Question

Let $S=\{\alpha :{\alpha }^2-5\alpha -6\le 0,\alpha \in R\}$ and $a,b\in S$. If the equation $x^2+7=4x-3\sin (ax+b)$ is satisfied for at least one real value of $x$, then

• A. minimum possible value of $2a+b$ is $-\frac{\pi }{2}$
• B. maximum possible value of $2a+b$ is $\frac{7\pi }{2}$
• C. minimum possible value of $2a+b$ is $\frac{\pi }{2}$
• D. maximum possible value of $2a+b$ is $\frac{11\pi }{2}$

Answer 1

Answer: (A), (D)

The correct options are
A minimum possible value of $2a+b$ is $-\frac{\pi }{2}$
D maximum possible value of $2a+b$ is $\frac{11\pi }{2}$

We have,
${\alpha }^2-5\alpha -6\le 0$
$\Rightarrow (\alpha -6)(\alpha +1)\le 0$
$\Rightarrow -1\le \alpha \le 6$
Hence, $a,b$ belong to $[-1,6]$
Now,
$x^2+7=4x-3\sin (ax+b)$
$\Rightarrow x^2-4x+7=-3\sin (ax+b)$
$L.H.S\ge 3$ and $R.H.S\le 3$
Hence $L.H.S=R.H.S=3$
Equality occurs at $x=2$ and $\sin (ax+b)=-1$
Hence,
$\sin (2a+b)=-1$
$\Rightarrow 2a+b=\frac{3\pi }{2}+2n\pi$
Now,
$2a+b$ belong to $[-3,18]$
Also,
$\Rightarrow 2a+b=\frac{3\pi }{2}+2n\pi$
Hence the maximum value of $n=2$ and minimum value of $n=-1$
So, options A, D are correct.