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Sine and Cosine Formulae and Their Applications — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSine and Cosine Formulae and Their Applications1 Mark
MCQ
In any $\triangle\text{ABC},\sum\text{a}^2(\sin\text{B}-\sin\text{C})=$
  • A
    $\text{a}^2+\text{b}^2+\text{c}^2$
  • B
    $\text{a}^2$
  • C
    $\text{b}^2$
  • ✓
    $0$
✓

Answer

Correct option: D.
$0$
Using sine rule, we have
$\sum\text{a}^2(\sin\text{B}-\sin\text{C})$
$\text{a}^2\Big(\frac{\text{b}}{\text{k}}-\frac{\text{c}}{\text{k}}\Big)+\text{b}^2\Big(\frac{\text{c}}{\text{k}}-\frac{\text{a}}{\text{k}}\Big)+\text{c}^2\Big(\frac{\text{a}}{\text{k}}-\frac{\text{b}}{\text{k}}\Big)$
$=\frac{1}{\text{k}}(\text{a}^2\text{b}-\text{a}^2\text{c}+\text{b}^2\text{c}-\text{b}^2\text{a}+\text{c}^2\text{a}-\text{c}^2\text{b})$
This expression cannot be simplified to match with any of the given options.
However, if the quesion is "In any $\triangle\text{ABC},\sum\text{a}^2(\sin\text{B}-\sin\text{C})=",$ then the solution is as follows.
Using sine rule, we have
$\sum\text{a}^2(\sin^2\text{B}-\sin^2\text{C})$
$\text{a}^2\Big(\frac{\text{b}^2}{\text{k}^2}-\frac{\text{c}^2}{\text{k}^2}\Big)+\text{b}^2\Big(\frac{\text{c}^2}{\text{k}^2}-\frac{\text{a}^2}{\text{k}^2}\Big)+\text{c}^2\Big(\frac{\text{a}^2}{\text{k}^2}-\frac{\text{b}^2}{\text{k}^2}\Big)$
$\frac{1}{\text{k}^2}(\text{a}^2\text{b}^2-\text{a}^2\text{c}^2+\text{b}^2\text{c}^2-\text{b}^2\text{a}^2+\text{c}^2\text{a}^2-\text{c}^2\text{b}^2)$
$=\frac{1}{\text{k}^2}\times0$
$=0$
Hence, the correct answer is option (d).

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