Question

If ${\sin }^{2}x+\sin x=(a+2)$, then which of the following statement(s) is(are) correct?

• A. Number of integral values of $a$ for real solution to exist is $3$.
• B. There exists no solution for $a<-\frac{9}{4}$ or $a>0$.
• C. The minimum value of $a$ for real solution is $-2$
• D. Number of prime values of a for real solution to exist is $1$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A Number of integral values of $a$ for real solution to exist is $3$.
B There exists no solution for $a<-\frac{9}{4}$ or $a>0$.
C The minimum value of $a$ for real solution is $-2$

Given ${\sin }^{2}x+\sin x=(a+2)$

$\Rightarrow {\sin }^{2}x+2\cdot \frac{1}{2}\cdot \sin x+\frac{1}{4}=a+2+\frac{1}{4}$

$\Rightarrow {(\sin x+\frac{1}{2})}^2=a+\frac{9}{4}$

Range of LHS is $[0,\frac{9}{4}]$

So for solution to exist for given equation , we must have

$0\le a+\frac{9}{4}\le \frac{9}{4}$

$\Rightarrow -\frac{9}{4}\le a\le 0$