Question

If $3\sin A+5\cos A=5$, then the value of ${(3\cos A-5\sin A)}^2$ is

• A. $4$
• B. $5$
• C. $2$
• D. $9$

Answer 1

The correct option is D

$9$

Explanation for the correct option:

Step 1. Find the value of ${(3\cos A-5\sin A)}^2$:

Given, $3\sin A+5\cos A=5$ …(1)

Let $3\cos A–5\sin A=x$ …(2)

Step 2. square and add equation (1) and (2),

$9{\sin }^{2}A+25{\cos }^{2}A+30\sin A\cos A+9{\cos }^{2}A+25{\sin }^{2}A–30\sin A\cos A=25+x^2$

$\Rightarrow$ $9({\sin }^{2}A+{\cos }^{2}A)+25({\sin }^{2}A+{\cos }^{2}A)=25+x^2$

$\Rightarrow$ $9+25=25+x^2$

$\Rightarrow$ $x^2=9$

$\mathbf{\therefore }{\mathbf{(}\mathbf{3}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{A}\mathbf{}\mathbf{–}\mathbf{}\mathbf{5}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{A}\mathbf{)}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{9}$

Hence, Option ‘D’ is Correct.