If `alpha,beta` are the roots of the equation `x^2-2x+3=0` obtain the equation whose roots are `alpha^3-3alpha^2+5alpha-2` and `beta^3-beta^2+beta=5`

Options

A. `x^(2) = 3x + 2 = 0`
B. `x^(2) - 3x - 2 = 0`
C. `x^(2) - 3x + 2 = 0`
D. none of these

Answer

3

Explanation

Given, `alpha, beta` are roots of equation
`x^(2) - 2x + 3 = 0`
`rArr alpha^(2) - 2alpha + 3 = 0" "(1)`
and `beta^(2) - 2beta + 3 = 0" "(2)`
`rArr alpha^(2) = 2alpha - 3` or `alpha^(3) = 2alpha^(2) - 3alpha`
`rArr P = (2alpha^(2) - 3alpha) - 3alpha^(2) + 5alpha - 2`
`= - alpha^(2) + 2alpha - 2 = 3 - 2 = 1" "` [Using (1)]
Similarly, we have Q = 2.
Now, sum of roots is 3 and product of roots is 2. Hence, the required equation is `x^(2) - 3x + 2 = 0`.