Algebra of Roots of Quadratic Equations

Q. The number of real solutions of the equation, $x^2-|x|-12=0$ is

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

Q.

If the sum of the roots of the equation $\lambda x^2+2x+3\lambda =0$ be equal to their product $\&\lambda \ne 0,$ then find the value of $\lambda .$

1. 6

2. -4

3. $-\frac{2}{3}$

4. 4

Q.

If $\alpha ,\beta$ be the roots of $x^2+px+q=0$ and $\alpha +h,\beta +h$ are the roots of $x^2+rx+s=0$, then

1. $p{r}^2=q{s}^2$

2. $2h=[\frac{p}{q}+\frac{r}{s}]$

3. $p^2-4q=r^2-4s$

4. $\frac{p}{r}=\frac{q}{s}$

Q. If roots of the given quadratic $2x^2-9x+7=0$ are $p,q.$ Then the equation whose roots are

1. $2x^2+9x+7=0$
2. $2x^2-43x+203=0$
3. $50x^2-45x+7=0$
4. $2x^2-x-3=0$

Q. If $\alpha$ is the greater root of $x^2-5x+4=0$ and $\alpha +m=2$, then the value of $m$ is

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

Q. If $a:b=1:5,$ then the roots of the equation $a{x}^2-bx+4a=0$ is/are

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

Q. If $r$ is the ratio of the roots of the equation $x^2-5x+8=0$, then $\frac{(r+1{)}^2}{r}$ is equal to

1. $\frac{64}{5}$
2. $-\frac{1}{40}$
3. $\frac{25}{8}$
4. $\frac{9}{8}$

Q. If the roots of the equation $4x^2+ax+3=0$ are in ratio of $1:2$, then value(s) of $a$ is/are

1. $-3\sqrt{6}$
2. $-\sqrt{6}$
3. $3\sqrt{6}$
4. $\sqrt{6}$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-5x+6=0$, then the value of ${\alpha }^3+{\beta }^3$ is

Q.

If α, β are the roots of $a{x}^2+bx+c$=0 then ${(a\alpha +b)}^{-3}$ +${(a\beta +b)}^{-3}$

1. $\frac{(b^3–3abc)}{a^3{c}^3}$

2. $a^3-2abc$

3. $b^3–3abc$

4. $\frac{(c^3–3abc)}{b^3{c}^3}$

Q.

For the equation $3x^2+px+3=0,p>0,$ if one of the roots is the square of the other, then find the value of $p.$

1. $\frac{1}{3}$

2. $1$

3. $3$

4. $\frac{2}{3}$

Q. The product of the roots of the equation $9x^2-18\left|x\right|+5=0$ is :

1. $\frac{25}{9}$
2. $\frac{5}{9}$
3. $\frac{25}{81}$
4. $\frac{5}{27}$

Q. If $a,b,c$ are distinct rational numbers, then the roots of the quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ are

1. imaginary
2. irrational
3. rational and equal
4. rational and distinct

Q. Let $\lambda \ne 0$ be in $R.$ If $\alpha$ and $\beta$ are the roots of the equation, $x^2-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3x^2-10x+27\lambda =0,$ then $\frac{\beta \gamma }{\lambda }$ is equal to

1. $18$
2. $36$
3. $9$
4. $27$

Q. If $m,n$ are the roots of the equation $x^2+7x-8=0,$ then the equation equation whose roots are $5m-11,5n-11$ is

1. $x^2-57x+306=0$
2. $x^2+306x-57=0$
3. $x^2+57x+306=0$
4. $x^2+57x-306=0$

Q.

If the sum of the roots of the equation $\lambda x^2+2x+3\lambda =0$ be equal to their product $\&\lambda \ne 0,$ then find the value of $\lambda .$

1. -4

2. 4

3. $-\frac{2}{3}$

4. 6

Q. Let $x_1$ and $x_2$ be the roots of the equation $2x^2+6x+b=0.$ If $\frac{x_1}{x_2}+\frac{x_2}{x_1}=k$ and $b<0,$ then the minimum integral value of $|k|$ is

Q. If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0,a\ne 0$ and $S_n={\alpha }^n+{\beta }^n,$ then

1. $S_5=-\frac{b}{a^5}(b^2-2ac{)}^2-\frac{(b^2-ac)bc}{a^4}$
2. $S_5=-\frac{b}{a^5}(b^2-2ac{)}^2+\frac{(b^2-ac)bc}{a^4}$
3. $a{S}_{n+1}+b{S}_n+c{S}_{n-1}=0$
4. $a{S}_{n+1}+b{S}_n+c{S}_{n-1}=1$

Q. If $\alpha$ and $\beta$ are the roots of $x^2+2x+2=0,$ then the value of ${\left(\frac{1}{\alpha +2}\right)}^3+{\left(\frac{1}{\beta +2}\right)}^3$ is

1. $-\frac{1}{2}$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $-\frac{1}{4}$

Q.

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x^2-1=0,$ then the value of $\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }=$

1. $-6$

2. $-7$

3. $-5$

4. $-2$

Q.

If $\alpha ,\beta$ are roots of $x^2-3ax+a^2=0$ such that $a^2+b^2=1.75,$ then possible values of $a$ are

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

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

3. -1 , - $\frac{1}{2}$

4. 1, $\frac{1}{2}$

Q. If $\alpha$ and $\beta$ are the roots of $x^2+2x+2=0,$ then the value of ${\left(\frac{1}{\alpha +2}\right)}^3+{\left(\frac{1}{\beta +2}\right)}^3$ is

1. $-\frac{1}{2}$
2. $\frac{1}{4}$
3. $-\frac{1}{4}$
4. $\frac{1}{2}$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-5x+6=0$, then the value of ${\alpha }^3+{\beta }^3$ is

Q. If the difference of the roots of $x^2-px+q=0$ is unity, then the value of $p^2-4q$ is

Q. If the roots of the equation $4x^2+ax+3=0$ are in ratio of $1:2$, then value(s) of $a$ is/are

1. $\sqrt{6}$
2. $-\sqrt{6}$
3. $-3\sqrt{6}$
4. $3\sqrt{6}$

Q. If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation,
$x^2+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

1. $2\sqrt{5}$
2. $20$
3. $2\sqrt{7}$
4. $4\sqrt{2}$

Q. If $\alpha ,\beta$ are the roots of $2x^2+6x+b=0$ and $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }<2$, then $b$ lies in the interval

1. $(-1,\infty )$
2. $b\in (-\infty ,0)\cup (\frac{9}{2},\infty )$
3. $(1,\infty )$
4. $(-\infty ,1]$

Q.

If α, β are the roots of the equation $a{x}^2+bx+c=0$,
then $\frac{\alpha }{a\beta +b}+\frac{\beta }{a\alpha +b}=$

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

2. $\frac{2}{a}$

3. $-\frac{2}{a}$

4. $\frac{2}{c}$

Q. For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then the roots are

1. $1$
2. $b$
3. $c$
4. $\frac{1}{c}$

Q. The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation $x^2+ax+b=0,{\alpha }^2-2$ is also a root of this equation, is

1. $2$
2. $8$
3. $4$
4. $6$