IB Mathematics SL 1.4 Financial applications AA SL Paper 1- Exam Style Questions- New Syllabus
Question
Most-appropriate topic codes (Mathematics: analysis and approaches):
• SL 1.9: The binomial theorem: expansion of \( (a+b)^n, n \in \mathbb{N} \)
• Prior learning: Simple applications of ratio, percentage and proportion; Rounding and decimal approximations
▶️ Answer/Explanation
(a) Finding \( k \)
Nominal annual interest rate = 4%
Quarterly interest rate = \( \frac{4\%}{4} = 1\% \)
As a decimal: \( 1\% = 0.01 = \frac{1}{100} \)
\( \boxed{k = 0.01} \) or \( \boxed{k = \frac{1}{100}} \)
(b) Expanding \( (1 + x)^4 \)
Using binomial theorem or Pascal’s triangle:
\( (1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 \)
\( \boxed{1 + 4x + 6x^2 + 4x^3 + x^4} \)
(c) Amount after one year
METHOD 1: Using binomial expansion
From part (b) with \( x = k = 0.01 \):
\( (1 + 0.01)^4 = 1 + 4(0.01) + 6(0.01)^2 + 4(0.01)^3 + (0.01)^4 \)
\( = 1 + 0.04 + 0.0006 + 0.000004 + 0.00000001 \)
\( = 1.04060401 \)
Amount = \( 1000 \times 1.04060401 = 1040.60401 \) dinar
\( \boxed{1041} \) dinar
METHOD 2: Direct calculation
\( (1.01)^2 = 1.0201 \)
\( (1.01)^4 = (1.0201)^2 = 1.04060401 \)
Amount = \( 1000 \times 1.04060401 = 1040.60401 \) dinar
\( \boxed{1041} \) dinar
Note: The “hence” in part (c) suggests using the binomial expansion from part (b), but both methods are acceptable.