G Tewani Solutions for Chapter: Theory of Equations, Exercise 2: Archives

If the roots of the equation $b{x}^2+cx+a=0$ are imaginary, then for all real values of $x$, the expression $3b^2{x}^2+6bcx+2c^2$ is

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$ $a,b,c\in R,$ have a common root, then $a:b:c$ is

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of equation ${\mathrm{px}}^2+\mathrm{qx}+\mathrm{r}=0,\mathrm{p}\ne 0.$ If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{A}.\mathrm{P}.$ and $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}=4,$ then the value of $\left|\mathrm{\alpha }-\mathrm{\beta }\right|$ is 

Let $\mathrm{p}$ and $\mathrm{q}$ be real numbers such that $\mathrm{p}\ne 0, {\mathrm{p}}^3\ne \mathrm{q}$ and ${\mathrm{p}}^3\ne -\mathrm{q}$. If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are non zero complex numbers satisfying $\mathrm{\alpha }+\mathrm{\beta }=-\mathrm{p}$ and ${\mathrm{\alpha }}^3+{\mathrm{\beta }}^3=\mathrm{q}$, then a quadratic equation having $\frac{\mathrm{\alpha }}{\mathrm{\beta }}$ and $\frac{\mathrm{\beta }}{\mathrm{\alpha }}$ as its roots is

A value of $b$ for which the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have one root in common is?

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2-6\mathrm{x}-2=0$, with $\mathrm{\alpha }>\mathrm{\beta }$. If ${\mathrm{a}}_{\mathrm{n}}={\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}$ for $\mathrm{n}\ge 1$, then the value of $\frac{{\mathrm{a}}_{10}-2{\mathrm{a}}_8}{2{\mathrm{a}}_9}$ is

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}.$ Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0.$ If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2,$ then ${\alpha }_1+{\beta }_2$ equals?

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation  $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $\left|x_1-x_2\right|<1$. Which of the following intervals is (are) a subset(s) of $S$?