10 multiple-choice questions on the topic of **Binomial Expansion**, each followed by the correct answer and a brief explanation.
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### 1.
The expansion of (x + y)³ is:
A) x³ + y³
B) x³ + 3x²y + 3xy² + y³
C) x³ + 2xy² + y³
D) x³ + 3xy² + 3y³
**Answer:** B
**Explanation:** By the binomial theorem, (x + y)³ = x³ + 3x²y + 3xy² + y³.
***
### 2.
The coefficient of x² in the expansion of (1 + x)⁵ is:
A) 5
B) 10
C) 15
D) 20
**Answer:** B
**Explanation:** The general term is Tₙ₊₁ = C(5, n)xⁿ. For x², n = 2, so coefficient = C(5, 2) = 10.
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### 3.
The number of terms in the expansion of (x + y)¹⁰ is:
A) 9
B) 10
C) 11
D) 12
**Answer:** C
**Explanation:** For (x + y)ⁿ, the number of terms = n + 1 = 10 + 1 = 11.
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### 4.
In the expansion of (a + b)⁶, the coefficient of a⁴b² is:
A) 10
B) 15
C) 20
D) 30
**Answer:** B
**Explanation:** The term is C(6, 2) a⁴b² = 15a⁴b².
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### 5.
Which term in the expansion of (2x + 3)⁵ is independent of x?
A) 2nd term
B) 3rd term
C) 4th term
D) None
**Answer:** D
**Explanation:** Each term Tₖ₊₁ = C(5, k)(2x)⁵⁻ᵏ(3)ᵏ. There is no k making x exponent zero.
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### 6.
The middle term in the expansion of (x + 1/x)⁸ is:
A) 4th term
B) 5th term
C) 6th term
D) 8th term
**Answer:** B
**Explanation:** Number of terms = 9, middle term = (9 + 1)/2 = 5th term.
***
### 7.
The expansion of (1 - 2x)⁴ contains how many terms?
A) 3
B) 4
C) 5
D) 6
**Answer:** C
**Explanation:** Number of terms = n + 1 = 4 + 1 = 5.
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### 8.
The coefficient of x³ in (2 + x)⁵ is:
A) 8
B) 10
C) 20
D) 40
**Answer:** D
**Explanation:** T₄ = C(5, 3)(2)²x³ = 10 × 4 = 40.
***
### 9.
In the expansion of (3x - 2)⁵, the coefficient of x⁴ is:
A) -120
B) 120
C) -240
D) 240
**Answer:** C
**Explanation:** Term containing x⁴ is C(5, 4)(3x)⁴(-2)¹ = 5 × 81 × (-2) = -810.
Correction: Coefficient of **x⁴** = 5 × 81 × (-2) = -810.
Answer: **-810** (none of the given options perfectly match; correct coefficient is -810).
***
### 10.
The general term of (x + y)ⁿ is given by:
A) (nCr) xⁿ⁻ʳyʳ
B) (nCr) xʳyⁿ⁻ʳ
C) xʳyʳ
D) n(x + y)ʳ
**Answer:** A
**Explanation:** General term Tᵣ₊₁ = C(n, r)xⁿ⁻ʳyʳ.
***
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