Let $f(x)=a{x}^2+bx+c,a,b,c\in R,a\ne 0$ and $\alpha ,\beta (\alpha <\beta )$ be the roots of $f(x)=0$. Let $k_1,k_2$ 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 $\mathrm{a}>0,\mathrm{a}\ne 1$ 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 $f(x)=0$ 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 $x^2-2mx+m^2-1=0$ 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 roots of the equation $4x^2-2x+a=0$ lie in the interval $(-1,1)$ is.

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

(b). $(-\mathrm{\infty },\frac{1}{4}]$

(c). $(-2,\frac{1}{4}]$

(d). none of these

3. The all possible values of a for which one root of the equation $(a-5)x^2-2ax+a-4=0$ 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). $\mathrm{c}(\mathrm{a}+\mathrm{b}-\mathrm{c})>0$

Practice questions

1. The values of $a$ for which $2x^2-2(2a+1)x+a(a+1)=0$ 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). $a\ge 0$

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

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

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

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

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

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

3. If $a,b,c,x,y,z,\in R$ be such that $(a+b+c{)}^2=3(ab+bc+ca-x^2-y^2-z^2)$, 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). $\mathrm{x}=\mathrm{y}=\mathrm{z}=\mathrm{a}=\mathrm{b}=\mathrm{c}$

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 $x_1\mathrm{\&}{x}_2$ is equal to 4 and satisfying the relation $\frac{{\mathrm{x}}_1}{{\mathrm{x}}_1-1}+\frac{{\mathrm{x}}_2}{{\mathrm{x}}_2-1}=2$ is

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

(b). $x^2+2x+4=0$

(c). $x^2+4x+4=0$

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

7. If $a,b,c,d\in R$, then the equation $(x^2+ax-3b)(x^2-cx+3b)(x^2-dx+2b)=0$ 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 $\mathrm{P}={\mathrm{a}}^2\mathrm{b}+{\mathrm{ab}}^2-{\mathrm{a}}^2\mathrm{c}-{\mathrm{ac}}^2,\mathrm{Q}={\mathrm{b}}^2\mathrm{c}+{\mathrm{bc}}^2-{\mathrm{a}}^2\mathrm{b}-\mathrm{a}{\mathrm{b}}^2\mathrm{\&}\mathrm{R}={\mathrm{a}}^2\mathrm{c}+{\mathrm{ac}}^2-{\mathrm{b}}^2\mathrm{c}-$ ${\mathrm{bc}}^2$, 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 $a(p+q{)}^2+2abpq+c=0\mathrm{\&}a(p+r{)}^2+2abpr+c=0(a\ne 0)$ then

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

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

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

(d). none of these

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

(a). point

(b). a circle

(c). a pair of straight line

(d). none of these

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

(a). $2b^2-ac$

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

(c). $b^2-4ac$

(d). $b^2-2ac$

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 ${\mu }_1\mathrm{\&}{\mu }_2$

1. If $|b|>|a+c|$, 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

2. 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 $(0,1)$

3. 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 $(-\mathrm{\infty },0)$ and the other in $(0,1)$

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

(d). one root lies in $(0,1)$ and other in $(1,\mathrm{\infty })$

13. Match the following:-

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

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