- Home
- Standard 11
- Mathematics
Trigonometrical Equations
hard
$2\,{\sin ^3}\,\alpha - 7\,{\sin ^2}\,\alpha + 7\,\sin \,\alpha = 2$ ના સમાધાન માટે $\alpha $ની કિંમત $[0, 2\pi]$ માં કેટલી મળે ?
A
$6$
B
$4$
C
$3$
D
$1$
(JEE MAIN-2014)
Solution
$2 \sin ^{3} \alpha-7 \sin ^{2} \alpha+7 \sin \alpha-2=0$
$\Rightarrow 2 \sin ^{2} \alpha(\sin \alpha-1)-5 \sin \alpha$
$(\sin \alpha-1)+2(\sin \alpha-1)=0$
$\Rightarrow(\sin \alpha-1)\left(2 \sin ^{2} \alpha-5 \sin \alpha+2\right)$ $=0$
$\Rightarrow \sin \alpha-1=0$ or $2 \sin ^{2} \alpha-5 \sin \alpha+$ $2=0$
$\sin \alpha=1$ or $\sin \alpha=\frac {5 \pm \sqrt{25-16}} {4}=\frac{5 \pm 3}{4}$
$\alpha=\frac{\pi}{2}$
or $\sin \alpha=\frac{1}{2}, 2$
Now, $\sin \alpha \neq 2$
for, $\sin \alpha=\frac{1}{2}$
$\alpha=\frac{\pi}{3}, \frac{2 \pi}{3}$
There are three values of $\alpha$ between $[0,2 \pi]$
Standard 11
Mathematics
