Roots of the equation $x^{3} - (a + b + c) x^{2} + (a b + b c + c a) x - a b c = 0$ , if $x^{2} + 2 x + 7 = 0$ and $a x^{2} + b x + c = 0$ have a common root, where a,b,c $\in$ R, can be

4, 8, 28

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1, 2, 7

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1, 4, 36

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None of the above

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Solution

The correct option is B $1, 2, 7$
Equation :
$x^{3} - (a + b + c) x^{2} + (a b + b c + c a) x - a b c = 0$
$x^{2} + 2 x + 7 = 0$
$a x^{2} + b x + c = 0$
they have common root
$α + β = - 2; + \frac{b}{a} = + \frac{2}{1}$
$α β = 7 = \frac{c}{a}$
$b = 2$
$a = 1$
$c = 7$
$a = 1; b = 2; c = 7$

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Roots of the equation x3−(a+b+c)x2+(ab+bc+ca)x−abc=0, if x2+2x+7=0 and ax2+bx+c=0 have a common root, where a,b,c∈R, can be

The correct option is B 1,2,7Equation :x3−(a+b+c)x2+(ab+bc+ca)x−abc=0x2+2x+7=0ax2+bx+c=0they have common rootα+β=−2;+ba=+21αβ=7=cab=2a=1c=7a=1;b=2;c=7

Roots of the equation $x^{3} - (a + b + c) x^{2} + (a b + b c + c a) x - a b c = 0$ , if $x^{2} + 2 x + 7 = 0$ and $a x^{2} + b x + c = 0$ have a common root, where a,b,c $\in$ R, can be

The correct option is B $1, 2, 7$
Equation :
$x^{3} - (a + b + c) x^{2} + (a b + b c + c a) x - a b c = 0$
$x^{2} + 2 x + 7 = 0$
$a x^{2} + b x + c = 0$
they have common root
$α + β = - 2; + \frac{b}{a} = + \frac{2}{1}$
$α β = 7 = \frac{c}{a}$
$b = 2$
$a = 1$
$c = 7$
$a = 1; b = 2; c = 7$