Problem solving skills.

• If one root of $a x^{2}+b x+c=0$ is $n$ times the other, then $(n+1)^{2} a c=n b^{2}$

• If one root of $a x^{2}+b x+c=0$ is square of the other, then $\left(a^{2} c\right)^{1 / 3}+\left(a c^{2}\right)^{1 / 3}+b=0$

• If $\alpha, \beta$ are roots of $a x^{2}+b x+c=0$ then

$\quad$ (i) $-\alpha,-\beta$ are roots of $\mathrm{ax}^{2}-\mathrm{bx}+\mathrm{c}=0$

$\quad$ (ii) $\frac{1}{\alpha}, \frac{1}{\beta}$ are roots of $\mathrm{cx}^{2}+\mathrm{bx}+\mathrm{a}=0 ; \mathrm{ac} \neq 0$

$\quad$ (iii) $\mathrm{k} \alpha, \mathrm{k} \beta$ are roots of $\mathrm{ax}^{2}+\mathrm{kbx}+\mathrm{k}^{2} \mathrm{c}=0$

$\quad$ (iv) $\alpha^{2}, \beta^{2}$ are roots of $a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0$

• If the sum of the coefficient of $f(\mathrm{x})=0$ is 0 , then 1 is always a root of $f(\mathrm{x})=0$. Also $\mathrm{x}-1$ is a facter of $f(\mathrm{x})$.

In particular, for $a x^{2}+b x+c=0$ if

$a+b+c=0$, then 1 is always a root and the other $\operatorname{root}=\frac{c}{a}\left(\because\right.$ product of roots $\left.=\frac{c}{a}\right)$.

• $f(\mathrm{x})=\left(\mathrm{x}-\mathrm{a} _{1}\right)^{2}+\left(\mathrm{x}-\mathrm{a} _{2}\right)^{2}+\ldots \ldots \ldots \ldots \ldots \ldots . .+\left(\mathrm{x}-\mathrm{a} _{\mathrm{n}}\right)^{2}$, where $\mathrm{a} _{\mathrm{i}} \in \mathrm{R} \forall \mathrm{i}$.

$f(x)$ assumes its least value when $x=\frac{a _{1}+a _{2}+\ldots \ldots+a _{n}}{n}$

• While solving an equation, if you have to square, then additional roots will occur as the degree of the equation will change. In such cases, you have to check whether the roots satisfy the original equation or not.

Solved examples

1. If $\alpha, \beta$ are roots of the equation $\mathrm{x}^{2}-2 \mathrm{x}+3=0$

Then the equation whose roots are

$\alpha^{3}-3 \alpha^{2}+5 \alpha-2$ and $\beta^{3}-\beta^{2}+\beta+5$ is

(a) $x^{2}+3 x+2=0$

(b) $x^{2}-3 x-2=0$

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

(d) None

2. If $\alpha, \beta$ are roots of the equatio $\mathrm{x}^{2}+\mathrm{px}-\frac{1}{2 \mathrm{p}^{2}}=0, \forall \mathrm{p} \in \mathrm{R}-\{0\}$, then the minimum value of $\alpha^{4}+\beta^{4}$ is

(a) $2 \sqrt{2}$

(b) $2-\sqrt{2}$

(c) $2$

(d) $2+\sqrt{2}$

3. Let $\mathrm{p}(\mathrm{x})$ be a polynomial of least possible degree with rational coefficients, having $7^{\frac{1}{3}}+49^{\frac{1}{3}}$ as one of its roots, then the product of all roots of $p(x)=0$ is

(a) 56

(b) 63

(c) 7

(d) 49

4. If $\alpha, \beta, \gamma, \delta$ are roots of $x^{4}+4 x^{3}-6 x^{2}+7 x-9=0$, then the value of $\left(1+\alpha^{2}\right)\left(1+\beta^{2}\right)\left(1+\gamma^{2}\right)\left(1+\delta^{2}\right)$ is

(a) 9

(b) 11

(c) 13

(d) 5

5. If $\alpha, \beta, \gamma$, are roots of $8 x^{3}+1001 x+2008=0$, then the value of $(\alpha+\beta)^{3}+(\beta+\gamma)^{3}+(\gamma+\alpha)^{3}$ is

(a) 251

(b) 751

(c) 735

(d) 753

6. Total number of integral values of ’ $n$ ’ so that the equation $x^{2}+2 x-n=0(n \in N)$ and $n \in[5,100]$ has integral roots is

(a) 2

(b) 4

(c) 6

(d) 8 and $\mathrm{n} \in[5,100]$

7. If the equation $p(q-r) x^{2}+q(r-p) x+r(p-q)=0$ has equal roots, then $\frac{2}{\mathrm{q}}$ is equal to

(a) $\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{r}}$

(b) $\mathrm{p}+\mathrm{r}$

(c) $\frac{1}{\mathrm{p}}+\mathrm{r}$

(d) $\mathrm{p}+\frac{1}{\mathrm{r}}$

Practice questions

1. The largest interval for which $\mathrm{x}^{12}-\mathrm{x}^{9}+\mathrm{x}^{4}-\mathrm{x}+1>0$ is

(a) $-4<x \leq 0$

(b) $0<x<1$

(c) $-100<\mathrm{x}<100$

(d) $-\infty<\mathrm{x}<\infty$

2. Read the following passage and answer the questions.

If a continuous function $f$ defined on the real line $\mathrm{R}$, assumes positive and negative values in $\mathrm{R}$, then the equation $f(\mathrm{x})=0$ has a root in $\mathrm{R}$, for example, if it is known that a continous function $f$ on $\mathrm{R}$ is positive at some point and its minimum value is negative, then the equations $f(x)=0$ has a root in $\mathrm{R}$. Consider $f(\mathrm{x})=\mathrm{ke}^{\mathrm{x}}-\mathrm{x}, \forall \mathrm{x} \in \mathrm{R}$ where $\mathrm{k} \in \mathrm{R}$ is a constant.

