Multiple choice

If ${ \lambda }{ 1 }$ and ${ \lambda }{ 2 }$ are the two values of $\lambda$ such that the roots $\alpha$ and $\beta$ of the quadratic equation, $\lambda \left( { x }^{ 2 }-x \right) +x+5=0$ satisfy $\cfrac { \alpha }{ \beta } +\cfrac { \beta }{ \alpha } +\cfrac { 4 }{ 5 } =0$, then $\cfrac { { \lambda }{ 1 } }{ { \lambda }{ 2 }^{ 2 } } +\cfrac { { \lambda }{ 2 } }{ { \lambda }{ 1 }^{ 2 } } =0$ is equal to:

  1. $536$
  2. $512$
  3. $504$
  4. $488$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation is lambda*x^2 - lambda*x + 1 + 5 = 0, which is lambda*x^2 - lambda*x + 6 = 0. Sum of roots alpha+beta = 1, product alpha*beta = 6/lambda. (alpha/beta + beta/alpha) = ((alpha+beta)^2 - 2*alpha*beta) / (alpha*beta) = (1 - 12/lambda) / (6/lambda) = (lambda-12)/6. Given (lambda-12)/6 + 4/5 = 0, lambda-12 = -24/5, lambda = 12 - 4.8 = 7.2. This leads to a specific value for the expression.

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