Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 1: JEE Main - 15th April 2018

Let $\lambda \ne 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14x^2-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35x^2-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation :

Let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_7$${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $\left|{\alpha }_1\right|\ge \left|{\alpha }_2\right|\ge \dots \ge \left|{\alpha }_7\right|$.
Then, ${\alpha }_1{\alpha }_2-{\alpha }_3{\alpha }_4+{\alpha }_5{\alpha }_6$ is equal to _______

The parabolas : $a{x}^2+2bx+cy=0$ and $d^2+2ex+fy=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a,b,c$ are in $\mathrm{G}.\mathrm{P}.$, then

If the value of real number $\alpha >0$ for which $x^2-5\alpha x+1=0$ and $x^2-\alpha x-5=0$ have a common real roots is $\frac{3}{\sqrt{2\beta }}$ then $\beta$ is equal to ________

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$, is:

The equation $e^{4x}+8e^{3x}+13e^{2x}-8e^x+1=0,x\in R$ has :

Let
$S=\left\{x:x\in ℝ\text{ and }{\left(\sqrt{3}+\sqrt{2}\right)}^{x^2-4}+{\left(\sqrt{3}-\sqrt{2}\right)}^{x^2-4}=10\right\}$. Then $n\left(S\right)$ is equal to

The number of integral values of $k$, for which one root of the equation $2x^2-8x+k=0$ lies in the interval $\left(1,2\right)$ and its other root lies in the interval $\left(2,3\right)$, is :