Trigonometry
System of equations; count ordered pairs
MJMT_Full_Test_04
Grade 12
Question:
Number of ordered pairs $(x,y)$ satisfying $\dfrac{4^{\sin x}\cdot16^{\sin y}}{(1+16^{\sin x})(1+256^{\sin y})}=\dfrac{1}{4}$ and $3^{1+\sqrt{\cos^2x}}+3^{1+\cos y}=10$; $x,y\in[0,2\pi]$ is
Step-by-Step Solution
Key Concept: Second equation: by AM-GM, $3^{1+|\cos x|}+3^{1+\cos y}\geq2\cdot3^{1+\sqrt{|\cos x|\cos y}}\geq... $ Try $\cos x=0,\cos y=1$: $3+9=12\neq10$. Try $\sin x=\sin y=0$ then first equation: $4/(1+1)(1+1)=1\neq1/4$. From solution: $\sin x=\sin y=0$, $\cos y=1$: $y=0,2\pi$, $x=0,\pi,2\pi$. Count.
3 ordered pairs.
Correct Answer: 3