Common Roots

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

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

Q. If every pair of the equation x^2+px+qr=0, x^2+qx+rp=0, x^2+rx+pq=0 have a non-zero common root, then the sum of the three common roots is (1) -(p+q+r)/2 (2) (-p+q+r)/2 (3)-(p-q+r) (4)-p+q+r

Q.

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

1. 3

2. 4

3. 1

4. 2

Q.

If $-4$ is a root of the quadratic equation$x^2+px–4=0$ and the quadratic equation $x^2+px+k=0$ has equal roots. The value of $k$ is

1. $\frac{9}{4}$

2. $\frac{4}{9}$

3. $3$

4. None of these

Q. 8 Complex Numbers { The complex number z_1, z_2 satisfies the { equation z+1+8i=\vert z\vert(1+i), where i=\sqrt{-1 , { then the equation whose roots are \vert z_1\vert and \vert z_2\vert{ (1) \vert z\vert^2-18\vert z\vert+65=0{ (2) \vert z\vert^2-7\vert z\vert+12=0{ (3) \vert z\vert^2-1=0{ (4) \vert z\vert^2-17\vert z\vert+60=0

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

Q.

If $S_n$ denotes the sum of first 'n' terms of an A.P. and$\frac{S_{3n}-S_{n-1}}{S_{2n}-S_{2n-1}}=31$ then the value of n is

1. 15

2. 16

3. 19

4. 21

Q.

If every pair of the equations $x^2+px+qr=0,x^2+qx+rp=0,x^2+rx+pq=0$ have a common root, then the sum of three common roots is

1. $\frac{-(p+q+r)}{2}$

2. $\frac{-p+q+r}{2}$

3. -(p+q+r)

4. -p+q+r

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

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. 0

2. a+c

3. ac

4. -ac

Q. The value(s) of k for which equations $3x^2+4kx+2=0and2x^2+3x-2=0$ will have a common root can be:

1. $\frac{-11}{8}$
2. $\frac{11}{8}$
3. $\frac{7}{4}$
4. $\frac{-7}{4}$

Q. For the equation Ix^2I + IxI - 6 = 0, the roots are 1. Real and equal 2. Real with sum 0 3. Real with sum 1 4. Real with product 0

Q. how come. X square + 1 = 0 has no real roots

Q. Number of values of $k$ so that the equations $x^2+kx+(k+2)=0\text{and}{x}^2+(1-k)x+3-k=0$ have exactly one common root, is -

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

Q. Maximum number of real roots does the equation ax¹⁰-bx⁸+cx²+e=0 , a, b, c, e>0 can have.

Q. if alpha, beta, gamma are thevroots of x^3+8=0 then the equation whose root are aplha^2, beta^2, gamma^2

Q.

Find the root by factorization method
$2x^2+x-6=0$

Q. If one of the roots of the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ is coincident, then the numerical value of $(a+b)$ is

1. \N
2. -1
3. 2
4. 5

Q. For two quadratic equations to have both roots common, the ratio of the corresponding coefficients must be equal.

1. True
2. False

Q. determine the set of values of p for which the given quadratic equation has real roots (i) 5x^2-2px+3=0

Q. If equation bx^2+2ax+c=0 and bx^2+2cx+a=0(c not equal to a)have a common root , then b+4c+4a=____________

Q. The equations $x^2+ax+b=0$ and $x^2+cx+d=0$ have a common root. If the roots of $x^2+ax+b=0$ are equal, then which one of the following options is/are always true?

1. $a^2=4b$
2. $c^2=4d$
3. $2(b+d)=ac$
4. $2(a+c)=bd$

Q. If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$, a, b, c ϵ R have a common root, then a : b : c is

1. 1 : 2 : 3
2. 3 : 2 : 1
3. 1: 3 : 2
4. 3 : 1 : 2

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

Q. The number of roots of the equation cotx = /2+x in [-, 3/2] is?

Q.

$x^2-11x+a$ and $x^2-14x+2a$ will have a common factor, if a =

1. 24

2. 0, 24

3. 3, 24

4. 0, 3

Q. If $x^2+x+1=0$ and $x^2+ax+b=0$ have a common root, then the minimum value of $(x-a{)}^2+2b$ is

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

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

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

Q.

If the equation $x^2+px+q=0$ and $x^2+qx+p=0$, have a common root, then p + q + 1 =

1. 0

2. 1

3. 2

4. -1

Q.

A value of b for which the equations $x^2+bx-1=0,x^2+x+b=0$ have one root in common is

1. $-\sqrt{2}$

2. $-i\sqrt{3}$

3. $i\sqrt{5}$

4. $\sqrt{2}$