The equation 2 sin 2 x-p-3 sin x-2 p-2-0 possesses a real solution- ifA- p - 6B- 0 - p - 1C- 4 - p - 6D- -1 - p - 3

The correct option is D 2 sin^2 x (p+3) sin x + 2p 2 =0 ⇒sin x = (p+3)±√((p+)^2 8(2p 2))/4 ⇒sin x = (p+3)±(p 5)/4 = p 1/2, 2 ⇒ 1 ≤p 1/2≤ 1 ⇒ 2 ≤ p 1 ...

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Standard XII

Mathematics

Properties Derived from Trigonometric Identities

The equation ...

Question

The equation $2 {sin}^{2} x - (p + 3) sin x + 2 p - 2 = 0$ possesses a real solution, if

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Solution

The correct option is B
$2 {sin}^{2} x - (p + 3) sin x + 2 p - 2 = 0$
$\Rightarrow sin x = \frac{(p + 3) \pm \sqrt{(p +)^{2} - 8 (2 p - 2)}}{4}$
$\Rightarrow sin x = \frac{(p + 3) \pm (p - 5)}{4} = \frac{p - 1}{2}, 2$
$\Rightarrow - 1 \leq \frac{p - 1}{2} \leq 1 \Rightarrow - 2 \leq p - 1 \leq 2$
$\Rightarrow - 1 \leq p \leq 3$

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The equation 2sin2x–(p+3)sinx+2p–2=0 possesses a real solution, if

The correct option is B 2sin2x−(p+3)sinx+2p−2=0 ⇒sinx=(p+3)±√(p+)2−8(2p−2)4 ⇒sinx=(p+3)±(p−5)4=p−12,2 ⇒−1≤p−12≤1⇒−2≤p−1≤2 ⇒−1≤p≤3

The equation $2 {sin}^{2} x - (p + 3) sin x + 2 p - 2 = 0$ possesses a real solution, if