G Tewani Solutions for Chapter: Theory of Equations, Exercise 1: Exercises

If $\alpha , \beta$ are the nonzero roots of $a{x}^2+bx+c=0$ and $\alpha +h, \beta +h$ are the roots of $p{x}^2+qx+r=0$, then $h$

Value of $a$ for which equations $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have a common root

If the equation $x^2+2\left(\lambda +1\right)x+{\lambda }^2+\lambda +7=0$ has only negative roots, then the least value of $\lambda$ equals

If the set of values of $a$ for which $f\left(x\right)=a{x}^2-\left(3+2a\right)x+6, a\ne 0$ is positive for exactly three distinct negative integral values of $x$ is $(c, d],$ then the value of $d$ is

If $a,b\in R$ such that $a+b=1$ and $\left(1-2ab\right)\left(a^3+b^3\right)=12.$ The value of $\left(a^2+b^2\right)$ is equal to

If the cubic equation $2x^3-9x^2+12x+k=0$ has two equal roots then maximum value of $\left|k\right|$ is

Let $a,b$ and $c$ be distinct non-zero real numbers such that $\frac{1-a^3}{a}=\frac{1-b^3}{b}=\frac{1-c^3}{c}.$ The value of $\left(a^3+b^3+c^3\right)$ is

If the fraction $\frac{x^3+\left(a-10\right)x^2-x+a-6}{x^3+\left(a-6\right)x^2-x+a-10}$ reduces to a quotient of two linear functions, then $a$ equals                          .