11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer If \[a=\cos \alpha +i\,\sin \alpha ,\,\,b=\cos \beta +i\,\sin \beta ,\]\[c=\cos \gamma +i\,\sin \gamma \,\,\text{and}\,\,\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1,\] then \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )\] is equal to [RPET 2001]

    A) 3/2

    B) - 3/2

    C) 0

    D) 1

    Correct Answer: D

    Solution :

    \[\frac{b}{c}=\frac{\cos \beta +i\sin \beta }{\cos \gamma +i\sin \gamma }\times \frac{\cos \gamma -i\sin \gamma }{\cos \gamma -i\sin \gamma }\] \[\Rightarrow \frac{b}{c}=\,\cos (\beta -\gamma )+i\sin (\beta -\gamma )\]              ......(i) Similarly,\[\,\frac{c}{a}=\cos (\gamma -\alpha )+i\sin \,(\gamma -\alpha )\]    ......(ii)  and    \[\,\,\frac{a}{b}=\cos (\alpha -\beta )+i\sin (\alpha -\beta )\]  .....(iii) from (i) + (ii) + (iii) \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )\]\[+i[\sin (\beta -\gamma )+\sin (\gamma -\alpha )+\sin (\alpha -\beta )]=1\] Equating real and imaginary parts, \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=1\].

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