Quadratic Equation Question 9
Question 9 - 2024 (31 Jan Shift 2)
The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$ is
(1) 2
(2) more than 2
(3) 1
(4) 0
Show Answer
Answer (4)
Solution
Take $e^{\sin x}=t(t>0)$
$\Rightarrow t-\frac{2}{t}=2$
$\Rightarrow \frac{t^{2}-2}{t}=2$
$\Rightarrow t^{2}-2 t-2=0$
$\Rightarrow t^{2}-2 t+1=3$
$\Rightarrow(t-1)^{2}=3$
$\Rightarrow t=1 \pm \sqrt{3}$
$\Rightarrow t=1 \pm 1.73$
$\Rightarrow t=2.73$ or $-0.73($ rejected as $t>0)$
$\Rightarrow e^{\sin x}=2.73$
$\Rightarrow \log _e e^{\sin x}=\log _e 2.73$
$\Rightarrow \sin x=\log _e 2.73>1$
So no solution.
