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Trigonometric Functions

Multiple Choice Questions

301.

$If (1 + \tan (α)) (1 + \tan (4 α)) = 2, α \in (0, \frac{π}{16}), then α = ?$

$\frac{π}{20}$
$\frac{π}{30}$
$\frac{π}{40}$
$\frac{π}{50}$

302.

$If \cos (θ) = \frac{\cos (α) - \cos (β)}{1 - \cos (α) \cos (β)}, then one of the values of \tan (\frac{θ}{2}) is$

$\cot (\frac{β}{2}) \tan (\frac{α}{2})$
$\tan (\frac{α}{2}) \tan (\frac{β}{2})$
$\tan (\frac{β}{2}) \cot (\frac{α}{2})$
$\tan^{2} (\frac{α}{2}) \tan^{2} (\frac{β}{2})$

303.

The value of the expression $\frac{1 + \sin (2 α)}{\cos (2 α - 2 π) \tan (α - \frac{3 π}{4})} - \frac{1}{4} \sin (2 α) [\cot (\frac{α}{2}) + \cot (\frac{3 π}{2} + \frac{α}{2})] is$

0
1
$\sin^{2} (\frac{α}{2})$
$\sin^{2} (α)$

304.

$If \frac{1}{6} \sin (θ), \cos (θ) and \tan (θ) are in geometric progression, then the solution set of θ is$

$2 nπ \pm (\frac{π}{6})$
$2 nπ \pm (\frac{π}{3})$
$nπ + {(- 1)}^{n} (\frac{π}{3})$
$nπ + (\frac{π}{3})$

305.
$In ∆ ABC if x = \tan (\frac{B - C}{2}) \tan (\frac{A}{2}), y = \tan (\frac{C - A}{2}) \tan (\frac{B}{2}) and z = \tan (\frac{A - B}{2}) \tan (\frac{C}{2}), then (x + y + z) = ?$

xyz

- xyz

2xyz

$\frac{1}{2} xyz$

- xyz

$Given, x = \tan (\frac{B - C}{2}) \tan (\frac{A}{2}) y = \tan (\frac{C - A}{2}) \tan (\frac{B}{2}) and z = \tan (\frac{A - B}{2}) \tan (\frac{C}{2}) \Rightarrow x = \frac{b - c}{b + c} [∵ \tan (\frac{B - C}{2}) = \frac{b - c}{b + c} \cot (\frac{A}{2})] Y = \frac{c - a}{c + a} and z = \frac{a - b}{a + b} Now, \frac{1 + x}{1 - x} = \frac{b + c + b - c}{b + c - b + c} [By componendo and dividendo] \Rightarrow \frac{1 + x}{1 - x} = \frac{b}{c} Similarly, \frac{1 + y}{1 - y} = \frac{c}{a} and \frac{1 + z}{1 - z} = \frac{a}{b} ∴ (\frac{1 + x}{1 - x}) (\frac{1 + y}{1 - y}) (\frac{1 + z}{1 - z}) = \frac{b}{c} \times \frac{c}{a} \times \frac{a}{b} = 1 \Rightarrow (1 + x) (1 + y) (1 + z) = (1 - x) (1 - y) (1 - z) \Rightarrow 1 + x + y + z + xy + yz + zx + xyz = 1 - (x + y + z) + xy + yz + zx - xyz \Rightarrow (x + y + z) = - xyz$

306.

$If A > 0, B > 0 and A + B = \frac{π}{3}, then the maximum value of Atan (B) is$

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

307.

$In ∆ ABC, if bcos (θ) = c - a, (where θ is an acute angle), then (c - a) \tan (θ) = ?$

$2 \sqrt{ca} \cos (\frac{B}{2})$
$2 \sqrt{ac} \sin (\frac{B}{2})$
$2 cacos (\frac{B}{2})$
$2casin (\frac{B}{2})$

308.

$If \tan (α) and \tan (β) are the roots of the equation x^{2} + px + q = 0, then the value of \sin^{2} (α + β) + pcos (α + β) \sin (α + β) + {qcos}^{2} (α + β)$