The set of all values of a for which the equation 2x + asin x = 2a - 7 has a solution is

Correct answer: B. [2,6] 2x + asin x = 2a - 7 i.e. 2 ^ 2 x - asin x + 2a - 8 = 0 sin x = a ± a^ 2 - 8(2a - 8) 4 = (a ± (a - 8))/(4) sin x = (a - 4)/(2)or2 Hence, - 1 ≤ (a - 4)^ 2 ≤ 1 The range of a is [2,6]

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The set of all values of a for which the equation 2x + asin x = 2a...

The set of all values of

a

for which the equation

\cos 2x + asin x = 2a - 7

has a solution is

Options

A $( - \infty, 2 )$
B $[2,6]$
C $( 6, \infty )$
D $( - \infty, \infty )$

Correct answer

B. $[2,6]$

Step-by-step solution

$\cos 2x + asin x = 2a - 7$

i.e. $2\sin^{2} x - asin x + 2a - 8 = 0$

$sin x = \dfrac{a \pm \sqrt{a^{2} - 8\left(2a - 8\right)}}{4} = \dfrac{a \pm \left(a - 8\right)}{4}$

$sin x = \dfrac{a - 4}{2}or2$

Hence, $- 1 \le \left(a - 4\right)^{2} \le 1$

The range of $a$ is $[2,6]$

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