Q115 / 155 · WBJEE 2025 · Mathematics

If the equation $\sin ^4 x-(p+2) \sin ^2 x-(p+3)=0$ has a solution, the $p$ must lie in the interval

  • Correct answer
    A
    $[-3,-2]$
  • Correct answer
    B
    $(-3,-2)$
  • C
    $(2,3)$
  • D
    $[-5,-3]$

Solution

$\begin{aligned} & \sin ^2 x=\frac{(P+2) \pm \sqrt{(P+2)^2+4.1(P+3)}}{2.1} \\ & =\frac{(P+2) \pm \sqrt{P^2+4 p+4+4 p+12}}{2} \\ & =\frac{(P+2) \pm \sqrt{(P+4)^2}}{2} \\ & =\frac{2 p+6}{2}=P+3 \\ & =\sin ^2 x \in[0,1] \\ 0 & \leq P+3 \leq 1 \\ & -3 \leq P \leq-2\end{aligned}$

Hence, the correct answer is options: 'A', 'B'.

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