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 $\mathrm{k} _{1}, \mathrm{k} _{2}$ be two real numbers such that $\mathrm{k} _{1}<\mathrm{k} _{2}$

Logarithmic Equations

If we have an equation of the form as $\log _{\mathrm{a}} f(\mathrm{x})=\mathrm{b}$ where $\mathrm{a}>0, \mathrm{a} \neq 1$ can be written as $f(\mathrm{x})=\mathrm{a}^{\mathrm{b}}$ when $f(\mathrm{x})>0$.

Logarithmic Inequalities

Descartes Rule of signs

The maximum number of positive real roots of a polynomial equation $f(\mathrm{x})=0$ is the number of changes of signs from positive to negative and negative to positive.

The maximum number of negative real roots of a polynomial equation $f(\mathrm{x})=0$ is the number of changed signs from positive to negative and negative to positive in $f(-\mathrm{x})=0$

Solved examples

1. The values of $m$ for which exactly one root of $x^{2}-2 m x+m^{2}-1=0$ lies in the interval $(-2,4)$ is

(a) $(-3,-1) \cup(3,5)$

(b) $(-3,-1)$

(c) $(3,5)$

(d) none

2. The values of a for which both the rootes of the equation $4 x^{2}-2 x+a=0$ lie in the interval $(-1,1)$ is.

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

(b) $\left(-\infty, \dfrac{1}{4}\right]$

(c) $\left(-2, \dfrac{1}{4}\right]$

(d) none

3. The all possible values of a for which one root of the equation $(a-5) x^{2}-2 a x+a-4=0$ is smaller than 1 and the other greater than 2 is

(a) $[5,24)$

(b) $(5,24]$

(c) $(5,24)$

(d) none of these

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

(a) $\mathrm{c}(\mathrm{a}+\mathrm{b}+\mathrm{c})>0$

(b) $\mathrm{c}+(\mathrm{a}+\mathrm{b}+\mathrm{c}) \mathrm{c}>0$

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

(d) $\mathrm{c}(\mathrm{a}+\mathrm{b}-\mathrm{c})>0$

EXERCISE

1. The values of a for which $2 x^{2}-2(2 a+1) x+a(a+1)=0$ may have one root less than $a$ and other root greater than a are given by

(a) $1>\mathrm{a}>0$

(b) $-1<\mathrm{a}<0$

(c) $\mathrm{a} \geq 0$

(d) $\mathrm{a}>0$ & $\mathrm{a}<-1$

2. The value of a for which the equation $\left(1-a^{2}\right) x^{2}+2 a x-1=0$ has roots belonging to $(0,1)$ is

(a) $\mathrm{a}>\dfrac{1+\sqrt{5}}{2}$

(b) $\mathrm{a}>2$

(c) $\dfrac{1+\sqrt{5}}{2}<\mathrm{a}<2$

(d) $\mathrm{a}>\sqrt{2}$

3. If $a, b, c, x, y, z, \in R$ be such that $(a+b+c)^{2}=3\left(a b+b c+c a-x^{2}-y^{2}-z^{2}\right)$, then

(a) $\mathrm{a}=\mathrm{b}=\mathrm{c}=0=\mathrm{x}=\mathrm{y}=\mathrm{z}$

(b) $x=y=z=0, a=b=c$

(c) $\mathrm{a}=\mathrm{b}=\mathrm{c}=0 ; \mathrm{x}=\mathrm{y}=\mathrm{z}$

(d) $x=y=z=a=b=c$

4. Number of positive integers $n$ for which $n^{2}+96$ is a perfect square is

(a) 8

(b) 12

(c) 4

(d) infinite

5. The curve $y=(\lambda+1) x^{2}+2$ intersects the curve $y=\lambda x+3$ is exactly one point, if $\lambda$ equals

(a) $\{-2,2\}$

(b) $\{1\}$

(c) $\{-2\}$

(d) $\{2\}$

6. A quadratic equation whose product of roots $x _{1}$ & $x _{2}$ is equal to 4 and satisfying the relation $\dfrac{\mathrm{x} _{1}}{\mathrm{x} _{1}-1}+\dfrac{\mathrm{x} _{2}}{\mathrm{x} _{2}-1}=2$ is

