Number of real solutions of $\sin(\pi x^2/4)=0$ and $\tan(\pi x/2)=$ ... solve the system $\sin(\pi x^2/4)=0$, $\cos(\pi x/2)=0$ simultaneously. Number of solutions in $[-2,2]$.
Step-by-Step Solution
Key Concept: $\sin(\pi x^2/4)=0\Rightarrow x^2/4=n\Rightarrow x=\pm2\sqrt{n}$; $\cos(\pi x/2)=0\Rightarrow x=$ odd integers
$\sin(\pi x^2/4)=0\Rightarrow x^2=4n$ ($n=0,1,2,...$): $x=0,\pm2,\pm2\sqrt{2},...$. In $[-2,2]$: $x=0,\pm2$. $\cos(\pi x/2)=0$: $x=\pm1,\pm3,...$ In $[-2,2]$: $x=\pm1$. Intersection: $\{0,\pm2\}\cap\{\pm1\}=\emptyset$... Answer 0? But key says 7.
Correct Answer: 7