Let α and β be two real roots of the equation
(k + 1)tan²x - √2 . λtanx = (1 - k), where k(≠ - 1) and λ are real numbers. if tan² (α + β) = 50, then a value of λ is:

Solution

Maths sir · 4 steps

Move everything to one side, read off the sum and product of the tan roots, and k cancels.

1. Make it quadratic. Put t = tan x and bring everything to one side: (k+1)t² - √2 λ t - (1-k) = 0. Its two roots are tan α and tan β.
2. Sum and product. From the coefficients, tan α + tan β = (√2 λ)/(k+1) and tan α tan β = -(1-k)/(k+1) = (k-1)/(k+1). So 1 - tan α tan β = ((k+1)-(k-1))/(k+1) = 2/(k+1).

   tan(α+β) = (tan α+tan β)/(1-tan α tan β) = (√2 λ/(k+1))/(2/(k+1)) = λ/√2

3. Use the condition. tan²(α+β) = λ²/2 = 50, so λ² = 100 and λ = ± 10.
4. So. A value of λ is 10, option (b).

The idea to keep. When roots of a quadratic in tan x appear inside tan(α+β), use sum and product of roots in the tan addition formula. Extra parameters like k often cancel.

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