If α and β are the roots of the equation x2-2x+3=0, then the sum of roots of the equation having roots as α 3-3α 2+5α -2 and β 3-β 2+β +5 is

Question

If α and β are the roots of the equation x2-2x+3=0, then the sum of roots of the equation having roots as α 3-3α 2+5α -2 and β 3-β 2+β +5 is (A) 1 (B) 3 (C) 5 (D) 7

Tardigrade Admin · Accepted Answer

Since, α 2-2α +3=0 ⇒ (α )3-3(α )2+5α -2=α ((α )2 - 2 α + 3)-((α )2 - 2 α + 3)+1=1 And β 2-2β +3=0 ⇒ (β )3-(β )2+β +5=β ((β )2 - 2 β + 3)+((β )2 - 2 β + 3)+2=2 So, the required equation is x2-(2 + 1)x+2.1=0 ⇒ x2-3x+2=0 Sum of roots =3

Q. If $α$ and $β$ are the roots of the equation $x^{2} - 2 x + 3 = 0,$ then the sum of roots of the equation having roots as $α^{3} - 3 α^{2} + 5 α - 2$ and $β^{3} - β^{2} + β + 5$ is

$1$

$3$

$5$

$7$

Q. If α and β are the roots of the equation x2−2x+3=0, then the sum of roots of the equation having roots as α3−3α2+5α−2 and β3−β2+β+5 is

1

3

5

7

Since, α2−2α+3=0 ⇒(α)3−3(α)2+5α−2=α((α)2−2α+3)−((α)2−2α+3)+1=1 And β2−2β+3=0 ⇒(β)3−(β)2+β+5=β((β)2−2β+3)+((β)2−2β+3)+2=2 So, the required equation is x2−(2+1)x+2.1=0 ⇒x2−3x+2=0 Sum of roots =3

Q. If $α$ and $β$ are the roots of the equation $x^{2} - 2 x + 3 = 0,$ then the sum of roots of the equation having roots as $α^{3} - 3 α^{2} + 5 α - 2$ and $β^{3} - β^{2} + β + 5$ is

$1$

$3$

$5$

$7$