If α and β are the distinct roots of the equation x² + (3)^(1/4)x + 3^(1/2) = 0, then the value of α ⁹⁶(α ¹² - 1) + β ⁹⁶(β ¹² - 1) is equal to :

Solution

Maths sir · 4 steps

Factor out 3^(1/4) and the roots become 3^(1/4)ω and 3^(1/4)ω². Then the big powers are easy.

1. Scale the variable. Put x = 3^(1/4)t. Then x² = 3^(1/2)t² and 3^(1/4)x = 3^(1/2)t, so the equation becomes 3^(1/2)(t² + t + 1) = 0.
2. Identify the roots. The roots of t² + t + 1 = 0 are ω and ω², the complex cube roots of unity, with ω³ = 1. So α = 3^(1/4)ω and β = 3^(1/4)ω².

   α¹² = 3³ ω¹² = 27, α⁹⁶ = 3²⁴ ω⁹⁶ = 3²⁴

3. Same for beta. The exponents 12 and 96 are multiples of 3, so the ω powers vanish for β too: β¹² = 27 and β⁹⁶ = 3²⁴. Each term is 3²⁴(27 - 1) = 26 × 3²⁴.
4. So. The sum is 2 × 26 × 3²⁴ = 52 × 3²⁴, which is option (c).

The idea to keep. When the coefficients are powers of one number, scale x to clear them. If t²+t+1=0 is left, the roots are ω, ω², and any power with an exponent that is a multiple of 3 equals 1.

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