(a) $x^{2}-2 x+4=0$

(b) $x^{2}+2 x+4=0$

(c) $x^{2}+4 x+4=0$

(d) $x^{2}-4 x+4=0$

7. If $a, b, c, d \in R$, then the equation $\left(x^{2}+a x-3 b\right)\left(x^{2}-c x+3 b\right)\left(x^{2}-d x+2 b\right)=0$ has

(a) 6 real roots

(b) at least 2 real roots

(c) 4 real roots

(d) 3 real roots

8. Suppose $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ are defined as $\mathrm{P}=\mathrm{a}^{2} \mathrm{~b}+\mathrm{ab}^{2}-\mathrm{a}^{2} \mathrm{c}-\mathrm{ac}^{2}, \mathrm{Q}=\mathrm{b}^{2} \mathrm{c}+\mathrm{bc}^{2}-\mathrm{a}^{2} \mathrm{~b}-\mathrm{ab}^{2}$ & $\mathrm{R}=\mathrm{a}^{2} \mathrm{c}+\mathrm{ac}^{2}-\mathrm{b}^{2} \mathrm{c}-$ $\mathrm{bc}^{2}$, where $\mathrm{a}>\mathrm{b}>\mathrm{c}$ and the equation $\mathrm{Px}^{2}+\mathrm{Qx}+\mathrm{R}=0$ has equal roots, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in

(a) A.P

(b) G.P

(c) H.P

(d) AGP

9. If $a(p+q)^{2}+2 a b p q+c=0$ & $a(p+r)^{2}+2 a b p r+c=0(a \neq 0)$ then

(a) $\mathrm{qr}=\mathrm{p}^{2}$

(b) $\mathrm{qr}=\mathrm{p}^{2}+\dfrac{\mathrm{c}}{\mathrm{a}}$

(c) $\mathrm{qr}=-\mathrm{p}^{2}$

(d) none of these

10. $x^{2}-x y+y^{2}-4 x-4 y+16=0$ represents

(a) point

(b) a circle

(c) a pair of straight line

(d) none of these

11. If the roots of the equation $a x^{2}+b x+c=0$ are of the form $\dfrac{k+1}{k}$ & $\dfrac{k+2}{k+1}$, then $(a+b+c)^{2}$ is equal to

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

(b) $\sum \mathrm{a}^{2}$

(c) $b^{2}-4 a c$

(d) $b^{2}-2 a c$

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

$\mathrm{a} f(\mu)<0$ is the necessary and sufficient condition for a particular real number $\mu$ to the between the roots of a quadratic equation $f(\mathrm{x})=0$, where $f(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$. Again if $f\left(\mu _{1}\right) f\left(\mu _{2}\right)<0$, then exactly one of the roots will lie between $\mu _{1}$ & $\mu _{2}$

i. If $|\mathrm{b}|>|\mathrm{a}+\mathrm{c}|$, then

(a) One root of $f(\mathrm{x})=0$ is positive, the other is negative.

(b) Exactly one of the roots of $f(x)=0$ lies in $(-1,1)$.

(c) 1 lies between the roots of $f(x)=0$.

(d) Both the roots of $f(\mathrm{x})=0$ are less than 1

ii. If $\mathrm{a}(\mathrm{a}+\mathrm{b}+\mathrm{c})<0<(\mathrm{a}+\mathrm{b}+\mathrm{c}) \mathrm{c}$, then

(a) one root is less than 0 , the other is greater than 1 .

(b) Exactly one of the roots lies in $(0,1)$

(c) Both the roots lie in $(0,1)$

(d) At least one of the roots lies in $(0,1)$

iii. If $(a+b+c) c<0<a(a+b+c)$, then

(a) one root is less than 0 , the other is greater than 1

(b) one root lies in $(-\infty, 0)$ and the other in $(0,1)$

(c) both roots lie in $(0,1)$

(d) one root lies in $(0,1)$ and other in $(1, \infty)$

13. Match the following:-

14. If $\alpha, \beta$ are the roots of $375 x^{2}-25 x-2=0 & S _{n}=\alpha^{n}+\beta^{n}$, then the value of $\dfrac{1}{3\left(\lim _{n \rightarrow \infty} \sum\limits _{r=1}^{n} S _{r}\right)}$ is…..

15. If $x, y, z$ are distinct positive number such that $x+\dfrac{1}{y}=y+\dfrac{1}{z}=z+\dfrac{1}{x}$, then $x y z=$.