Introduction to Quadratic Equations and Polynomials

Q.

How many forms of quadratic equations are there?

Q.

What type of polynomial is $x^2-2x+1$?

Q. If one of the roots of the equation $x^2+rx-s=0$ is the square of other, then $r^3+s^2+3sr-s=$

1. $0$
2. $s$
3. $1$
4. $r$

Q.

Define constant polynomial and give its degree.

Q.

How to convert quadratic equation from standard form to vertex form ?

Q.

The maximum value of the function $x^3+x^2+x-4$ is

1. $127$

2. $4$

3. Does not have a maximum value

4. None of the above

Q. If the ratio of the roots of the equation $x^2-px+q=0,$ is $a:b,$ then the value of $p^{2}ab=$

1. $qab$
2. $q(a^2+b^2)$
3. $q(a+b)$
4. $q(a+b{)}^2$

Q.

Which of the following is equal to $x$?

1. $\sqrt[12]{(x^4{)}^{\frac{1}{3}}}$

2. $(\sqrt{x^3}{)}^{\frac{2}{3}}$

3. $x^{\frac{12}{7}}-x^{\frac{5}{7}}$

4. $x^{\frac{12}{7}}\times x^{\frac{7}{12}}$

Q. $\alpha ,\beta ,\gamma ,\delta$ are the roots of $a{x}^4+b{x}^3+c{x}^2+dx+e=0,a\ne 0$.
Choose the correct pair

1. $\alpha \beta \gamma +\alpha \beta \delta +\beta \gamma \delta +\alpha \gamma \delta$
2. $\alpha \beta \gamma \delta$
3. $-\frac{b}{a}$
4. $\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta$
5. $\frac{c}{a}$
6. $-\frac{d}{a}$
7. $\alpha +\beta +\gamma +\delta$
8. $\frac{e}{a}$

Q.

The maximum value of $\frac{x^2-x+1}{x^2+x+1}$ for real values of $x$ is

1. $\frac{1}{2}$

2. $1$

3. $2$

4. $3$

Q.

The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is $2$. Then, the roots of the equation are

1. $-3,7,9$

2. $-3,–7,–9$

3. $-3,–5,7$

4. $-3,-7,9$

Q.

If $\alpha$ and $\beta$ are the roots of the equation $l{x}^2+mx+n=0$, then the equation, whose roots are ${\alpha }^3\beta$ and $\alpha {\beta }^3$, is

1. $l^4{x}^2–nl(m^2–2nl)x+n^4=0$

2. $l^4{x}^2+nl(m^2–2nl)x+n^4=0$

3. $l^4{x}^2+nl(m^2–2nl)x–n^4=0$

4. $l^4{x}^2-nl(m^2+2nl)x+n^4=0$

Q. $x^2+\frac{1}{x}$ is a polynomial with degree:

1. $-1$
2. None of these
3. $2$
4. $1$

Q. If $f\left(x\right)$ is a polynomial of degree $n\ge 1$ and $a$ is any real number, then, $(x-a)$ is a factor of $f\left(x\right)$, if $f\left(a\right)=0$.

1. True
2. False

Q. $\text{The value of}a\text{for which}$ $(a^2-1)x^2-(a-1)x+a^2-4a+3=0$ $\text{is an identity in}x$

Q. If $(P^2-1)x^2+(P-1)x+(P^2-4P+3)=0$ is an identity in $x$, then the value of $P$ is

1. $2$
2. $0$
3. $1$
4. $-1$

Q. Which of the following equation is an identity?

1. $x^2-4x+4=0$
2. $2x-1=0$
3. ${(x-2)}^2=x^2-4x+4$
4. ${(x+3)}^2=x^2+5x+8$

Q. The equation $a{x}^2+bx+c=0$ will be an identity iff

1. $a=b=c=0$
2. $a=b=c\ne 0$
3. $a\ne b\ne c=0$

Q. Which of the following are polynomials?

1. $2x^3+2x+4x^{\frac{3}{2}}$
2. $4x^2+6x+4x^3+2x^2$
3. $3x^2+2x+4x^9$
4. $3x^2+2x+4x^{-9}$

Q. A polynomial expression in $x$ can be of the form:
$y=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$where, $n\in R,a_i$ are constants.

1. False
2. True

Q.

The degree of the polynomial $(4x^3+2x+3)(3x^3+5x^4+1)$ is .

Q. $\text{The degree of the polynomial}3x^2+5x-7x-7x^6+3\text{is}$

1. $0$
2. $6$
3. $4$
4. $2$

Q.

If $2$ is the root of the quadratic equation $x^2+p^{2}x+6=0$, then the value of $(pi{)}^2$ (where $i=\sqrt{-1}$) is

__

Q. The degree of the polynomial $x^2-6x+2x^8-12$ is

1. $2$
2. $4$
3. $8$
4. $1$

Q. If the degree of the polynomial $3x^{a-2}+2x^4+3x^3+x^2+10$ is $7$ then the value of $a$ is

1. $5$
2. $3$
3. $9$
4. $0$

Q. Find the value of $m$ such that roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are positive

1. $[9,\infty )$
2. $[1,9]$
3. $(3,9)$
4. $(-\infty ,9]$

Q. $x^2+\frac{1}{x}$ is a polynomial with degree:

1. $2$
2. None of these
3. $-1$
4. $1$

Q. For $a\ne 0$,
$f\left(x\right)=a{x}^2+bx+c$ is a quadratic expression whereas
$a{x}^2+bx+c=0$ is a quadratic equation.

1. False
2. True

Q. Which of the following is/are identities?

1. $6x+1=7$
2. $(x+3{)}^2=x^2+6x+9$
3. $(x+3)(x-3)=x^2-9$
4. $x^2-3x+2=0$

Q. If $a_1,a_2,a_3,\cdots ,a_n;(n\ge 2)$ are real and $(n-1)a_1^2-2n{a}_2<0,$ then atleast roots of the equation $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_n=0$, are imaginary.

1. $2$
2. $4$
3. $5$
4. $1$