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Q.

If α,β are the roots of the equation $x^{2}$ −3x + 1 = 0,then the equation with roots 1/(α−2) ,1/(β−2) is:

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a

x^{2} - x - 1

b

{2 x}^{2} - x - 1

c

x^{2} - 2 x - 1

d

x^{2} - x - 2

answer is A.

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Detailed Solution

Concept- Once we know a quadratic equation's two roots, we can move on to write its general formulation. The sum and product of roots for the given quadratic equation will then be determined. Finally, we must include these numbers in a new quadratic equation's simplified expression.
Given a quadratic equation is

x^{2} - 3 x + 1 = 0

and α, and β are the roots of this equation.
The quadratic equation with roots

\frac{1}{(α - 2)}, \frac{1}{(β - 2)}

is given by:

\frac{(x – 1)}{(α - 2)}) \frac{(x – 1)}{(β - 2)}) = \frac{(x (α - 2) - 1)}{(α - 2)} \times \frac{(x (β - 2) - 1)}{(β - 2)}

= x^{2} (αβ - 2 (α + β) + 4 - x (α + β - 4) + \frac{1}{αβ} - 2 (α + β) + 4 .

The above expression is in the form of α+β and αβ,α+β and αβ.
For quadratic equation

x^{2}

−3x + 1 =0:
Sum of roots

= α + β = \frac{- b}{a} = \frac{- (- 3)}{1} = 3

And product of roots

= \frac{α}{β} = \frac{c}{a} = \frac{1}{1} = 1

Putting the values of

α + β  and αβ, α + β and αβ

in above equation, we get:

\frac{}{}

= \frac{{- x}^{2} + x + 1}{- 1}

=

x^{2}

- x - 1

Hence, the answer is option 1.

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