Binomial Theorem

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Q1. What is the value of the term containing x^5 in the expansion of (x + 1/2)^8? (2020)

Solution:

The term containing x^5 is C(8,5)(1/2)^3 = 56 * 1/8 = 7.

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Q2. Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

Solution:

The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.

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Q3. Calculate the term independent of x in the expansion of (x/2 - 3)^8.

Solution:

The term independent of x occurs when k = 4. Thus, the term is C(8,4) * (x/2)^4 * (-3)^4 = 70 * (1/16) * 81 = 256.

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Q4. What is the coefficient of x^2 in the expansion of (x - 2)^5?

Solution:

The coefficient of x^2 is given by 5C2 * (-2)^3 = 10 * (-8) = -80.

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Q5. What is the coefficient of x^3 in the expansion of (x - 2)^5?

Solution:

The coefficient of x^3 is C(5,3) * (-2)^2 = 10 * 4 = 40.

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Q6. Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

Solution:

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

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Q7. Calculate the term containing x^3 in the expansion of (2x + 5)^6. (2000)

Solution:

The term containing x^3 is C(6,3) * (2)^3 * (5)^(6-3) = 20 * 8 * 125 = 20000.

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Q8. What is the coefficient of x^4 in the expansion of (3x + 2)^6?

Solution:

The coefficient of x^4 is C(6,4) * (3)^4 * (2)^2 = 15 * 81 * 4 = 4860.

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Q9. What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

Solution:

The coefficient of x^2 is C(5,2) * (2)^2 * (-1)^3 = 10 * 4 * (-1) = -40.

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Q10. What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

Solution:

The coefficient of x^3 is C(6,3) * (5)^3 * (2)^3 = 20 * 125 * 8 = 20000.