(i) The line $\mathrm{y}=\mathrm{x}$ meets $\mathrm{y}=\mathrm{ke}^{\mathrm{x}}$ for $\mathrm{k} \leq 0$ at

(a) no point

(b) one point

(c) two point

(d) more than two points

(ii) The value of $k$ for which $\mathrm{ke}^{\mathrm{x}}-\mathrm{x}=0$ has only one root is

(a) $\frac{1}{\mathrm{e}}$

(b) $e$

(c) $\log _{\mathrm{e}} 2$

(d) $1$

(iii) For $\mathrm{k}>0$, the set of all values of $\mathrm{k}$ for which $\mathrm{ke}^{\mathrm{x}-\mathrm{x}}=0$ has two distinct roots is

(a) $\left(0, \frac{1}{\mathrm{e}}\right)$

(b) $\left(\frac{1}{\mathrm{e}}, 1\right)$

(c) $\left(\frac{1}{\mathrm{e}}, \infty\right)$

(d) $(0, 1)$

3. If $\frac{\sqrt{26-15 \sqrt{3}}}{5 \sqrt{2}-\sqrt{(38+5 \sqrt{3})}}=\mathrm{a}^{2}$, then $\mathrm{a}$ is

(a) $\frac{1}{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\sqrt{3}$

(d) None of these

4. Solution of $2 \log _{x} a+\log _{a x} a+3 \log _{b} a=0$, where $a>0, b=a^{2} x$ is

(a) $\mathrm{a}^{-1 / 2}$

(b) $\mathrm{a}^{-4 / 3}$

(c) $\mathrm{a}^{1 / 2}$

(d) None of these

5. Solution of the system of equations

$x+\frac{3 x-y}{x^{2}+y^{2}}=3, \quad y-\frac{x+3 y}{x^{2}+y^{2}}=0$ is $……….$or $……..$

6. The number of ordered 4-tuple $(x, y, z, w)$ where $x, y, z, w \in[1,10]$, which satisfies the inequality $2^{\sin ^{2} x} 3^{\cos ^{2} y} 4^{\sin ^{2} z} 5^{\cos ^{2} w} \geq 120$ is

(a) 0

(b) 144

(c) 81

(d) infinite.

7. The number of solutions of the following inequality

$2^{\frac{1}{\sin ^{2} x _{2}}} \cdot 3^{\frac{1}{\sin ^{2} x _{3}}} \cdot 4^{\frac{1}{\sin ^{2} x _{4}} \cdot \ldots \ldots \ldots . . .} n^{\frac{1}{\sin ^{2} x _{n}}} \leq n$ ! where

$\mathrm{x} _{\mathrm{i}} \in(0,2 \pi)$ for $\mathrm{i}=1,2,3 \ldots \ldots \ldots \ldots \ldots . \mathrm{n}$ is

(a) $1$

(b) $2^{\mathrm{n}-1}$

(c) $\mathrm{n}^{\mathrm{n}}$

(d) infinite

8. The number of solutions of $|[x]-2 x|=4$ is

(a) infinite

(b) 4

(c) 3

(d) 2

9. How many roots does the equation $3^{|x|}|2-| x||=1$ possess?

(a) 1

(b) 2

(c) 3

(d) 4

10. Let $S$ be the set of values of ’ $a$ ’ for which 2 lie between the roots of the quadratic equation $x^{2}+(a+2) x-$ $(a+3)=0$, then $S$ is gives by

(a) $(-\infty,-5)$

(b) $(5, \infty)$

(c) $(-\infty,-5]$

(d) $[5, \infty)$

11. Match the following

For what values of $m$, the equation $2 x^{2}-2(m+1) x+m(m+1)=0$ has $(m \in R)$

12 The real roots of the equation

$ \sqrt{x+2 \sqrt{x+2 \sqrt{x+\ldots \ldots .+2 \sqrt{x+2 \sqrt{3 x}}}}}=x \text { is } $

(n radical signs)

(a) 0

(b) 3

(c) 1

(d) None of these

13. Solution of the equation $1+3^{x / 2}=2^{x}$ is……

14. The number of real solutions of the system of equations

$\mathrm{x}=\frac{2 \mathrm{z}^{2}}{1+\mathrm{z}^{2}}$, $\quad$ $y=\frac{2 x^{2}}{1+x^{2}}$, $\quad$ $\mathrm{z}=\frac{2 \mathrm{y}^{2}}{1+\mathrm{y}^{2}}$ is

(a) 1

(b) 2

(c) 3

(d) 4

15. If $a, b, c>0, a^2=b c$ and $a+b+c=a b c$, then the least value of $a^4+a^2+7$ must be equal to

(a) 19

(b) 20

(c) 21

(d) 18