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If α,β,γ are the roots of cubic equation $x^{3} - 3 x^{2} + 2 x + 4 = 0$ and

y= $1 + \frac{\propto}{x - \propto} + \frac{βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}$ then the value of y at x=2 is

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answer is B.

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Detailed Solution

The given equation of y is y=

1 + \frac{\propto}{x - \propto} + \frac{βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

First, we will simplify this equation of y by simply taking the common denominator and solve it. Therefore taking the common denominator in the first two terms of the equation, we get
⇒y=

\frac{(x - \propto) \propto}{x - \propto} + \frac{βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

⇒y=

\frac{x}{x - \propto} + \frac{βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

Again taking the common denominator for the first two terms of the equation, we get
⇒y=

\frac{x (x - β) + βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

\frac{x^{2} - βx + βx}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

Simplifying the expression, we get
⇒y=

\frac{x^{2}}{(x - α) (x - β)} + \frac{γ x^{2}}{(x - α) (x - β) (x - γ)}

Now again taking the common denominator in the equation, we get
⇒y=

\frac{x^{2} (x - γ) + γ x^{2}}{(x - α) (x - β) (x - γ)}

\frac{x^{3} - γ x^{2} + γ x^{2}}{(x - α) (x - β) (x - γ)}

On further simplification, we get
⇒y=

\frac{x^{3}}{(x - α) (x - β) (x - γ)}

…………………….(1)
It is given that α,β,γ are the roots of the equation

x^{3} - 3 x^{2} + 2 x + 4 = 0

which means that

x^{3} - 3 x^{2} + 2 x + 4

=(x−α)(x−β)(x−γ).
So substituting

x^{3} - 3 x^{2} + 2 x + 4 = 0

=(x−α)(x−β)(x−γ)
in the equation (1), we get
⇒y=

\frac{x^{3}}{x^{3} - 3 x^{2} + 2 x + 4}

Now we will find the value of y at x=2. So we will put the value of x=2
in the equation of y. Therefore, we get
⇒y=

\frac{2^{3}}{2^{3} - 3 \times 2^{2} + 2 \times 2 + 4}

⇒y=

\frac{8}{8 - 3 \times 4 + 4 + 4}

Multiplying the terms, we get
⇒y=

\frac{8}{8 - 12 + 4 + 4}

Adding the terms, we get
⇒y=

\frac{8}{16 - 12}

On further simplification, we get
⇒y=

\frac{8}{4}

=2
Hence, the value of y at x=2 is 2.
So, the option 3 is the correct option

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