Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 1: Exercise 1

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2x^2+2qx+1=0,$ then $\left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)\left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)$ is equal to :

If $\alpha , \beta$ are the real and distinct roots of $x^2+px+q=0$ and ${\alpha }^4, {\beta }^4$ are the roots of $x^2-rx+s=0$, then the equation $x^2-4qx+2q^2-r=0$ always has $\left(\alpha \ne -\beta \right)$

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 $S$?

Number of solution(s) satisfying the equation $3x^2-2x^3={\log }_2\left(x^2+1\right)-{\log }_{2}x$ is

Let $P\left(x\right)=x^{32}-x^{25}+x^{18}-x^{11}+x^4-x^3+1$. Which of the following are CORRECT?

If $(x-1{)}^2$ is a factor of $a{x}^3+b{x}^2+c,$ then roots of the equation $c{x}^3+bx+a=0$ may be

How many ordered pairs of integers $(x,y)$ satisfy the equation $x^2+y^2=2(x+y)+xy$?

Let $a>0$, and let $P\left(x\right)$ be a polynomial with integer coefficients such that
$P\left(1\right)=P\left(3\right)=P\left(5\right)=P\left(7\right)=a$, and
$P\left(2\right)=P\left(4\right)=P\left(6\right)=P\left(8\right)=-a$
The smallest possible value of $a=5\times Q$. What is $Q$ ?