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BITSAT 2022
Exam
BITSAT
Year
2022
Subject
Mathematics
Chapter
Properties of Triangles

BITSAT 2022 · Mathematics · Properties of Triangles

If in a A B C, 2 b^2=a^2+c^2 , then ( 3B)/( B) is equal to

If in a △ABC,2b2=a2+c2\triangle A B C, 2 b^2=a^2+c^2, then sin⁡3Bsin⁡B\frac{\sin 3B}{\sin B} is equal to

Options

  1. Ac2−a22ca\frac{c^2-a^2}{2 c a}
  2. Bc2−a2ca\frac{c^2-a^2}{c a}
  3. C(c2−a2)2(ca)2\frac{\left(c^2-a^2\right)^2}{(c a)^2}
  4. D[c2−a22ca]2\left[\frac{c^2-a^2}{2 c a}\right]^2

Correct answer

D. [c2−a22ca]2\left[\frac{c^2-a^2}{2 c a}\right]^2

Step-by-step solution

sin⁡3Bsin⁡B=3sin⁡B−4sin⁡3Bsin⁡B=3−4sin⁡2B=3−4(1−cos⁡2B)=−1+4(a2+c2−b2)24(ac)2=−1+(a2+c22)2(ac)2=−1+(a2+c2)24(ac)2=(c2−a22ac)2\frac{\sin 3 B}{\sin B}=\frac{3 \sin B-4 \sin ^3 B}{\sin B}=3-4 \sin ^2 B=3-4\left(1-\cos ^2 B\right)=-1+\frac{4\left(a^2+c^2-b^2\right)^2}{4(a c)^2}=-1+\frac{\left(\frac{a^2+c^2}{2}\right)^2}{(a c)^2}=-1+\frac{\left(a^2+c^2\right)^2}{4(a c)^2}=(\frac{c^2-a^2}{2 a c})^2
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