Binomial Theorem

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Q1. Calculate the value of (1 + 3)^5 using the binomial theorem.

Solution:

(1 + 3)^5 = 4^5 = 1024.

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Q2. What is the value of the term containing x^4 in the expansion of (x + 1/2)^8? (2020)

Solution:

The term containing x^4 is C(8,4) * (1/2)^4 = 70 * 1/16 = 4.375.

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Q3. In the expansion of (2x + 3)^4, what is the coefficient of x^0?

Solution:

The coefficient of x^0 is given by 4C4 * (2x)^0 * (3)^4 = 1 * 81 = 81.

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Q4. Find the constant term in the expansion of (3x - 4/x)^5.

Solution:

The constant term occurs when the power of x is zero. The term is given by 5C2 * (3x)^2 * (-4/x)^3 = 10 * 9 * (-64) = -5760.

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Q5. What is the coefficient of x^2 in the expansion of (x + 4)^5?

Solution:

The coefficient of x^2 is C(5,2) * 4^3 = 10 * 64 = 640.

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Q6. In the expansion of (3 + 2x)^4, what is the coefficient of x^2? (2023)

Solution:

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

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Q7. 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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Q8. What is the 5th term in the expansion of (x - 1)^8? (2022)

Solution:

The 5th term corresponds to k=4. Using the binomial theorem, it is given by 8C4 * (x)^4 * (-1)^4 = 70x^4.

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Q9. What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

Solution:

The coefficient of x^2 is C(8,2) * (1/2)^2 = 28 * 1/4 = 7.

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Q10. Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)

Solution:

The term independent of x occurs when the powers of x cancel out. The term is 81.