Let the set of all a R such that the equation 2x + a x = 2a - 7 has a solution be [p,q].

Question

Let the set of all $a \in \mathbb{R}$ such that the equation $\cos 2x + a\sin x = 2a - 7$ has a solution be $[p,q]$.

Answer: 48

Accepted Answer

*(The source PDF's text for this question is truncated after "...be $[p, q]$ and", so the exact final quantity being asked for could not be reliably recovered. From the underlying equation, solving $\cos2x+a\sin x=2a-7$ gives $a = 2\sin x+4$ for $\sin x \in[-1,1]$, so $[p,q]=[2,6]$; the stated answer of 48 corresponds to a further expression built from $p$ and $q$ that is not recoverable from the source text with confidence --- for instance $4pq = 4(2)(6) = 48$ is consistent, but the exact original wording could not be confirmed.)*

Solutions · 1