Question

Solve the following system of simultaneous equations for x and y.
$4^{\sin x}+3^{1/\cos y}=11$
$5\times {16}^{\sin x}-2\times 3^{1/\cos y}=2$

• A. $x=n\pi \pm \frac{\pi }{6}$ and $y=2m\pi \pm \frac{\pi }{3},$ where $m,nϵz$
• B. $x=n\pi +(-1{)}^n\frac{\pi }{6}$ and $y=2m\pi \pm \frac{\pi }{3},$ where $m,nϵz$
• C. $x=n\pi +(-1{)}^n\frac{\pi }{3}$ and $y=2m\pi \pm \frac{\pi }{6},$ where $m,nϵz$
• D. $x=n\pi \pm \frac{\pi }{3}$ and $y=2m\pi \pm \frac{\pi }{6},$ where $m,nϵz$

Answer 1

The correct option is B $x=n\pi +(-1{)}^n\frac{\pi }{6}$ and $y=2m\pi \pm \frac{\pi }{3},$ where $m,nϵz$

Given $4^{\sin x}+3^{1/\cos y}=11$ ............(i)

And $5\times {16}^{\sin x}-2\times 3^{1/\cos y}=2\Rightarrow 5.4^{2\sin x}-{2.3}^{1/\cos y}=2$

$\Rightarrow 5.4^{2\sin x}-2(11-4^{\sin x})=2$ using (i)

$\Rightarrow 5.4^{2\sin x}+2.4^{\sin x}-24=0$

$\Rightarrow 4^{\sin x}=2,-\frac{12}{5}$ (not possible)

$\Rightarrow \sin x=\frac{1}{2}\Rightarrow x=n\pi +(-1{)}^n\frac{\pi }{6}$

Now from (i) $3^{1/\cos y}=9\Rightarrow \cos y=\frac{1}{2}y=2m\pi \pm \frac{\pi }{3}$ $m,n\in I$