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If alpha,beta are roots of the equation ...

If `alpha,beta` are roots of the equation `x^2-2x+3=0.` Then the equation whose roots are `P=alpha^3-3alpha^2+5alpha-2a ln dQ=beta^3-beta^2+beta+5` is `x^2+3x+2=0` b. `x^2-3x-2=0` c. `x^2-3x+2=0` d. none of these

A

`x^(2) + 3x + 2 = 0`

B

`x^(2) - 3x + 2 = 0`

C

`x^(2) - 3x - 2 = 0`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given `x^(2) - 2x + 3 = 0overset(alpha)underset(beta)(lt)`
`alpha + beta = 2` and `alphabeta = 3`
`:. alpha^(2) - 2alpha + 3 = 0 rArr alpha^(2) = 2alpha - 3`
`alpha^(3) = 2alpha^(2) - 3alpha`
Similarly `beta^(3) = 2beta^(2) - 3beta`
Now `P = alpha^(3) - 3alpha^(2) + 5alpha - 2`
`=2alpha^(2) - 3alpha - 3alpha^(2) + 5alpha - 2 = -alpha^(2) + 2alpha - 2`
`= 3 - 2 = 1`
`:. Q = beta^(2) - beta^(2) + beta + 5`
`= (2beta^(2) - 3beta) - beta^(2) + beta + 5 = beta^(2) - beta + 5`
`= -3 + 5 = 2`
`:.` Sum of roots `= P + Q = 1 + 2 = 3`
`:.` Product `PQ = 2`
`:.` Required equation `x^(2) - 3x + 2 = 0`.
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