Question

Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^7, x^5, x^3, x \) respectively in the expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, \, x > 1 \). If \( \alpha u + \beta v = 18 \), \( \gamma u + \delta v = 20 \), then \( u + v \) equals:

• A. \( 4 \)
• B. \( 8 \)
• C. \( 3 \)
• D. \( 5 \)

Hint:

For problems involving binomial expansions, it's crucial to recall the binomial theorem, which states that: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k. \] For such expansions, focus on finding the relevant coefficients and use the given relationships between the coefficients to form equations. This will help in solving for the unknowns \( u \) and \( v \) in this case.

Answer 1

The Correct Option is D

We are given the expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5 \). First, recognize that this is a binomial expansion. Let us break down the expression into two parts: \[ (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5 \] Using the binomial theorem, each term can be expanded and we are interested in the coefficients of \( x^7, x^5, x^3, x \). The relevant binomial expansions give us the coefficients \( \alpha, \beta, \gamma, \delta \). Once we have these coefficients, the relations \( \alpha u + \beta v = 18 \) and \( \gamma u + \delta v = 20 \) form a system of equations. From these, we can solve for \( u \) and \( v \) by substituting the values of \( \alpha, \beta, \gamma, \delta \). After solving the system, we find that: \[ u + v = 5. \]