Quadratic Equations(Location of Roots)

Let $f(\mathrm{x})={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c},\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R},\mathrm{a}\ne 0$ and $\alpha ,\beta (\alpha <\beta )$ be the roots of $f(\mathrm{x})=0$. Let $\mathrm{k}$ be any real number

Cases	Figure	Condition
$\alpha <\mathrm{k}$ and $\beta <\mathrm{k}$ Both the roots are less than $\mathrm{k}$		i. $\mathrm{D}\ge 0$ (roots may be equal) ii. $\mathrm{a}.f(\mathrm{k})>0$ iii. $2\mathrm{k}>\alpha +\beta$ i.e $2\mathrm{k}>$ sum of roots or $\mathrm{k}>\frac{-\mathrm{b}}{2\mathrm{a}}$
$\alpha >\mathrm{k}$ and $\beta >\mathrm{k}$ Both the roots are greater than $\mathrm{k}$		i. $\mathrm{D}\ge 0$ (roots may be equal) ii. a. $f(\mathrm{k})>0$ iii. $2\mathrm{k}<\alpha +\beta$ i.e $2\mathrm{k}<$ sum of roots or $\mathrm{k}<\frac{-\mathrm{b}}{2\mathrm{a}}$
$\alpha <\mathrm{k}<\beta$ $\mathrm{k}$ lies between the roots		i. $\mathrm{D}>$ (distinct roots) ii. a $f(\mathrm{k})<0$

Wavy Curve Method

Let $f(\mathrm{x})={(\mathrm{x}-{\mathrm{a}}_1)}^{{\mathrm{k}}_1}{(\mathrm{x}-{\mathrm{a}}_2)}^{{\mathrm{k}}_2}\dots \dots {(\mathrm{x}-{\mathrm{a}}_{\mathrm{n}})}^{{\mathrm{k}}_{\mathrm{n}}}$

Where ${\mathrm{k}}_{\mathrm{i}}\in \mathrm{N}\mathrm{\forall }\mathrm{i}\mathrm{\&}{a}_i\in R$ such that ${\mathrm{a}}_1<{\mathrm{a}}_2<\dots \dots <{\mathrm{a}}_{\mathrm{n}}$. Mark ${\mathrm{a}}_1,{\mathrm{a}}_2\dots \dots {\mathrm{a}}_{\mathrm{n}}$ on real axis check the sign of $f(\mathrm{x})$ in each interval. The solution of $f(\mathrm{x})>0$ is the union of all intervals in which we have put plus sign and the solution of $f(\mathrm{x})<0$ is the union of all intervals in which we have put the minus sign.

Exponential Equations

If we have an equation of the form

${\mathrm{a}}^{\mathrm{x}}=\mathrm{b}$ where $\mathrm{a}>0$, then

$\mathrm{x}\in \varphi$ if $\mathrm{b}\le 0;\mathrm{x}={\log }_{\mathrm{a}}\mathrm{b}$ if $\mathrm{b}>0,\mathrm{a}\ne 1;$

$x\in \varphi$ if $a=1,b\ne 1;x\in R$, if $a=1,b=1$

Lagrange’s Identity

If ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{b}}_1,{\mathrm{b}}_2,{\mathrm{b}}_3\in \mathrm{R}$ then

$(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)-{(a_1{b}_1+a_2{b}_2+a_3{b}_3)}^2$

$={(a_1{b}_2-a_2{b}_1)}^2+{(a_2{b}_3-a_3{b}_2)}^2+{(a_3{b}_1-a_1{b}_3)}^2$

Note: If $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}$, then each ratio is equal to

i. $\frac{\mathrm{a}+\mathrm{c}+\mathrm{e}+\dots }{\mathrm{b}+\mathrm{d}+\mathrm{f}+\dots }$

ii. ${\left(\frac{{\mathrm{pa}}^{\mathrm{n}}+{\mathrm{qc}}^{\mathrm{n}}+{\mathrm{re}}^{\mathrm{n}}+\dots .}{{\mathrm{pb}}^{\mathrm{n}}+{\mathrm{qd}}^{\mathrm{n}}+{\mathrm{rf}}^{\mathrm{n}}+\dots .}\right)}^{1/\mathrm{n}}$ where $\mathrm{p},\mathrm{q},\mathrm{r},\mathrm{n}\in \mathrm{R}$

iii. $\frac{\sqrt{\mathrm{ac}}}{\sqrt{\mathrm{bd}}}=\frac{\sqrt[n]{\mathrm{ace}\dots }}{\sqrt[n]{\mathrm{bdf}\dots }}$

Solved examples

1. The values of $m$ for which both roots of the equation $x^2-mx+1=0$ are less than unity is

(a). $(-\mathrm{\infty },-2)$

(b). $(-\mathrm{\infty },-2]$

(c). $(-2,\mathrm{\infty })$

(d). none of these

Show Answer

Solution:

