Let A and B denote the statements A: α+ β+ γ=0 B: α+ β+ γ=0 If (β-γ)+ (γ-α)+ (α-β)=-(3)/(2) , then

Correct answer: C. both A and B are true aligned & (β-γ)+ (γ-α)+ (α-β)=-(3)/(2) & 2[ (β-γ)+ (γ-α)+ (α-β)]+3=0 & 2[ (β-γ)+ (γ-α)+ (α-β)]+ ^2 α+ ^2 α+ ^2 β+ ^2 β+ ^2 γ+ ^2 γ=0 & ( α+ β+ γ)^2+( α+ β+ γ)^2=0 aligned

JEE Main 2009

Exam: JEE Main
Year: 2009
Subject: Mathematics
Chapter: Determinants

JEE Main 2009 · Mathematics · Determinants

Let A and B denote the statements A: α+ β+ γ=0 B: α+ β+ γ=0 If...

Let

A

and

B

denote the statements A:

\cos \alpha+\cos \beta+\cos \gamma=0

\sin \alpha+\sin \beta+\sin \gamma=0

\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}

, then

Options

A $A$ is true and $B$ is false
B $A$ is false and $B$ is true
Cboth $A$ and $B$ are true
Dboth $A$ and $B$ are false

Correct answer

C. both $A$ and $B$ are true

Step-by-step solution

\begin{aligned}& \cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}\\ & \Rightarrow 2[\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)]+3=0 \\ & \Rightarrow 2[\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)]+\sin ^2 \alpha+\cos ^2 \alpha+\sin ^2 \beta+\cos ^2 \beta+\sin ^2 \gamma+\cos ^2 \gamma=0 \\ & \Rightarrow(\sin \alpha+\sin \beta+\sin \gamma)^2+(\cos \alpha+\cos \beta+\cos \gamma)^2=0 \end{aligned}

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