Question

Which of the following cannot be the value of $(7sin\theta +24cos\theta +7)$ ?

• A. -9
• B. 22
• C. -11
• D. 33

Answer 1

Answer: (D)

33

$7\sin \theta +24\cos \theta +7$

$=25(\frac{7}{25}\sin \theta +\frac{24}{25}\cos \theta )+7$

$=25(\sin \alpha \sin \theta +\cos \alpha \cos \theta )+7$

$=25\cos (\alpha -\theta )+7$

Here,

$\sin \alpha =\frac{7}{25}$

$\cos \alpha =\frac{24}{25}$

Now, from the inequality,

$-1\le \sin (\alpha +\theta )\le 1$

$\Rightarrow -25\le 25\cos (\alpha -\theta )\le 25$

$\Rightarrow -25+7\le 25\cos (\alpha -\theta )+7\le 25+7$

$\Rightarrow -18\le 25\cos (\alpha -\theta )+7\le 32$

$\Rightarrow -18\le (7\sin \theta +24\cos \theta +7)<32[\because 7\sin \theta +24\cos \theta +7=25\cos (\alpha -\theta )+7]$

Hence the value of $(7\sin \theta +24\cos \theta +7)$ will only lie in the interval $[-18,32]$.

Since $33$ does not lie in the interval $[-18,32]$, thus $33$ cannot be the value of given expression.