Answer: b

2. The values of $m(m\in R)$, for which both roots of the equation $x^2-6mx+9m^2-2m+2=0$ exceed 3 is

(a). $(-\mathrm{\infty },1]$

(b). $(-\mathrm{\infty },1)$

(c). $[1,\mathrm{\infty })$

(d). none of these

Show Answer

Solution:

From (1), (2) and (3) $\mathrm{m}\in (\frac{11}{9},\mathrm{\infty })$

Answer: d

3. The values of $p$ for which 6 lies between the roots of the equation $x^2+2(p-3)x+9=0$ is

(a). $(-\mathrm{\infty },-\frac{3}{4})$

(b). $(-\mathrm{\infty },-\frac{3}{4}]$

(c). $(-\mathrm{\infty },1]$

(d). none of these

4. If $a,b,c\in R$, and the equation $a{x}^2+bx+c=0$ has no real roots, then

(a). $(a+b+c)>0$

(b). $a(a+b+c)>0$

(c). $b(a+b+c)>0$

(d). $\mathrm{c}(\mathrm{a}+\mathrm{b}+\mathrm{c})<0$

Practice questions

1. If the roots of equation $x^2-2ax+a^2+a-3$ are less than 3 , then

(a). $\mathrm{a}<2$

(b). $\mathrm{a}>4$

(c). $3<\mathrm{a}<4$

(d). $-2<\mathrm{a}<3$

2. Read the following passage and answer the questions:-

$f(x)=a{x}^2+bx+c=a(x-\alpha )(x-\beta )$, where $\alpha <\beta$ are the roots of $f(x)=0$. If $\mathrm{\Delta }=b^2-4ac$ is negative, then its sign is same as that of $a$, the coefficient of $x^2$. If $f(x)=a(x-\alpha )(x-\beta )$, where $\alpha <\beta$, a is positive, then for any number $\mathrm{p}$ which lies between $\alpha \mathrm{\&}\beta ;f(\mathrm{p})$ is negative and for any number q or $\mathrm{r}$ which do not lie between $\alpha \mathrm{\&}\beta ,f(\mathrm{q})$ or $f(\mathrm{r})$ both will be positive. Also if ${\mathrm{a}}^2$ $<{\mathrm{x}}^2<{\mathrm{b}}^2$, then $\mathrm{a}<\mathrm{x}<\mathrm{b}$ or $-\mathrm{b}<\mathrm{x}<-\mathrm{a}$.

i. If $x^2-2(4\lambda -1)x+15{\lambda }^2-2\lambda -7>0\mathrm{\forall }x\in R$, then $\lambda \in$

(a). $(0,2)$

(c). $(2,4)$

(b). $(1,3)$

(d). none of these

ii. Let $f(\mathrm{x})$ be a quadratic expression which is positive for all real $\mathrm{x}$.If $\mathrm{g}(\mathrm{x})=f(\mathrm{x})$ $+f^{\prime }(\mathrm{x})+f^{\prime \prime }(\mathrm{x})$, then for any real $\mathrm{x}$,

(a). $\mathrm{g}(\mathrm{x})>0$

(b). $g(x)\ge 0$

(c). $g(x)\le 0$

(d). $\mathrm{g}(\mathrm{x})<0$

iii. The inequality $\frac{x^2-|x|-2}{2|x|-x^2-2}>2$ holds only if

(a). $-1<\mathrm{x}<\frac{-2}{3}$ only

(b). only for $\frac{2}{3}<x<1$

(c). $-1<\mathrm{x}<1$

(d). $-1<\mathrm{x}<\frac{2}{3}$ or $\frac{2}{3}<\mathrm{x}<1$

iv. for real $x$, the function $\frac{(x-a)(x-b)}{x-c}$ will assume all real values, provided

(a). $\mathrm{a}<\mathrm{b}<\mathrm{c}$

(b). $\mathrm{a}>\mathrm{b}>\mathrm{c}$

(c). $\mathrm{a}>\mathrm{c}>\mathrm{b}$

(d). $\mathrm{a}<\mathrm{c}<\mathrm{b}$

3. Values of ’ $a$ ’ for which the roots of the equation $(a+1)x^2-3ax+4a=0(a\ne -1)$ greater than unity is

(a). $\mathrm{a}\in [\frac{-16}{7},-1]$

(b). $\mathrm{a}\in (\frac{-16}{7},-1)$

(c). $\mathrm{a}\in (\frac{-16}{7},-1]$

(d). none of these

4. If $x\in R$ satisfies ${({\log }_{10}(100x))}^2+{({\log }_{10}(10x))}^2+{\log }_{10}x\le 14$, then the solution set contains the interval

