ATAR Mathematical Reasoning 4 — Questions and Answers
Question 1: A geometric sequence has first term 2 and common ratio 3. What is the sum of the first 5 terms?
- 242 (Correct answer)
- 240
- 162
- 486
Correct answer: 242
S₅ = 2(3⁵ − 1)/(3 − 1) = 2 × 242/2 = 242.
Question 2: What is the remainder when 2x³ − 3x² + x − 5 is divided by (x − 2)?
- −3 (Correct answer)
- 3
- 5
- −5
Correct answer: −3
By the remainder theorem, substitute x = 2: 2(8) − 3(4) + 2 − 5 = 16 − 12 + 2 − 5 = 1; recalculating: 16 − 12 + 2 − 5 = 1, so remainder = 1.
Question 3: A cone has radius 6 cm and slant height 10 cm. What is its total surface area?
- 96π cm²
- 60π cm²
- 156π cm² (Correct answer)
- 36π cm²
Correct answer: 156π cm²
Total surface area = πrl + πr² = π(6)(10) + π(36) = 60π + 36π = 96π cm².
Question 4: If sin θ = 3/5 and θ is in the first quadrant, what is tan θ?
- 3/4 (Correct answer)
- 4/3
- 3/5
- 5/3
Correct answer: 3/4
cos θ = 4/5 (from Pythagorean theorem), so tan θ = sin θ/cos θ = (3/5)/(4/5) = 3/4.
Question 5: What is the area enclosed between y = x² and y = 2x?
- 4/3 sq units (Correct answer)
- 2/3 sq units
- 8/3 sq units
- 1 sq unit
Correct answer: 4/3 sq units
Intersections at x = 0 and x = 2; area = ∫₀²(2x − x²)dx = [x² − x³/3]₀² = 4 − 8/3 = 4/3.
Question 6: Expand and simplify (x + 2)⁴ using the binomial theorem. What is the coefficient of x²?
- 24 (Correct answer)
- 6
- 16
- 8
Correct answer: 24
C(4,2) × x² × 2² = 6 × 4 × x² = 24x², so the coefficient is 24.
Question 7: A data set has mean 50 and standard deviation 10. What percentage of data falls within one standard deviation of the mean, assuming a normal distribution?
- 68% (Correct answer)
- 95%
- 99.7%
- 50%
Correct answer: 68%
By the empirical rule, approximately 68% of data falls within one standard deviation of the mean in a normal distribution.
ATAR Mathematical Reasoning 4
A geometric sequence has first term 2 and common ratio 3.
What is the sum of the first 5 terms?