . . (iii) ∴ \cos (θ) = \frac{1}{2} [from Eqs . (ii) and (iii)] \Rightarrow θ = \cos^{- 1} (\frac{1}{2}) \Rightarrow θ = \frac{π}{3} = 60 °$