Question

send step by step solution

Answer 1

Solution :

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Given :

a + b + c = 1,

a² + b² + c² = 9

a³ + b³ + c³ = 1

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To find :

The value of :

⇒ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = ?$

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We know that,

we can also write,

⇒ $\frac{1}{a}= \frac{1}{a} ( \frac{bc}{bc})$

⇒ $\frac{1}{a}= \frac{bc}{abc}$

,

⇒ $\frac{1}{b}= \frac{1}{b} ( \frac{ac}{ac})$

⇒ $\frac{1}{b}= \frac{ac}{abc}$

&

⇒ $\frac{1}{c}= \frac{1}{c} ( \frac{ab}{ab})$

⇒ $\frac{1}{a}= \frac{ab}{abc}$

And hence,

⇒ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

⇒ $\frac{bc}{abc} + \frac{ac}{abc} + \frac{ab}{abc}$

⇒ $\frac{(ab + bc + ac)}{abc}$ ...(1)

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And so,

We also know that,

⇒ (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

⇒ (a + b + c)² = a² + b² + c² + 2(ab + bc + ac)

By, substituting the known values,

We get,

⇒ (1)² = 9 + 2(ab + bc + ac)

⇒ 1= 9 + 2(ab + bc + ac)

⇒ 1 - 9 = 2(ab + bc + ac)

⇒ -8 = 2 (ab + bc + ac)

⇒ ab + bc + ac = -8/2

⇒ ab + bc + ac = -4 ....(2)

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(a + b + c)³ = a³ + b³ + c³ + 3(a + b + c)(ab + bc + ac) - 3abc

⇒ (1)³ = 1 + 3(1)(-4) - 3abc

⇒ 1 = 1 - 12 - 3abc

⇒ 1 = - 11 - 3abc

⇒ 1 + 11 = - 3abc

⇒ 12 = -3abc

⇒ abc = 12/-3

⇒ abc = -4 ....(3)

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⇒ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

⇒ $\frac{(ab + bc + ac)}{abc}$

⇒ $\frac{-4}{-4}$

⇒ 1

         ⇒ ∴  $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ = 1

         ⇒ ∴ d) is correct,.

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