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

प्रश्न

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

पर्याय

\[a^2 + b^2 - c^2\]
\[c^2 + a^2 - b^2\]
\[b^2 - c^2 - a^2\]
\[c^2 - a^2 - b^2\]

MCQ

उत्तर

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).

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Sine and Cosine Formulae and Their Applications

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 7 Q 6 Q 8

पाठ 10: Sine and cosine formulae and their applications - Exercise 10.4 [पृष्ठ २७]

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आरडी शर्मा Mathematics [English] Class 11

पाठ 10 Sine and cosine formulae and their applications
Exercise 10.4 | Q 7 | पृष्ठ २७

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