IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Functions–Representation and shape of more complex functions
Topic :Function- Weightage : 21 %
All Questions for Topic : Representation and shape of more complex functions,Transformation of quadratic functions,Rational functions,Graphing trigonometric functions,Linear programming, including inequalities,Networks-edges and arcs, nodes/ vertices, paths,Calculating network pathways,Weighted networks,Domain and range
Question : Machine Learning Success Rates [22 marks]
This question explores success rates of three face recognition methods:
• Eigenface (E) – Linear model
• Fisherface (F) – Quadratic model
• LBPH (L) – Quadratic model
n = number of face observations (0 ≤ n ≤ 10)
a Question a [2 marks] – Linear Model Parameters
For Eigenface method E = bn + c, determine values of b and c from the graph.
b (slope):
c (y-intercept):
▶️Answer/Explanation
Step 1: Find y-intercept (c)
When n=0, E=50 ⇒ c=50
Step 2: Calculate slope (b)
Using point (2,55): 55 = b(2) + 50 ⇒ 5 = 2b ⇒ b=2.5
Verification with (4,60): 60 = 2.5(4) + 50 = 10 + 50 = 60 ✓
Final Model:
E = 2.5n + 50
Complete Answers:
b = 2.5
c = 50
b Question b [1 mark] – Maximum Success Rate
For Fisherface method F = -1.5(n-8)² + 96, find the maximum success rate and corresponding n value.
Fmax (%):
n value:
▶️Answer/Explanation
Analyzing the Quadratic Function:
F = -1.5(n-8)² + 96 is in vertex form: f(x) = a(x-h)² + k
Vertex Properties:
• Maximum occurs at vertex since a = -1.5 < 0 (parabola opens downward)
• Vertex at (h,k) = (8,96)
Interpretation:
Maximum success rate of 96% occurs when n=8 observations are made
Complete Answers:
Fmax = 96%
n = 8
c Question c [5 marks] – Intersection Point
Find the value of n where Fisherface (F) and Eigenface (E) methods have equal success rates.
▶️Answer/Explanation
Method 1: Algebraic Solution
Set E = F:
2.5n + 50 = -1.5(n-8)² + 96
Step 1: Expand quadratic
2.5n + 50 = -1.5(n² – 16n + 64) + 96
2.5n + 50 = -1.5n² + 24n – 96 + 96
Step 2: Rearrange terms
1.5n² – 21.5n + 50 = 0
Step 3: Quadratic formula
n = [21.5 ± √(21.5² – 4×1.5×50)] / (2×1.5)
Discriminant = 462.25 – 300 = 162.25
√162.25 = 12.74
n = (21.5 ± 12.74)/3
Solutions:
n = (21.5 + 12.74)/3 ≈ 11.41 (invalid: n≤8)
n = (21.5 – 12.74)/3 ≈ 2.92
Method 2: Graphical Verification
Testing integer values:
At n=2: E=55%, F=42%
At n=3: E=57.5%, F=61.5%
Intersection occurs between n=2 and n=3
Final Answer:
n ≈ 2.92 or 3 (when rounded)
d Question d [4 marks] – LBPH Model
Find the quadratic equation for LBPH method (L) given vertex (8,86) and point (2,50).
▶️Answer/Explanation
Step 1: Vertex Form
L = a(n-h)² + k where (h,k)=(8,86)
⇒ L = a(n-8)² + 86
Step 2: Use given point
At n=2, L=50:
50 = a(2-8)² + 86
50 = 36a + 86
Step 3: Solve for a
36a = 50 – 86 = -36
a = -1
Final Equation:
L = -(n-8)² + 86
or expanded form: L = -n² + 16n – 64 + 86 = -n² + 16n + 22
Verification:
At n=8: L = -0 + 86 = 86 ✓
At n=2: L = -36 + 86 = 50 ✓
Complete Answer:
L = -(n-8)² + 86
e Question e [10 marks] – Method Analysis
Analyze all three methods to recommend the best one considering:
- Factors affecting success rate
- Intersection calculations
- Accuracy considerations
- Justified recommendation
▶️Answer/Explanation
1. Key Factor:
The primary factor affecting success rate is the number of observations (n). All methods improve with more observations but have different patterns.
2. Intersection Analysis:
E vs F: Intersect at n≈2.92 (E catches up to F)
E vs L: Solve 2.5n+50 = -n²+16n+22 ⇒ n≈3.53
F vs L: Solve -1.5(n-8)²+96 = -(n-8)²+86 ⇒ n≈10.58 (outside domain)
3. Accuracy Considerations:
• Eigenface (E): Steady linear improvement, simplest model
• Fisherface (F): Rapid improvement peaking at n=8 (96%)
• LBPH (L): Moderate improvement peaking at n=8 (86%)
4. Recommendation:
For n ≤ 3: Eigenface performs best (simple, adequate results)
For 3 < n ≤ 8: Fisherface dominates (highest success rate)
Beyond n=8: Not applicable (F decreases, L plateaus)
5. Limitations:
• Models are approximations of real-world behavior
• Actual performance may vary with image quality and lighting
Exemplar Conclusion:
Fisherface is generally best when 4+ observations can be made, providing the highest maximum success rate (96%). For limited observations (n≤3), Eigenface is preferable due to its consistent performance.
Syllabus Reference
Unit 2: Algebra
- Linear functions
- Quadratic functions
- Modeling real-world data
Unit 3: Function
- Function transformations
- Graph interpretation
- Comparative analysis
Assessment Criteria: A (Knowledge & Understanding), D (Applying mathematics in real-life contexts)