Quadratic Equations with Exactly One Root Common

Q. If the equation $2x^2+3x+5\lambda =0$ and $x^2+2x+3\lambda =0$ have a common root, then $\lambda =$

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

Q. If the quadratic equations $a{x}^2+2cx+b=0$ and $a{x}^2+2bx+c=0,(b\ne c)$ have a common root, then $a+4b+4c$ equals to

Q.

If the two equations $x^2-cx+d=0$ and $x^2-ax+b=0$ have one common root and the second has equal roots, then $2(b+d)=$)

1. a+c

2. ac

3. -ac

4. 0

Q. If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is
$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

1. $[\frac{-10}{3},-3)$
2. $(3,\frac{10}{3}]$
3. $[\frac{-11}{3},\frac{-10}{3})$
4. $(\frac{-10}{3},-3]$

Q.

If the equations $x^2-x-12=0$ and $k{x}^2+10x+3=0$ may have one common root, then $k$ is:

1. $3$ or $-\frac{43}{16}$

2. $-3$ or $-\frac{43}{16}$

3. $3$ or $\frac{13}{16}$

4. None of these

Q. If $x^2-11x+a$ and $x^2-14x+2a$ have common factor, then value(s) of $a$ is/are

1. $a=0$
2. $a=16$
3. $a=24$
4. $a=4$

Q. If equations $x^6+a{x}^4-4x=0$ and $x^7+a{x}^5-1=0$ have a common root, then which of the following statements is (are) CORRECT ?

1. Sum of all possible values of $a$ is $\frac{-1}{2}.$
2. Sum of all possible common roots is $0.$
3. Sum of all possible common roots is $1.$
4. Sum of all possible values of $a$ is $0.$

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. None of the above

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

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

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

Q. If the equations $x^2+b^2=1-2bx$ and $x^2+a^2=1-2ax$ have one and only one common root, then the value of $|a-b|$ is

1. $0$
2. $2$
3. None of these
4. $1$

Q.

Two distinct polynomial f(x) and g(x) are defined as follows:

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$

If the equation f (x) = 0 and g(x) = 0 have a common root, then the sum of the roots of the equation f (x) + g(x) = 0 is

1. 1

2. 0

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

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

Q.

If $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have a common root, then $a$ is equal to:

1. $-2$

2. $-1$

3. $2$

4. $1$

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. $9$

2. $6$

3. $12$

4. $3$

Q. If two quadratic equations $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ have exactly one root $\alpha$ in common, then select the correct statement.

1. $\frac{{\alpha }^2}{br-qc}=\frac{\alpha }{ar-pc}=\frac{1}{aq-pb}$
2. $\frac{{\alpha }^2}{br-qc}=\frac{\alpha }{ar-pc}=\frac{-1}{aq-pb}$
3. $\frac{-{\alpha }^2}{br-qc}=\frac{\alpha }{ar-pc}=\frac{1}{aq-pb}$
   
4. $\frac{{\alpha }^2}{br-qc}=\frac{-\alpha }{ar-pc}=\frac{1}{aq-pb}$

Q.

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

Q. If every pair among the equations $x^2+ax+bc=0,x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then which of the following is/are correct?

1. Product of the common roots is $abc$
2. Product of the common roots is $-abc$
3. Sum of the common roots is $\frac{1}{2}(a+b+c)$
4. Sum of the common roots is $-\frac{1}{2}(a+b+c)$

Q.

The factors when $x^3-9x^2+24x-20$ is factorised using Rational Root Theorem, are

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

Q.

If the equations $x^2+bx+c=0$ and $x^2+cx+b=0(b\ne c)$ have a common root, then :

1. None

2. $b+c+1=0$

3. $b+c=0$

4. $b+c=1$

Q. If the two equations $x^2-11x+a=0$ and $x^2-14x+2a=0$ have a common root and $a\ne 0,$ then the value of the common root is

Q. Find the value of $m$ such that the quadratic equations $3x^2-2mx-4=0$ and $x(x-4m)+2=0$ have a common root.

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

Q.

Let $p\left(x\right)=x^2–5x+a$ and $q\left(x\right)=x^2–3x+b$, where a and b are positive integers. Suppose hof(p(x), q(x)) = x – 1 and k(x) = 1cm (p(x), q(x)). If the coefficient of the highest degree term of k(x) is 1, the sum of the roots of (x – 1) + k(x) is.

1. 5

2. 4

3. 7

4. 6

Q. If $a,b,c$ are rational number $(a>b>c>0)$ and quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval $(-1,0)$, then which of the following statement(s) is/are correct?

1. $a{x}^2+2bx+c=0$ has both roots negative
2. Both roots are rational
3. $c{x}^2+2bx+a=0$ has both roots negative
4. $a+c<2b$

Q.

If the equation $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$ have a non - zero common root, then $\lambda$ is equal to:

1. None

2. $-1$

3. $1$

4. $3$

Q. If the equations $x^2+ax+12=0,$ $x^2+bx+15=0$ and $x^2+(a+b)x+36=0$ have a common positive integral root, then the value of $a-b$ is

1. $-15$
2. $1$
3. $15$
4. $-1$

Q.

If $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root and $a,b,c$ are non-zero real numbers, then$\frac{a^3+b^3+c^3}{abc}$ is equal to:

1. $1$

2. None

3. $2$

4. $3$

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

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

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 equation $x^2+px+q=0$ and $x^2+qx+p=0,$ have a common root, where $p\ne q,$ then find the value of $p+q+1?$

1. $-1$

2. $0$

3. $1$

4. $2$

Q.

Given two quadratic equations $x^2-1=-b^2-2bx\&x^2-1=-a^2-2ax$ have exactly one root in common then select the correct statements.

1. $a+b+2=0$

2. $a=b^2+2$

3. $a-b=0$

4. $a=b+2$

Q. The value of $a+b$ if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root is

Q.

If the equation $x^2+px+q=0$ and $x^2+qx+p=0$ have exactly one root in common, then equation with other roots is

1. $x^2+x-pq=0$

2. $x^2-x-pq=0$

3. $x^2-x+pq=0$

4. $x^2+x+pq=0$