Question

The sum of all integral values of k(k ≠ 0) for which the equation `2/(x - 1), 1/(x - 2) = 2/k` in x has no real roots, is ______.

Options

• 65

• 66

• 67

• 68

Answer 1

The sum of all integral values of k(k ≠ 0) for which the equation `2/(x - 1), 1/(x - 2) = 2/k` in x has no real roots, is 66.

Explanation:

`2/(x - 1), 1/(x - 2) = 2/k` 

⇒ `(2x - 4 - x + 1)/((x - 1)(x - 2)) = 2/k`

⇒ 2x² – (6 + k)x + 3k + 4 = 0

For non-real roots D < 0

⇒ (6 + k)² – 8(3k + 4) < 0

⇒ k² + 12k + 36 – 24k – 32 < 0

⇒ (k – 6)² – 32 < 0

Integral value of k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Sum of k = 66