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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

If x+ y ≥ α x x ∈ R then y+ α is equal to

If sin⁡x+sin⁡y≥cos⁡αcos⁡x∀x∈R\sin x+\sin y \geq \cos \alpha \cos x \forall x \in R then sin⁡y+cos⁡α\sin y+\cos \alpha is equal to

Options

  1. A12\frac{1}{2}
  2. B1
  3. C2
  4. D-1

Correct answer

B. 1

Step-by-step solution

 As sin⁡x+sin⁡y≥cos⁡αcos⁡x∀x∈R Let x=−π2⇒sin⁡y≥1⇒sin⁡y=1⇒1+sin⁡x≥cos⁡αcos⁡x⇒cos⁡αcos⁡x−sin⁡x≤1⇒cos⁡2α+1=1⇒cos⁡α=0⇒sin⁡y+cos⁡α=1+0=1\begin{aligned} & \text { As } \sin x+\sin y \geq \cos \alpha \cos x \forall x \in \mathbb{R} \\ & \text { Let } x=-\frac{\pi}{2} \Rightarrow \sin y \geq 1 \Rightarrow \sin y=1 \\ & \Rightarrow 1+\sin x \geq \cos \alpha \cos x \\ & \Rightarrow \cos \alpha \cos x-\sin x \leq 1 \\ & \Rightarrow \sqrt{\cos ^2 \alpha+1}=1 \Rightarrow \cos \alpha=0 \Rightarrow \sin y+\cos \alpha=1+0=1\end{aligned}
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