Chapter 1

If \(f(x+y, x-y)=x y\), then \(\frac{f(x, y)+f(y, x)}{2}=\) (a) \(x\) (b) \(y\) (c) 0 (d) None of these

Let \(f(1)=1\) and \(f(n)=2 \sum_{r=1}^{n-1} f(r) .\) Then \(\sum_{n=1}^{m} f(n)\) is equal to (a) \(3^{n}-1\) (b) \(3^{m}\) (c) \(3^{n-1}\) (d) None of these

If \(f(x)=\frac{x-1}{x+1}\), then \(f(2 x)\) in terms of \(f(x)\) is (a) \(\frac{f(x)+1}{f(x)+3}\) (b) \(\frac{3 f(x)+1}{f(x)+3}\) (c) \(\frac{f(x)+3}{f(x)+1}\) (d) \(\frac{f(x)+3}{3 f(x)+1}\)

If for a function \(f(x), f(x+y)=f(x)+f(y)\) for all reals \(x\) and \(y\) then \(f(0)\) is equal to (a) 1 (b) 0 (c) \(f(x) \forall x \in R\) (d) None of these

If \(f(n+1)+f(n-1)=2 f(n)\) and \(f(0)=0\), then \(f(n), n \in N\) is (a) \(n f(1)\) (b) \((f(1))^{n}\) (c) 0 (d) None of these

Let \(f\) be a function satisfying \(f(x+y)=f(x)+\) \(f(y)\) for all \(x, y \in R .\) If \(f(1)=K\), then \(f(n), n \in N\) is equal to (a) \(K^{n}\) (b) \(n K\) (c) \(K^{-n}\) (d) None of these

If \(f(x)=x^{2}-3 x+1\) and \(f(2 \alpha)=2 f(\alpha)\), then \(\alpha\) is equal to (a) \(\frac{1}{\sqrt{2}}\) (b) \(-\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{\sqrt{2}}\) or \(-\frac{1}{\sqrt{2}}\) (d) None of these

If a function \(F\) is such that \(F(0)=2, F(1)=3\), \(F(n+2)=2 F(n)-F(n+1)\) for \(n \geq 0\), then \(F(5)\) is equal to (a) \(-7\) (b) \(-3\) (c) 7 (d) 13

If \(f(x)=\frac{4^{x}}{4^{x}+2}\), then \(f\left(\frac{1}{1997}\right)+\) \(f\left(\frac{2}{1997}\right)+\ldots+\left(\frac{1996}{1997}\right)\) is equal to (a) 998 (b) 1997 (c) 0 (d) None of these

If \(f(\theta)=\tan \theta\), then \(\frac{f(\theta)-f(\phi)}{1+f(\theta) f(\phi)}\) is equal to (a) \(f(\theta-\phi)\) (b) \(f(\phi-\theta)\) (c) \(f(\theta+\phi)\) (d) None of these

If \(e^{f(x)}=\frac{10+x}{10-x}, x \in(-10,10)\) and \(f(x)\) \(=k f\left(\frac{200 x}{100+x^{2}}\right)\), then \(k\) is equal to (a) \(0.8\) (b) \(0.7\) (c) \(0.6\) (d) \(0.5\)

If \(f(x)=\log \left(\frac{1+x}{1-x}\right)\) when \(-1

If \(f: R \rightarrow R, f(x+y)=f(x)+f(y), \forall x, y \in R\) and \(f(1)=7\), then \(\sum_{r=1}^{n} f(r)\) is equal to (a) \(\frac{7 n(n+1)}{2}\) (b) \(\frac{7 n}{2}\) (c) \(\frac{7(n+1)}{2}\) (d) \(7 n(n+1)\)

Let \(f\) be a function satisfying \(2 f(x y)=[f(x)]^{y}+\) \([f(y)]^{x}\) and \(f(1)=k \neq 1\), then \(\sum_{x=1}^{n} f(x)\) is equal to: (a) \(\frac{k\left(k^{n}-1\right)}{k-1}\) (b) \(\frac{k\left(k^{n}+1\right)}{k-1}\) (c) \(\frac{k^{n}+1}{k-1}\) (d) \(\frac{k\left(k^{n}-1\right)}{k+1}\)

If \(e^{x}=y+\sqrt{1+y^{2}}\), then \(y\) is equal to (a) \(e^{x}+e^{-x}\) (b) \(e^{x}-e^{-x}\) (c) \(\frac{1}{2}\left(e^{x}-e^{-x}\right)\) (d) \(\frac{1}{2}\left(e^{x}+e^{-x}\right)\)

If \(f(x)\) is an even function and \(g(x)\) is an odd function, and \(x^{2} f(x)-2 f(1 / x)=g(x)\), then \(f(5)\) is equal to (a) 5 (b) \(1 / 75\) (c) 0 (d) \(g(5)\)

The function \(f\) satisfies the functional equation \(3 f(x)+2 f\left(\frac{x+59}{x-1}\right)=10 x+30\) for all real \(x \neq 1\). The value of \(f(7)\) is (a) 8 (b) 4 (c) \(-8\) (d) 11

If \(f(x) \cdot f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)\) and \(f(4)=65\), then \(f(6)\) is (a) 215 (b) 217 (c) 220 (d) None of these

Let \(f: R \rightarrow R\) be defined by \(f(x)=2 x+|x|\) then \(f(2 x)+f(-x)-f(x)\) is equal to (a) \(2 x\) (b) \(2|x|\) (c) \(-2 x\) (d) \(-2|x|\)

If \(x, y, z\) are distinct positive numbers different from 1 such that \(\left(\log _{y} x, \log _{z} x-\log _{x} x\right)+(\log y\) \(\log y-\log y)+\left(\log _{x} z \log _{y}-\log _{z} z\right)=0\), what is the value of \(x y z\) (a) 2 (b) 1 (c) \(-1\) (d) 0

If \(x\) satisfies \(|x-1|+|x-2|+|x-3| \geq 6\) then (a) \(0 \leq x \leq 4\) (b) \(x \leq-2\) or \(x \geq 4\) (c) \(x \leq 0\) or \(x \geq 4\) (d) None of these