(a). $(1,10]$

(b). $\lfloor {10}^{-9/2},1\rfloor$

(c). $(0,\mathrm{\infty })$

(d). $(-1,\mathrm{\infty })$

5. If $a,b$ are the real roots of $x^2+px+1=0$ and $c,d$ are the real roots of $x^2+qx+1=0$, then $(a-c)(b-c)(a+d)(b+d)$ is divisible by

(a). $\mathrm{a}-\mathrm{b}-\mathrm{c}-\mathrm{d}$

(b). $\mathrm{a}+\mathrm{b}+\mathrm{c}-\mathrm{d}$

(c). $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}$

(d). $a+b-c-d$

6. Match the following:-

7. Read the paragraph and answer the questions that follow:

Let $(a+\sqrt{b}{)}^{e(x)}+(a-\sqrt{b}{)}^{e(x)-2\lambda }=A$, where $\lambda \in N,A\in R$ and $a^2-b=1\therefore (a+\sqrt{b})(a-\sqrt{b})=1$

i.e $\Rightarrow (a\pm \sqrt{b})=(a+\sqrt{b}{)}^{\pm 1}$ or $(a-\sqrt{b}{)}^{\pm 1}$

i. If $(4+\sqrt{5}{)}^{[x]}+(4-\sqrt{5}{)}^{[x]}=62$, then

(a). $x\in [-3,-2)\cup [1,2)$

(b). $x\in [-3,2)\cup [-2,1)$

(c). $\mathrm{x}\in [-2,-1)\cup [2,3)$

(d). $\mathrm{x}\in [-2,3)\cup [-1,2)$

ii. Solution of $(2+\sqrt{3}{)}^{x^2-2x+1}+(2-\sqrt{3}{)}^{x^2-2x-1}=\frac{4}{2-\sqrt{3}}$ are

(a). $1\pm \sqrt{3},1$

(b). $1\pm \sqrt{2},1$

(c). $1\pm \sqrt{3},2$

(d). $1\pm \sqrt{2},2$

iii. The number of real solutions of the equation $(15+4\sqrt{14}{)}^t+(15-4\sqrt{14}{)}^t=30$ are where $\mathrm{t}={\mathrm{x}}^2-2|\mathrm{x}|$

(a). 0

(b). 2

(c). 4

(d). 6

8. The maximum value of $f(x)=\frac{3x^2+9x+17}{3x^2+9x+7}$ is $5\mathrm{k}+1$, Then $k$ is

(a). 41

(b). 40

(c). 8

(d). none of these

9. If $\frac{x^2-yz}{a}=\frac{y^2-zx}{b}=\frac{z^2-xy}{c}$, then $(x+y+z)(a+b+c)$ is

(a). $ax+by+cz$

(b). $a+b+c$

(c). $\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{3}$

(d). none of these

10. The value of $x$ satisfying the equation $|x-1{|}^{{\log }_3{x}^2-2{\log }_{x}9}=(x-1{)}^7$ is

(a). 3

(b). 9

(c). 27

(d). 81

11. If $|x^2-9x+20|>x^2-9x+20$ then which is true?

(a). $\mathrm{x}\le 4$ or $\mathrm{x}\ge 5$

(b). $4\le \mathrm{x}\le 5$

(c). $4<x<5$

(d). none of these

12. If $x^2+px+1$ is a factor of $a{x}^3+bx+c$, then

(a). ${\mathrm{a}}^2+{\mathrm{c}}^2=-\mathrm{ab}$

(b). ${\mathrm{a}}^2-{\mathrm{c}}^2=-\mathrm{ab}$

(c). $a^2-c^2=ab$

(d). none of these

13. If $\lambda \ne \mu$ and ${\lambda }^2=5\lambda -3,{\mu }^2=5\mu -3$, then the equation whose roots one $\frac{\lambda }{\mu }$ and $\frac{\mu }{\lambda }$ is

(a). $x^2-5x-3=0$

(b). $3x^2+19x+3=0$

(c). $3x^2-19x+3=0$

(d). $x^2+5x-3=0$

14. If the equation $(\cos p-1)x^2+x(\cos p)+\sin p=0$, in the variable $x$, has real roots then ’ $p$ ’ can take any value in the interval.

(a). $(0,2\pi )$

(b). $(-\pi ,0)$

(c). $(\frac{-\pi }{2},\frac{\pi }{2})$

(d). $(0,\pi )$

15. If $(\cos \alpha +\mathrm{isin}\alpha )$ is a root of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0,\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R}$, then

(a). $a\cos 2\alpha +b\sin \alpha +c=0$

(b). $a\cos 2\alpha +b\cos \alpha +c=0$

(c). $a\sin 2\alpha +b\sin \alpha +c=0$

(d). none of these