QJEEM12Maf6ec: JEE Main 2012 Mathematics - Class 11 JEEM12Maf6ec Solution: In a If and then the angle R is equal to :

Question

Verified Solution • 2025–26 Session moderate

In a

\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}

If

3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6

and

4\sin Q + 3\cos P = 1,

then the angle R is equal to :

Accepted Answer

Given

3

\sin \,P + 4\cos Q = 6

\,\,\,\,\,\,\,\,...\left( i \right)

4\sin Q + 3\cos P = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)

Squaring and adding

(i)

&

(ii)

we get

9\,{\sin ^2}P + 16{\cos ^2}Q + 24\sin P\cos Q

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 16\,{\sin ^2}Q + 9{\cos ^2}P + 24\sin Q\cos P

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,

= 36 + 1 = 37

\Rightarrow 9\left( {{{\sin }^2}p + {{\cos }^2}P} \right) + 16\left( {{{\sin }^2}Q + {{\cos }^2}q} \right)

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 24\left( {\sin P\cos Q + \cos P\sin Q} \right) = 37

\Rightarrow 9 + 16 + 24\sin \left( {P + Q} \right) = 37

[ As

{\sin ^2}\theta + {\cos ^2}\theta = 1

and

\sin A\cos B + \cos A\sin B

= \sin \left( {A + B} \right)

]

\Rightarrow \sin \left( {P + Q} \right) = {1 \over 2}

\Rightarrow P + Q = {\pi \over 6}

or

{{5\pi } \over 6}

\Rightarrow R = {{5\pi } \over 6}

or

{\pi \over 6}

(as

P + Q + R = \pi

)

If

R = {{5\pi } \over 6}

then

0 &lt; P,Q &lt; {\pi \over 6}

\Rightarrow \cos Q &lt; 1

and

\sin P &lt; {1 \over 2}

\Rightarrow 3\sin P + 4\cos Q &lt; {{11} \over 2}

which is not true.

So

R = {\pi \over 6}

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