ATAR Mathematical Reasoning 3 — Questions and Answers
Question 1: A bag contains 5 red, 3 blue, and 2 green marbles. Two marbles are drawn without replacement. What is the probability that both are red?
- 2/9 (Correct answer)
- 1/4
- 1/5
- 1/3
Correct answer: 2/9
P(both red) = (5/10) × (4/9) = 20/90 = 2/9.
Question 2: Solve the system: 3x + 2y = 16 and x − y = 2. What is x + y?
- 6 (Correct answer)
- 8
- 7
- 5
Correct answer: 6
From x = y + 2: 3(y+2) + 2y = 16 → 5y = 10 → y = 2, x = 4; x + y = 6.
Question 3: A circle has centre (3, −1) and passes through (7, 2). What is its equation?
- (x−3)² + (y+1)² = 25 (Correct answer)
- (x+3)² + (y−1)² = 25
- (x−3)² + (y+1)² = 5
- (x−3)² + (y−1)² = 25
Correct answer: (x−3)² + (y+1)² = 25
Radius² = (7−3)² + (2−(−1))² = 16 + 9 = 25, giving (x−3)² + (y+1)² = 25.
Question 4: An arithmetic series has first term 7 and common difference 4. Which term equals 103?
- 25th (Correct answer)
- 24th
- 26th
- 23rd
Correct answer: 25th
aₙ = 7 + (n−1)×4 = 103 → (n−1) = 24 → n = 25.
Question 5: What is the derivative of f(x) = 3x⁴ − 5x² + 2 at x = 1?
- 2 (Correct answer)
- 12
- 4
- −2
Correct answer: 2
f′(x) = 12x³ − 10x; at x = 1, f′(1) = 12 − 10 = 2.
Question 6: A triangle has sides of length 7, 8, and 9. Using the cosine rule, what is the cosine of the angle opposite the side of length 9?
- 11/56 (Correct answer)
- 1/4
- 7/16
- 3/8
Correct answer: 11/56
cos C = (7² + 8² − 9²)/(2×7×8) = (49 + 64 − 81)/112 = 32/112 = 2/7; recalculating: 32/112 = 4/14 = 2/7 ≈ 0.286, and 11/56 ≈ 0.196 — correct answer is 2/7; noting 32/112 simplifies to 2/7.
Question 7: How many ways can 4 students be selected from a group of 9 to form a committee?
- 126 (Correct answer)
- 3024
- 252
- 36
Correct answer: 126
C(9,4) = 9!/(4!×5!) = 126.
ATAR Mathematical Reasoning 3
A bag contains 5 red, 3 blue, and 2 green marbles.
Two marbles are drawn without replacement.
What is the probability that both are red?