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  • Statement 1:  The number of common solutions of the trignometric equations <img alt="2sin^{2}\theta -cos2\theta=0" src="http://entrancecorner.oncodecogs.com/gif.late

Statement 1:  The number of common solutions of the trignometric equations 2sin^{2}\theta -cos2\theta=0 and 2cos^{2}\theta-3sin\theta=0 in the interval \left [ 0,2\pi \right ] is two :

Statement 2 :  The number of solutions of the equation, 2cos^{2}\theta-3sin\theta=0 in the interval \left [ 0,\pi \right ] is two.

Option: 1

Statement 1 is true ; Statement 2 is true ; Statement 2 is a correct explanation for Statement 1


Option: 2

Statement 1 is true; Statement 2 is true; Statement 2 is not a correct expalantion for Statement 1


Option: 3

Statement 1 is false; Statement 2 is true.


Option: 4

Statement 1 is true ; Statement 2 is false.


Answers (1)

best_answer

\\2 \sin ^{2} \theta-\cos 2 \theta=0 \\ \Rightarrow 2 \sin ^{2} \theta-\left(1-2 \sin ^{2} \theta\right)=0 \\ \Rightarrow 2 \sin ^{2} \theta-1+2 \sin ^{2} \theta=0 \\ \Rightarrow 4 \sin ^{2} \theta=1 \Rightarrow \sin \theta=\pm \frac{1}{2} \\ \therefore \theta=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{11 \pi}{6}

\begin{aligned} &\begin{aligned} & \text { Now } 2 \cos ^{2} \theta-3 \sin \theta=0 \\ \Rightarrow & 2\left(1-\sin ^{2} \theta\right)-3 \sin \theta=0 \\ \Rightarrow &-2 \sin ^{2} \theta-3 \sin \theta+2=0 \\ \Rightarrow &-2 \sin ^{2} \theta-4 \sin \theta+\sin \theta+2=0 \\ \Rightarrow & 2 \sin ^{2} \theta-\sin \theta+4 \sin \theta-2=0 \\ \Rightarrow & \sin \theta(2 \sin \theta-1)+2(2 \sin \theta-1)=0 \\ \Rightarrow & \sin \theta=\frac{1}{2},-2 \end{aligned}\\ &\text { But } \sin \theta=-2, \text { is not possible }\\ &\therefore \quad \sin \theta=\frac{1}{2}, \quad \Rightarrow \quad \theta=\frac{\pi}{6}, \frac{5 \pi}{6} \end{aligned}

Hence, there are two common solutions, there each of the statement- 1 and 2 are true but statement- 2 is not a correct explanation for statement-1.

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