MHTCET 16 May 2024 Shift 1 Paper

Concept:Both sets are defined by trigonometric equations; we transform the equation of one set into the other to check whether the two sets are equal.Explanation:Consider the equation defining set $P$:$$\sin \theta - \cos \theta = \sqrt{2} \cos \theta$$Add $\cos \theta$ to both sides to isolate $\sin \theta$:$$\sin \theta = (\sqrt{2}+1) \cos \theta$$Divide both sides by $(\sqrt{2}+1)$ and rationalize the denominator:$$\frac{\sin \theta}{\sqrt{2}+1} = \frac{\sin \theta(\sqrt{2}-1)}{(\sqrt{2})^2 - 1^2}$$This simplifies to:$$\sqrt{2} \sin \theta - \sin \theta = \cos \theta$$Rearranging the terms gives:$$\sin \theta + \cos \theta = \sqrt{2} \sin \theta$$This is exactly the equation that defines set $Q$.Hence, every $\theta$ in $P$ satisfies the condition for $Q$, so $P \subseteq Q$.The steps are reversible, so every $\theta$ in $Q$ also satisfies the condition for $P$, giving $Q \subseteq P$.Therefore, the two sets have identical elements.Answer:Option D: $P = Q$