Home › Worksheets › A Level Year 12 Math › Binomial Expansion Prepared by Mohsim Digital Education
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(a + b)² expands to a² + 2ab + b² and the symbol ⁿCᵣ is read as n choose r, two building blocks that carry this Year 12 expansion sheet. The key also records (1 + x)³ as 1 + 3x + 3x² + x³ and counts n + 1 terms in (x + a)ⁿ.
Key Takeaways
(a + b)² expands to a² + 2ab + b² (Q1).
ⁿCᵣ is read as n choose r (Q2).
(1 + x)³ expands to 1 + 3x + 3x² + x³.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.
The binomial expansion of (a + b)² is
(a) a² + 2ab + b² (b) a² + b² (c) a² − 2ab + b² (d) a² + ab + b²
2.
The binomial coefficient ⁿCᵣ is also written as
(a) n choose r (b) n ÷ r (c) n − r (d) n!
3.
The expansion of (1 + x)³ is
(a) 1 + 3x + 3x² + x³ (b) 1 + x³ (c) 1 + 3x (d) 1 + 3x² + x³
4.
In the expansion of (a + b)ⁿ, the number of terms is
(a) n + 1 (b) n (c) n − 1 (d) 2n
5.
The binomial coefficient ⁿC₀ is equal to
(a) 1 (b) n (c) 0 (d) n!
6.
In the expansion of (1 + x)⁴, the coefficient of x² is
(a) 6 (b) 4 (c) 8 (d) 12
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.
Write the expansion of (a + b)².
8.
Write the general term of the binomial expansion.
9.
Evaluate ⁵C₂.
10.
How many terms are there in the expansion of (a + b)¹⁰?
11.
Write the first three terms of (1 + x)ⁿ using binomial coefficients.
12.
What is the sum of all the binomial coefficients in (1 + 1)ⁿ?
13.
State the value of ⁿCₙ.
14.
Give one application of the binomial expansion.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.
Expand (x + 2)².
16.
Expand (2x + 1)³.
17.
Using Pascal's triangle, expand (1 + x)⁴.
18.
Find the coefficient of x in the expansion of (1 + 3x)³.
19.
Evaluate ⁶C₃.
20.
Expand (3 + x)².
Answer Key
1. a) a² + 2ab + b²
2. a) n choose r
3. a) 1 + 3x + 3x² + x³
4. a) n + 1
5. a) 1
6. a) 6
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the expansion of (a + b)²?
a² + 2ab + b², option (a) in this worksheet (Q1).
How is ⁿCᵣ read?
n choose r, option (a) in this worksheet (Q2).
What is the expansion of (1 + x)³?
1 + 3x + 3x² + x³.