Quadratic Equations(Location of Roots)

Let $f(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}, \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R}, \mathrm{a} \neq 0$ 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. $\mathrm{D} \geq 0$ (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. $\mathrm{D} \geq 0$ (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 $f(\mathrm{x})=\left(\mathrm{x}-\mathrm{a} _{1}\right)^{\mathrm{k} _{1}}\left(\mathrm{x}-\mathrm{a} _{2}\right)^{\mathrm{k} _{2}} \ldots \ldots\left(\mathrm{x}-\mathrm{a} _{\mathrm{n}}\right)^{\mathrm{k} _{\mathrm{n}}}$

Where $\mathrm{k} _{\mathrm{i}} \in \mathrm{N} \forall \mathrm{i} \& a _{i} \in R$ such that $\mathrm{a} _{1}<\mathrm{a} _{2}<\ldots \ldots<\mathrm{a} _{\mathrm{n}}$. Mark $\mathrm{a} _{1}, \mathrm{a} _{2} \ldots \ldots \mathrm{a} _{\mathrm{n}}$ 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

$\mathrm{x} \in \phi$ if $\mathrm{b} \leq 0 ; \quad \mathrm{x}=\log _{\mathrm{a}} \mathrm{b}$ if $\mathrm{b}>0, \mathrm{a} \neq 1 ;$

$x \in \phi$ if $a=1, b \neq 1 ; x \in R$, 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

$\left(a _{1}^{2}+a _{2}^{2}+a _{3}^{2}\right)\left(b _{1}^{2}+b _{2}^{2}+b _{3}^{2}\right)-\left(a _{1} b _{1}+a _{2} b _{2}+a _{3} b _{3}\right)^{2}$

$=\left(a _{1} b _{2}-a _{2} b _{1}\right)^{2}+\left(a _{2} b _{3}-a _{3} b _{2}\right)^{2}+\left(a _{3} b _{1}-a _{1} b _{3}\right)^{2}$

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

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

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

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

Solved examples

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

(a). $(-\infty,-2)$

(b). $(-\infty,-2]$

(c). $(-2, \infty)$

(d). none of these

Show Answer

Solution:

Answer: b

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

(a). $(-\infty, 1]$

(b). $(-\infty, 1)$

(c). $[1, \infty)$

(d). none of these

Show Answer

Solution:

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

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). $\left(-\infty,-\frac{3}{4}\right)$

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

(c). $(-\infty, 1]$

(d). none of these

4. If $a, b, c \in R$, and the equation $a x^{2}+b x+c=0$ 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 $x^{2}-2 a x+a^{2}+a-3$ 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:-

$f(x)=a x^{2}+b x+c=a(x-\alpha)(x-\beta)$, where $\alpha<\beta$ are the roots of $f(x)=0$. If $\Delta=b^{2}-4 a c$ 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 $\alpha \& \beta ; f(\mathrm{p})$ is negative and for any number q or $\mathrm{r}$ which do not lie between $\alpha \& \beta, f(\mathrm{q})$ 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 $x^{2}-2(4 \lambda-1) x+15 \lambda^{2}-2 \lambda-7>0 \forall x \in R$, 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). $g(x) \geq 0$

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

(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 $(a+1) x^{2}-3 a x+4 a=0(a \neq-1)$ greater than unity is

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

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

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

(d). none of these

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

(a). $(1,10]$

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

(c). $(0, \infty)$

(d). $(-1, \infty)$

5. If $a, b$ are the real roots of $x^{2}+p x+1=0$ and $c, d$ are the real roots of $x^{2}+q x+1=0$, 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 $(2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{4}{2-\sqrt{3}}$ 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 $f(x)=\frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7}$ is $5 \mathrm{k}+1$, Then $k$ is

(a). 41

(b). 40

(c). 8

(d). none of these

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

(a). $a x+b y+c z$

(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 $\left|x^{2}-9 x+20\right|>x^{2}-9 x+20$ then which is true?

(a). $ \mathrm{x} \leq 4$ or $\mathrm{x} \geq 5$

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

(c). $4<x<5$

(d). none of these

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

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

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

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

(d). none of these

13. If $\lambda \neq \mu$ 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). $x^{2}-5 x-3=0$

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

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

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

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

(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