Let $f(x)=\left\{\begin{array}{cc}3 \sin x+a^2-10 a+30 & , x \notin Q \\ 4 \cos x & , x \in Q\end{array}\right.$. Which one of the following statements is correct ?
Answer & explanation
Correct answer: option 3
If f(x) is continuous for all x, then
$3 \sin x+a^2-10 a+30=4 \cos x$ for all x
$\Rightarrow (a-5)^2+5=4 \cos x-3 \sin x$ for all x
$\Rightarrow (a-5)^2+5=5 \cos (x+\theta)$, where $\theta=\tan ^{-1} \frac{3}{4}$ for all x
We observe that LHS $\geq 5$ and RHS $\leq 5$. So, the two sides of the above equality are equal if
a = 5 and $\cos (x+\theta)=1$
$\Rightarrow a =5$ and $x=2 n \pi-\tan ^{-1} \frac{3}{4}$