If α and β are the roots of the equation
x² + px + 2 = 0 and 1/α and 1/β are the
roots of the equation 2x² + 2qx + 1 = 0, then
( α - 1/α )( β - 1/β )( α + 1/β )( β + 1/α ) is equal to :

Solution

Maths sir · 4 steps

Expand each pair of brackets so that only αβ and α + β appear, then let Vieta finish it.

1. Read off Vieta. From x² + px + 2 = 0 we get α + β = -p and αβ = 2. So α² + β² = (α+β)² - 2αβ = p² - 4.
2. First pair. Multiply out (α - 1/α)(β - 1/β) = αβ + 1/αβ - (α² + β²)/αβ. This is 2 + 1/2 - (p² - 4)/2 = (9 - p²)/2.
3. Second pair. Multiply out (α + 1/β)(β + 1/α) = αβ + 1 + 1 + 1/αβ. This is 2 + 2 + 1/2 = 9/2.

   (9 - p²)/2 × 9/2 = 9/4(9 - p²)

4. So. The expression equals 9/4(9 - p²), which is option (c). The second equation only links the two parameters: the sum of its roots gives -p/2 = -q, so p = 2q. That relation is not needed here.

The idea to keep. Any symmetric expression in the roots can be rewritten using only the sum and the product. Expand it fully first, then substitute the two values from Vieta.

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