Set of values of ‘a’ for which equation, cos, 2x + a, sin, x = 2a - 7 possess a solution

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Trigonometrical Equations

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The set of values of $‘a’$ for which the equation, $cos\, 2x + a\, sin\, x = 2a - 7$ possess a solution is :

A

$(-\infty , 2)$

B

$[2, 6]$

C

$(6, \infty )$

D

$(-\infty, \infty )$

Solution

$cos 2x + a\, sinx = 2a – 7$

i.e. $2sin^2x – a\, sinx + 2a – 8 = 0$

$sinx = \frac{{a \pm \sqrt {{a^2} – 8(2a – 8)} }}{4} = \frac{{a \pm (a – 8)}}{4}$

$sinx = \frac{{a – 4}}{2}\,or\,2$

Hence $-1 \leq (a- 4)/ 2 \leq 1$

$\Rightarrow$ the range of $a$

Standard 11

Mathematics

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