Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $3x^2-6x+1=0$, form an equation whose roots are (i) $\dfrac{1}{\alpha }, \dfrac{1}{\beta }$

  1. $x^2-6x+3=0$
  2. $x^2+6x+3=0$
  3. $x^2-6x-3=0$

  4. None of these
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A Correct answer
Explanation

For the equation 3x^2 - 6x + 1 = 0, the sum of roots alpha + beta = 6/3 = 2 and product alpha*beta = 1/3. The new roots are 1/alpha and 1/beta. Their sum is (alpha + beta) / (alpha*beta) = 2 / (1/3) = 6. Their product is 1 / (alpha*beta) = 1 / (1/3) = 3. The new equation is x^2 - (sum)x + (product) = 0, which is x^2 - 6x + 3 = 0.

AI explanation

For the given equation 3x^2-6x+1=0, the sum of roots is 6/3 = 2 and the product is 1/3. For the new roots 1/a and 1/b, the new sum is (a+b)/ab = 2/(1/3) = 6 and the new product is 1/(ab) = 1/(1/3) = 3. The new quadratic equation is x^2 - (new sum)x + (new product) = 0, which is x^2-6x+3=0.

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