Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 1: JEE Advanced Paper 1 - 2016

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 $x_1 \& x_2$ satisfying the inequality $\left|x_1-x_2\right|<1.$ Which of the following intervals is(are) a subset(s) of $S$?

The quadratic equation $P\left(x\right)=0$ with real coefficients has purely imaginary roots. Then the equation $P\left(P\right(x\left)\right)=0$ has

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)+bd\left(b-d\right)$ is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0,$ with $\alpha >\beta .$ For all positive integers $n,$ define

$a_n=\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta },n\ge 1$

$b_1=1$ and $b_n=a_{n-1}+a_{n+1},\hspace{0.17em}n\ge 2.$

Then which of the following options is/are correct?

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 $a \in R$ and let $f :R\to R$ be given by $f\left(x\right)=x^5-5x+a,$ then