ATAR Mathematical Reasoning 2 — Questions and Answers
Question 1: A rectangle has a perimeter of 54 cm and a length that is twice its width. What is the area of the rectangle?
- 162 cm²
- 108 cm² (Correct answer)
- 81 cm²
- 54 cm²
Correct answer: 108 cm²
Width = 9 cm, length = 18 cm, so area = 9 × 18 = 162 cm².
Question 2: If the probability of rain on any given day is 0.3, what is the probability that it does NOT rain on exactly 2 consecutive days?
- 0.49 (Correct answer)
- 0.09
- 0.42
- 0.51
Correct answer: 0.49
P(no rain on 2 consecutive days) = 0.7 × 0.7 = 0.49.
Question 3: A car travels 120 km at 60 km/h, then 80 km at 40 km/h. What is the average speed for the whole journey?
- 50 km/h
- 48 km/h (Correct answer)
- 52 km/h
- 55 km/h
Correct answer: 48 km/h
Total distance = 200 km, total time = 2 + 2 = 4 hours, so average speed = 200/4 = 50 km/h.
Question 4: What is the value of x if log₂(x) + log₂(x − 2) = 3?
- x = 4 (Correct answer)
- x = 2
- x = 6
- x = 8
Correct answer: x = 4
log₂(x(x−2)) = 3 means x(x−2) = 8, giving x² − 2x − 8 = 0, so (x−4)(x+2) = 0; x = 4 (x must be positive).
Question 5: A sequence is defined by aₙ = 3n − 1. What is the sum of the first 10 terms?
- 155 (Correct answer)
- 140
- 145
- 165
Correct answer: 155
Sum = Σ(3n−1) for n=1 to 10 = 3(55) − 10 = 165 − 10 = 155.
Question 6: Two angles of a triangle are 35° and 75°. A line is drawn parallel to the longest side. What is the angle between this parallel line and the side adjacent to the 35° angle?
- 35° (Correct answer)
- 75°
- 70°
- 105°
Correct answer: 35°
The third angle is 70°, and the longest side is opposite this angle; by alternate interior angles, the angle equals 35°.
Question 7: If f(x) = 2x² − 3x + 1, what is the minimum value of f(x)?
- −1/8 (Correct answer)
- 1/8
- −7/8
- 7/8
Correct answer: −1/8
The vertex is at x = 3/4, giving f(3/4) = 2(9/16) − 3(3/4) + 1 = 9/8 − 9/4 + 1 = −1/8.
ATAR Mathematical Reasoning 2
A rectangle has a perimeter of 54 cm and a length that is twice its width.
What is the area of the rectangle?