IB Mathematics SL 1.7 Laws of exponents with rational exponents AA SL Paper 2- Exam Style Questions- New Syllabus
Question
Consider the expansion of \( (x + k)^{11} \), where \( k > 0 \).
(a) Write down the number of terms in the expansion.
(b) In the expansion, the coefficient of \( x^7 \) is 1320. Find the value of \( k \).
(b) In the expansion, the coefficient of \( x^7 \) is 1320. Find the value of \( k \).
Most-appropriate topic codes (IB Mathematics AA SL 2025):
• SL 1.9: The binomial theorem: expansion of \( (a + b)^n \) — parts (a), (b)
• SL 1.7: Laws of exponents — part (b)
• SL 1.7: Laws of exponents — part (b)
▶️ Answer/Explanation
(a)
For \( (x + k)^{11} \), the number of terms is \( 11 + 1 = 12 \).
(b)
The general term in the expansion is:
\( T_{r+1} = \binom{11}{r} x^{11-r} k^r \)
For the coefficient of \( x^7 \), we need \( 11 – r = 7 \) so \( r = 4 \).
The coefficient is \( \binom{11}{4} k^4 = 330 k^4 \).
Given this equals 1320:
\( 330 k^4 = 1320 \)
\( k^4 = \frac{1320}{330} = 4 \)
\( k = \sqrt[4]{4} = \sqrt{2} \approx 1.41 \) (since \( k > 0 \))