91Ó°ÊÓ

Let \(f:(-1,1) \rightarrow B\), be a function defined by \(f(x)=\tan ^{-1} \frac{2 x}{1-x^{2}}\), then \(f\) is both one-one and onto when \(B\) is the interval (A) \(\left(0, \frac{\pi}{2}\right)\) (B) \(\left[0, \frac{\pi}{2}\right)\) (C) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) (D) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

A real valued function \(f(x)\) satisfies the functional equation \(f(x-y)=f(x) f(y)-f(a-x) f(a+y)\) where \(a\) is a given constant and \(f(0)=1, f(2 a-x)\) is equal to (A) \(-f(x)\) (B) \(f(x)\) (C) \(f(A)+f(a-x)\) (D) \(f(x)\)

The largest interval lying in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for which the function \(f(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\) is defined, is (A) \([0, \pi]\) (B) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (C) \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)\) (D) \(\left[0, \frac{\pi}{2}\right)\)

For real \(x\), let \(f(x)=x^{3}+5 x+1\), then \(\quad\) (A) \(f\) is one-one but not onto \(R\) (B) \(f\) is onto \(R\) but not one-one (C) \(f\) is one-one and onto \(R\) (D) \(f\) is neither one-one nor onto \(R\)

The domain of the function \(f(x)=\frac{1}{\sqrt{|x|-x}}\) is (A) \((0, \infty)\) (B) \((-\infty, 0)\) (C) \((-\infty, \infty)-\\{0\\}\) (D) \((-\infty, \infty)\)

If \(f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0\), and \(S=\\{x \in R: f(x)=f(-x)\\} ;\) then \(\mathrm{S}:\) (A) contains more than two elements. (B) is an empty set. (C) contains exactly one element (D) contains exactly two elements