Binomial Theorem
Binomial Expansion Application
nta_pyq_2025_apr
Grade 11
Question:
Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$, $x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals:
Step-by-Step Solution
Key Concept: Sum of two conjugate binomials eliminates odd-powered terms of $\sqrt{x^3-1}$; only even-power terms survive, yielding a polynomial whose coefficients give $\alpha, \beta, \gamma, \delta$.
The sum equals $2[x^5 + {}^5C_2 x^3(x^3-1) + {}^5C_4 x(x^3-1)^2] = 10x^7 + 20x^6 + 2x^5 - 20x^4 - 20x^3 + 10x$. So $\alpha = 10, \beta = 2, \gamma = -20, \delta = 10$. System: $10u + 2v = 18$ and $-20u + 10v = 20$. Solving: $u = 1, v = 4$, so $u + v = 5$.
Correct Answer: 5