Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

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Question: Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

Options:

Correct Answer: 15

Exam Year: 2019

Solution:

The coefficient of x^2 is given by 6C2 * (1)^4 = 15.

Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

Step 1: Identify the expression to expand, which is (x + 1/x)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = x, b = 1/x, and n = 6.
Step 4: We need to find the term where the power of x is 2. This means we want x^(6-k) * (1/x)^k = x^(6-k-k) = x^(6-2k).
Step 5: Set the exponent equal to 2: 6 - 2k = 2.
Step 6: Solve for k: 6 - 2k = 2 → 2k = 4 → k = 2.
Step 7: Now, substitute k = 2 into the binomial coefficient: 6C2.
Step 8: Calculate 6C2, which is 6! / (2!(6-2)!) = 15.
Step 9: The term corresponding to k = 2 is 6C2 * (x)^(6-2) * (1/x)^2 = 15 * x^4 * (1/x^2) = 15 * x^2.
Step 10: The coefficient of x^2 in the expansion is 15.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to expand expressions of the form (a + b)^n.
Coefficient Extraction – It assesses the ability to identify and calculate specific coefficients from the expanded form of a binomial expression.