Quadratic Equations with Exactly One Root Common

Q.

If $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root. And $a,b\ne 0,$ then find the value of $\frac{a^3+b^3+c^3}{abc}$

1. 2

2. 3

3. 4

4. 1

Q.

If $a{x}^2+bx+c=0andb{x}^2+cx+a=0$ have a common root and a, b, c are non-zero real numbers, then find the value of $\frac{a^3+b^3+c^3}{abc}$

1. 2

2. 3

3. 1

4. 4

Q.

The roots of the equation $\left(q-r\right)x^2+\left(r-p\right)x+\left(p-q\right)=0$ are

1. $\frac{\left[r-p\right]}{\left[q-r\right]},1$

2. $\frac{\left[p-q\right]}{\left[q-r\right]},1$

3. $\frac{\left[p-q\right]}{\left[q-r\right]},2$

4. $\frac{\left[q-r\right]}{\left[p-q\right]},2$

5. $\frac{\left[r-p\right]}{\left[p-q\right]},1$

Q.

If $x^2-hx-21=0,x^2-3hx+35=0(h>0)$ have a common root, then the value of h is equal to

1. 1

2. 4

3. 3

4. 2

Q. If $2x^2+5x+2b=0$ and $2x^3+7x^2+5x+1=0$ have atleast one common root for three values of $b$, then the sum of all three values of $b$ is

1. $0$
2. $\frac{1}{2}$
3. $\frac{3}{2}$
4. $\frac{5}{2}$

Q.

If the equations $2x^2+kx-5=0$ and $x^2-3x-4=0$ have a common root, then the value of k is

1. $-2$

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

3. $-3$

4. $\frac{27}{4}$

Q. If $2x^2+5x+2b=0$ and $2x^3+7x^2+5x+1=0$ have atleast one common root for three values of $b$, then the sum of all three values of $b$ is

1. $\frac{5}{2}$
2. $\frac{3}{2}$
3. $\frac{1}{2}$
4. $0$

Q. If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Q. If $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root, then

1. $a+b+1=0$
2. $a=b$
3. $a+b=1$
4. $a+b=0$

Q. If $(x+\alpha)$ is a common factor of $(x^2 +ax+b)$ and $(x^2 +cx+d)$ and $a:b:c:d=2:3:4:7, $ then the value of $\alpha$ is

Q. If the two polynomials $x^2-11x+a$ and $x^2-14x+2a$ have a common factor, then $a=$

1. $0,24$
2. $24$
3. $0$
4. $0,3$

Q.

If $a,b,c$ are in G.P, then the equation $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root if $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

1. G.P.

2. A.P.

3. H.P.

4. None of these

Q.

If the equations $2x^2+kx-5=0$ and $x^2-3x-4=0$ have a common root, then the value of k is

1. $\frac{27}{4}$

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

3. $-3$

4. $-2$

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

Q. Two distinct polynomials $f\left(x\right)$ and $g\left(x\right)$ are defined as follows:
$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a.$
If the equations $f\left(x\right)=0$ and $g\left(x\right)=0$ have a common root then the sum of the roots of the equation $f\left(x\right)+g\left(x\right)=0$ is

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

Q. The value of $a$ for which $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have exactly one common root is

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

Q.

If equation $a{x}^2+2cx+b=0$ and $a{x}^2+2bx+c=0$ have one root in common, then $a+4b+4c$ equals

1. $0$

2. $1$

3. $-1$

4. $-2$

Q. The sum of the values of '$m$' for which the equations $3x^2+4mx+2=0$ and $2x^2+3x-2=0$ may have a common root, is

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

Q. If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x–1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

1. $0$
2. $\alpha \gamma$
3. $\beta \gamma$
4. $\alpha \beta$

Q.

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a common root and $a,b,c\in N$ then minimum value of $a+b+c$ is equal to:

1. $6$

2. $9$

3. $12$

4. $3$

Q.

Let $f\left(x\right)=a{x}^2+bx+c,g\left(x\right)=a{x}^2+px+q$ where $a,b,c,q,p,\in R\&b\ne p.$ If their discriminants are equal and $f\left(x\right)=g\left(x\right)$ has a root $\alpha$ then

1. $\alpha$ will be $A.M$ of the roots of $g\left(x\right)=0$

2. $\alpha$ will be $A.M.$ of the roots of $f\left(x\right)=0andg\left(x\right)=0$

3. $\alpha$ will be $A.M$ of the roots of $f\left(x\right)=0$

4. None of the above

Q. The value(s) of $k$ for which the quadratic equations $(1-2k)x^2-6kx-1=0$ and $k{x}^2-x+1=0$ have at least one root in common, is (are)

1. $\left\{\frac{2}{9}\right\}$
2. $\{\frac{1}{2},\frac{2}{9}\}$
3. $\{\frac{1}{3},\frac{2}{9}\}$
4. $\left\{\frac{1}{2}\right\}$

Q. If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Q. If the equations $k(6x^2+3)+rx+(2x^2-1)=0$ and $6k(2x^2+1)+px+(4x^2-2)=0,k\ne 0$ have both roots common, then the value of $\frac{p}{r}$ is

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

Q. If equations $x^2+bx+c=0$ and $b{x}^2+cx+1=0$ have a common root then

1. $b^2+c^2+1=bc$
2. $b+c+1=0$
3. $(b-c{)}^2+(c-1{)}^2+(b-1{)}^2=0$
4. $b+c+1=-bc$

Q.

The value of $a$ for which the equations $x^2-3x+a=0$ and $x^2+ax-3=0$have a common root is

1. $2$

2. $-2$

3. $3$

4. $1$

Q.

Find the value of '$a$' so that $x^2-11x+a=0$ and $x^2-14x+2a=0$ have a common root.

1. $12$

2. $0$

3. $48$

4. $24$

Q. If every pair of equations $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $x^2+cx+ab=0$ has a common root, then product of these common roots is

1. $\frac{ab}{c}$
2. $\frac{c}{ab}$
3. $abc$
4. $a^2{b}^2{c}^2$

Q. If $x^2+bx-a=0$ and $x^2-ax+b=0$ have only one common root, then

1. $a+b=0$
2. $a-b=1$
3. $a=b$
4. $a+b=1$

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