Algebra of Roots of Quadratic Equations

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. $\frac{2}{3}$

2. $3$

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

4. $1$

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

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

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

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. The sum of the roots of the equation, $x+1-2{\log }_2(3+2^x)+2{\log }_4(10-2^{-x})=0,$ is

1. ${\log }_{2}13$
2. ${\log }_{2}14$
3. ${\log }_{2}11$
4. ${\log }_{2}12$

Q.
If $m$ is chosen in the quadratic equation $(m^2+1)x^2–3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is -

1. $8\sqrt{3}$
2. $8\sqrt{5}$
3. $4\sqrt{5}$
4. $4\sqrt{3}$

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

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

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

Q.

Express $r$ in terms of $p$ and $q$, if the roots of the equation $x^2+px+q=0$ are $\alpha ,\beta$ and the roots of the equation
$x^2-rx+s=0$ are ${\alpha }^4,{\beta }^4$

1. $2q^2$ - $(p^2-2q{)}^2$

2. $p-2q^2-2q$

3. $2q-p-2q^2$

4. $(p^2-2q{)}^2$ - $2q^2$

Q.

If $x^2+px+q=0$ is the quadratic equation whose roots are
$a–2$ and $b–2$ where $a$ and $b$ are the roots of $x^2-3x+1=0$, then

1. None of these

2. $p=-1,q=1$

3. $p=1,q=-5$

4. $p=1,q=5$

Q. Let $a,b\in R,a\ne 0,$ such that the equation, $a{x}^2-2bx+5=0$ has a repeated root $\alpha ,$ which is also a root of the equation $x^2-2bx-10=0.$ If $\beta$ is the other root of this equation, then ${\alpha }^2+{\beta }^2$ is equal to:

1. $28$
2. $26$
3. $25$
4. $24$

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. $5$
4. $-1$

Q. Let $\alpha ,\beta$ be two roots of the equation $x^2+(20{)}^{\frac{1}{4}}x+(5{)}^{\frac{1}{2}}=0$. Then ${\alpha }^8+{\beta }^8$ is equal to

1. 10
2. 160
3. 100
4. 50

Q. The set of all values of $k>-1,$ for which the equation $(3x^2+4x+3{)}^2-(k+1\left)\right(3x^2+4x+3\left)\right(3x^2+4x+2)+k(3x^2+4x+2{)}^2=0$ has real roots, is

1. $(\frac{1}{2},\frac{3}{1}]-\left\{1\right\}$
2. $[-\frac{1}{2},1)$
3. $(1,\frac{5}{2}]$
4. $[2,3)$

Q. If the difference between the roots of the equation $x^2+ax+1=0$ is less than $\sqrt{5}$, then choose the set of possible values of $a$ is:

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

Q. The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

1. $2$
2. $5$
3. $6$
4. $4$

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2x^2+2qx+1=0$, then $(\alpha -\frac{1}{\alpha })(\beta -\frac{1}{\beta })(\alpha +\frac{1}{\beta })(\beta +\frac{1}{\alpha })$ is equal to

1. $\frac{9}{4}(9-p^2)$
2. $\frac{9}{4}(9-q^2)$
3. $\frac{9}{4}(9+q^2)$
4. $\frac{9}{4}(9+p^2)$

Q. If the difference of the roots of the equation $(k-2)x^2-(k-4)x-2=0,k\ne 2$ is $3$, then the sum of all the values of $k$ is

1. $0$
2. $-\frac{3}{2}$
3. $\frac{7}{2}$
4. $\frac{9}{2}$

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. $3$

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

3. $1$

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

Q. The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has both the roots at infinity is

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

Q.

If $\alpha ,\beta$ are the real and distinct roots of $x^2+px+q=0$ and ${\alpha }^4,{\beta }^4$ are the roots of $x^2-rx+5=0$, then the equation $x^2-4qx+2a^2-r=0$ always has:

1. one positive root and one negative root

2. two negative roots

3. two real roots

4. two positive roots

Q. The sum of the roots of the equation, $x+1-2{\log }_2(3+2^x)+2{\log }_4(10-2^{-x})=0,$ is

1. ${\log }_{2}14$
2. ${\log }_{2}13$
3. ${\log }_{2}11$
4. ${\log }_{2}12$

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

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-px+q=0$ then the quadratic equation whose roots are
$(\alpha -\beta {)}^2(\alpha +\beta \left)\right({\alpha }^2+{\beta }^2+\alpha \beta )$ and $({\alpha }^3{\beta }^2+{\alpha }^2{\beta }^3)$ is where $S=p[p^4-5p^{2}q+5q^2],P=p^2{q}^2(p^4-5p^{2}q+4q^2)$

1. None of these
2. $x^2-Sx+P=0$
3. $x^2+Sx-P=0$
4. $x^2+Sx+P=0$

Q. Let $\alpha ,\beta$ be the roots of $x^2-x-1=0$ $(\alpha >\beta )$ and $m,n\in Z,k\in W$ such that $a_k=m{\alpha }^k+n{\beta }^k.$ If $a_4=35,$ then the value of $3m+2n$ is equal to

1. $85$
2. $40$
3. $25$
4. $75$

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

Q.

If α, β are the roots of the equation $x^2$ + 15x + 17 = 0, find ${(\alpha -\beta )}^2$.

1. 147

2. 138

3. 157

4. 128

Q.

The value of $a$ for which one root of the quadratic equation $(a^2-5a+3)x^2+(3a-1)x+2=0$ is twice as large as the other, is

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

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

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

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