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Mark the Correct Alternative in Each of the Following: in Any ∆Abc, the Value of 2 a C Sin ( a − B + C 2 ) is - Mathematics

MCQ

Mark the correct alternative in each of the following:

In any ∆ABC, the value of  \[2ac\sin\left( \frac{A - B + C}{2} \right)\]  is 

Options

  • \[a^2 + b^2 - c^2\] 

  • \[c^2 + a^2 - b^2\] 

  • \[b^2 - c^2 - a^2\] 

  • \[c^2 - a^2 - b^2\] 

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Solution

In ∆ABC,  

\[A + B + C = \pi \left( \text{ Angle sum property } \right)\]
\[ \Rightarrow A + C = \pi - B\] 

\[\therefore 2ac\sin\left( \frac{A - B + C}{2} \right)\]
\[ = 2ac\sin\left( \frac{\pi - 2B}{2} \right)\]
\[ = 2ac\sin\left( \frac{\pi}{2} - B \right)\]
\[ = 2ac\cos B\]

\[= 2ac\left( \frac{c^2 + a^2 - b^2}{2ca} \right) \left( \text{ Using cosine rule } \right)\]
\[ = c^2 + a^2 - b^2\] 

Hence, the correct answer is option (b).

Concept: Sine and Cosine Formulae and Their Applications
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 10 Sine and cosine formulae and their applications
Q 7 | Page 27
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