Question

Statement 1 : The number of common solutions of the trigonometric equations $2si{n}^2\theta -cos2\theta =0$ and $2co{s}^2\theta -3sin\theta =0$ is the interval $[0,2\pi ]$ is 2.
Statement 2 : The number of solutions of the equation $2co{s}^2\theta -3sin\theta =0$ in $[0,\pi ]$ is 2.

• A. Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of statement
• B. Statement 1 is true, Statement 2 is true, Statement 2 is not correct explanation of statement 1
• C. Statement 1 is true and Statement 2 is false
• D. Statement 1 is false and Statement 2 is true

Answer 1

The correct option is A Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of statement

For statement : $1,\theta \in [0,2\pi ]$
$2sin2\theta cos2\theta =\theta \Rightarrow (2sin\theta -1)(2sin\theta +1)=0$
$2cos2\theta -3sin\theta =\theta \Rightarrow (2sin\theta -1)(sin\theta -2)=0$
Both equations have common factor is $(2sin\theta -1)$
$\therefore 2sin\theta -1=0\Rightarrow sin\theta =\frac{1}{2}\Rightarrow \theta$ has 2 values in $[0,2\pi ]$
For statement $2:\theta \in [0,\pi ]$
The equation $2co{s}^2\theta -3sin\theta =\theta \Rightarrow (2sin\theta -1)(sin\theta +2)=0$
$\Rightarrow sin\theta =\frac{1}{2}\Rightarrow \theta$ has 2 values in $[0,\pi ]$