BITSAT 2010
- Exam
- BITSAT
- Year
- 2010
- Subject
- Mathematics
- Chapter
- Matrices
BITSAT 2010 · Mathematics · Matrices
The value of ( ^ -1 x+ ^ -1 ((x)/(2)+(1)/(2) 3-3 x^2 )); ((1)/(2) ≤...
The value of (\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)); (\frac{1}{2} \leq x \leq 1) is
Options
- A(-\frac{\pi}{3})
- B(\frac{\pi}{3})
- C(\frac{3}{\pi})
- D(-\frac{3}{\pi})
Correct answer
B. (\frac{\pi}{3})
Step-by-step solution
Let (\cos ^{-1} x=y)
(\Rightarrow x=\cos y), so that (\frac{1}{2} \leq x \leq 1) or (\quad 0 \leq y \leq \frac{\pi}{3})
and (\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}=\frac{1}{2} \cos y+\frac{\sqrt{3}}{2} \sin y)
(=\cos \frac{\pi}{3} \cos y+\sin \frac{\pi}{3} \sin y=\cos \left(\frac{\pi}{3}-y\right))
(\Rightarrow \cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)=\frac{\pi}{3}-y)
(\therefore) the given expression is equal to
(y+\frac{\pi}{3}-y, \text{ i.e., } \frac{\pi}{3})