Arun Sharma Solutions for Exercise 2: Review Test

If $a,b,c$ are positive numbers and it is known that $a^2+b^2+c^2=8$ then:

Find the value of the expression given in terms of variables $\text{'}x\text{'}$ and $\text{'}y\text{'}$.

$\frac{\left(x^2+\left(a-c\right)x-ac\right)\left(x^2-ax-bx+ab\right)\left(x+c\right)}{\left(x^2-a^2\right)\left(x^3-b{x}^2-c^{2}x+b{c}^2\right)}$

Raman derived an equation to denote distance of a Haley's comet $\left(x\right)$ in the form of a quadratic equation. Distance is given by the solution of quadratic equation $x^2+Bx+C=0$. To determine constants of the above equation for Haley's Comet, two separate series of experiments were conducted by Raman. Based on the data of the first series, value of $x$ obtained is $\left(1,8\right)$and based on the second series of data, value of $x$ obtained is $\left(2,10\right)$. Later on it is discovered that first series of data gave incorrect value of constant $C$ while second series of data gave incorrect value of constant $B$. What is the set of actual distance of Haley' Comet found by Raman?

$\text{'}a\text{'},\text{'}b\text{'} \text{and} \text{'}c\text{'}$ are three real numbers. Which of the following statements is/are always true?

$\left(A\right) \left(a-1\right)\left(b-1\right)\left(c-1\right)<abc$

$\left(B\right) \frac{\left(a^2+b^2+c^2\right)}{2}\ge ca+cb-ab$

$\left(C\right) a^{2}b\sqrt{c}$ is a real number

If we have $f\left[g\left(y\right)\right]=g\left[f\left(y\right)\right]$, then which of the following is true?

$f\left(a\right)=\frac{a^8-1}{a^2+1}$ and $g\left(a\right)=\frac{a^4-3}{{\left(a+1\right)}^2}$, what is $f\left(\frac{1}{g\left(2\right)}\right)?$

Dev was solving a question from his mathematics book when he encountered the expression $\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}$ then $a^a{b}^b{c}^c$ is

The number of integral solutions of $\frac{\log 5+\log \left(a^2+1\right)}{\log \left(a-2\right)}=2$ is