The set of all values of a for which the equation cos2x+asinx=2a-7 has a solution is

Question

The set of all values of a for which the equation cos2x+asinx=2a-7 has a solution is (A) (- ∞ , 2) (B) [2 , 6] (C) (6 , ∞ ) (D) (- ∞ , ∞ )

Tardigrade Admin · Accepted Answer

cos2x+asinx=2a-7 i.e. 2sin2x-asinx+2a-8=0 sinx=(a ± √a2 - 8 (2 a - 8)/4)=(a ± (a - 8)/4) sinx=(a - 4/2) or 2 Hence, -1≤ ((a - 4)/2)≤ 1 The range of a is [2 , 6] .

Q. The set of all values of $a$ for which the equation $cos 2 x + a s in x = 2 a - 7$ has a solution is

$(- \infty, 2)$

$[2, 6]$

$(6, \infty)$

$(- \infty, \infty)$

$cos 2 x + a s in x = 2 a - 7$
i.e. $2 s i n^{2} x - a s in x + 2 a - 8 = 0$
$s in x = \frac{a \pm a ^{2} - 8 ( 2 a - 8 )}{4} = \frac{a \pm ( a - 8 )}{4}$
$s in x = \frac{a - 4}{2}$ or $2$
Hence, $- 1 \leq \frac{( a - 4 )}{2} \leq 1$
The range of $a$ is $[2, 6]$ .

Q. The set of all values of a for which the equation cos2x+asinx=2a−7 has a solution is

(−∞,2)

[2,6]

(6,∞)

(−∞,∞)