Algebra of Roots of Quadratic Equations

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

2. -4

3. 6

4. 4

Q.

Let α & β be the roots of \({x^2}\) – 6x – 2= 0 which α >β . if $a_n$= ${\alpha }^n$ –${\beta }^n$ for n$\ge$1, then the

value of $\frac{a_{10}-2a_8}{2a_9}$ is

1. 4

2. 3

3. 1

4. 2

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

Q. If $\alpha ,\beta$ are the roots of the equation $2x^2-35x+2=0,$ then the value of $\sqrt{(2\alpha -35{)}^3(2\beta -35{)}^3}$ is

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

Q. The set of values of $a$ for which the sum and product of roots of the equation $x^2-(a^2-5a+5)x+(2a^2-3a-4)=0$ is less than $1$, is

1. $(-1,\frac{5}{2})$
2. $(1,4)$
3. $(1,\frac{5}{2})$
4. $(-1,4)$

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

Q.

If the roots of the equation $l{x}^2+nx+n$ = 0, l ¹ 0 are in the ratio p : q, then ${\left(p/q\right)}^{1/2}+{\left(q/p\right)}^{1/2}+{\left(n/l\right)}^{1/2}$

1. $\frac{n}{l}$​

2. 2${\left(\frac{n}{l}\right)}^{1/2}$

3. 0

4. None of these

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 $\alpha ,\beta ,\gamma$ 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. $-5$

3. $-2$

4. $-7$

Q.

If the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocals, then $\frac{b^2}{ac}+\frac{bc}{a^2}=$

1. 2

2. 1

3. 0

4. 3

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

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

Q.

α, β are the roots of the equation $x^2$ + 4x + 5. Find ${\alpha }^3$ + ${\beta }^3$

1. -4

2. 4

3. -6

4. 6

Q.

If \(\alpha~, \beta\) are the roots of the equation \(x^2 - px+q=0~\text{and}~\alpha\) > \(0, \beta\) >\(0\) , then the value of \(\alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}~\text{is}~\left ( p+6 \sqrt{q}+4q^{\frac{1}{4}} \sqrt{p+2\sqrt{q}} \right )^k\) , where k is equal to

Q. Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{3a_9}$ is:

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

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

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

Q.

$\alpha ,\beta$ are the roots of the equation $x^2+14x+10=0.$
Find the value of (${\alpha }^2$ + ${\beta }^2$).

1. $196$

2. $206$

3. $186$

4. $176$

Q.

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

1. 147

2. 157

3. 138

4. 128

Q.

If $x,y,z$ are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy=$

1. Non-negative

2. Non-positive

3. Zero

4. positive only

Q.

If a root of the equation $a{x}^2+bx+c=0$ be reciprocal of the equation then $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$, then

1. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

2. $(c{c}^{\prime }+a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

3. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

4. $(b{b}^{\prime }-a{a}^{\prime }{)}^2=(c{a}^{\prime }-b{c}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

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 distinct
4. rational and equal

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

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$ and $A_n={\alpha }^n+{\beta }^n$, then which of the following is true?

1. $A_{n+1}=a{A}_n-b{A}_{n-1}$
2. $A_{n+1}=b{A}_n-a{A}_{n-1}$
3. $A_{n+1}=b{A}_n+b{A}_{n-1}$
4. $A_{n+1}=a{A}_n+b{A}_{n-1}$

Q.

If α, β are the roots of the quadratic equation $x^2$ – (a – 2)x –(a+1) =0, where 'a' is a variable then the least value of ${\alpha }^2$+${\beta }^2$

1. 5

2. 9

3. 3

4. 7

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. two real roots

2. one positive root and one negative root

3. two positive roots

4. two negative roots

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

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

Q.

The sum of roots of the polynomial equation $(x-1)(x-2)(x-3)=2(x-2)(x-3)$ is

1. $5$

2. $3$

3. $1$

4. $8$

Q.

If $\alpha ,\beta$ are the roots of the equation $8x^2-3x+27=0$, then the value of ${\left(\frac{{\alpha }^2}{\beta }\right)}^{\frac{1}{3}}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{\frac{1}{3}}$ is

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

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

3. $\frac{7}{2}$

4. $4$