If α and β are the roots of the equation,
7x² – 3x – 2 = 0, then the value of
α/(1 - α ²) + β/(1 - β ²) is equal to :

Solution

Maths sir · 4 steps

Add the two fractions first, and then everything you need is just α+β and αβ.

1. Sum and product. For 7x² - 3x - 2 = 0, α+β = 3/7 and αβ = -2/7.
2. Combine the fractions. Over a common denominator, the numerator is α(1-β²) + β(1-α²) = (α+β) - αβ(α+β). The denominator expands to 1 - (α²+β²) + α²β².

   α/(1-α²)+β/(1-β²) = (α+β)(1-αβ)/(1-(α²+β²)+α²β²)

3. Put in numbers. Numerator: 3/7 × 9/7 = 27/49. Next, α²+β² = (α+β)² - 2αβ = 9/49 + 28/49 = 37/49 and α²β² = 4/49. So the denominator is (49 - 37 + 4)/49 = 16/49.
4. So. The value is (27/49)/(16/49) = 27/16, which is option (c).

The idea to keep. Any symmetric expression in the roots can be rewritten using only α+β and αβ. Combine fractions first, then substitute, and watch the signs.

Answer from the NTA final answer key. Solution by PrepWiser.