Nishit K Sinha Solutions for Chapter: X+2 Maths, Exercise 4: Practice Exercises

If x is a real number, then $\sqrt{{\log }_{\mathrm{e}}\frac{4\mathrm{x}-{\mathrm{x}}^2}{3}}$ is a real number if and only if:

Let f be a function such that $\mathrm{f}\left(\mathrm{mn}\right)=\mathrm{f}\left(\mathrm{m}\right) \mathrm{f}\left(\mathrm{n}\right)$ for every positive integers m and n. If $\mathrm{f}\left(1\right), \mathrm{f}\left(2\right)$ and $\mathrm{f}\left(3\right)$ are positive integers, $\mathrm{f}\left(1\right)<\mathrm{f}\left(2\right)$, and $\mathrm{f}\left(24\right)=54$, then equals _____.

$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)=\frac{5\mathrm{x}+2}{3\mathrm{x}-5}$ and $\mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2-2\mathrm{x}-1$, then the value of $\mathrm{g}\left(\mathrm{f}\right(\mathrm{f}\left(3\right)\left)\right)$ is.

Let $\mathrm{f}\left(\mathrm{x}\right)$ be a function satisfying $\mathrm{f}\left(\mathrm{x}\right) \mathrm{f}\left(\mathrm{y}\right)=\mathrm{f}(\mathrm{x} \mathrm{y})$ for all real x, y. If $\mathrm{f}\left(2\right)=4$, then what is the value of $\mathrm{f}\left(\frac{1}{2}\right)$?

Let $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2-\mathrm{b}\left|\mathrm{x}\right|$, where a and b are constants. Then at $\mathrm{x}=0, \mathrm{f}\left(\mathrm{x}\right)$ is

If $\mathrm{f}\left(\mathrm{x}\right)=\log \left\{\frac{(1+\mathrm{x})}{(1-\mathrm{x})}\right\}$ then $\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right)$ is:

Which of the following expressions is indeterminate?

The set of all positive integers is the union of two disjoint subsets:

$\left\{\mathrm{f}\right(1),\mathrm{f}(2),\dots .,\mathrm{f}(\mathrm{n}),\dots .\}$ and $\left\{\mathrm{g}\right(1),\mathrm{g}(2),\dots .,,\mathrm{g}(\mathrm{n})\dots .,$ where

$\mathrm{f}\left(1\right)<\mathrm{f}\left(2\right)<\dots \mathrm{f}\left(\mathrm{n}\right)\dots ,$ and $\mathrm{g}\left(1\right)<\mathrm{g}\left(2\right)<\dots .<\mathrm{g}\left(\mathrm{n}\right)$

$\mathrm{G}\left(\mathrm{n}\right)=\mathrm{f}\left(\mathrm{f}\right(\mathrm{n}\left)\right)+1$ for all $\mathrm{n}\ge 1$