Embibe Experts Solutions for Chapter: Functions, Exercise 1: Exercise 1

Consider a function $f\left(x\right)=sgn\left(x\right)+sgn\left(\left[x\right]\right)+\mathrm{sgn}\left(\left\{\mathrm{x}\right\}\right)$ defined in the interval $\left(-5,5\right)$, where  $\left[·\right]$and $\left\{·\right\}$ represent the greatest integer and the fractional part functions respectively. Then

If $n\left(f_1\left(x\right),f_2\left(x\right)\right)$ denotes the number of solutions of the equation $f_1\left(x\right)=f_2\left(x\right)$, then which of the following is/are CORRECT?

The set of real values of '$x$' satisfying the equality $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$ (where $\left[\right]$ denotes the greatest integer function) belongs to the interval $\left(a,\frac{b}{c}\right]$ where $a, b, c\in N$ and $\frac{b}{c}$is in its lowest form, then the value of $a+b+c+abc$ is

Let $\left\{x\right\}$ & $\left\{x\right\}$ denote the fractional and integral part of a real number $x$, respectively, then number of solutions of the equations $4\left\{x\right\}=x+\left[x\right]$, is

What is the sum of the squares of the roots of the equation $x^2-7\left[x\right]+5=0?$

(Here $\left[x\right]$ denotes the greatest integer less than or equal to $x.$ For example $\left[3.4\right]=3$ and $\left[-2.3\right]=-3.)$

Let $f\left(x\right)=1-x-x^3$. If the set of values of $x$ satisfying the inequality, $1-f\left(x\right)-f^3\left(x\right)>f\left(1-5x\right)$ is $\left(a,0\right)\cup \left(b,\infty \right)$. Find $a+b$.

Suppose $p\left(x\right)$ is a polynomial with integer coefficients. The remainder when $p\left(x\right)$ is divided by $x-1$ is $1$ and the remainder when $p\left(x\right)$ is divided by $x-4$ is $10$. If $r\left(x\right)$ is the remainder when $p\left(x\right)$ is divided by $\left(x-1\right)\left(x-4\right)$, then the value of $r\left(2006\right)$, is

Let $f\left(x\right)=\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+5$ where $x\in \left[-6,6\right]$. If the range of the function is $\left[a,b\right]$ where $a, b\in N$, then the value of $\left(a+b\right)$, is