Mathematical Reasoning Flashcards

A rectangle has a perimeter of 54 cm and a length that is twice its width. What is the area of the rectangle?

Answer: 108 cm²

Width = 9 cm, length = 18 cm, so area = 9 × 18 = 162 cm².

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?

Answer: 0.49

P(no rain on 2 consecutive days) = 0.7 × 0.7 = 0.49.

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?

Answer: 48 km/h

Total distance = 200 km, total time = 2 + 2 = 4 hours, so average speed = 200/4 = 50 km/h.

What is the value of x if log₂(x) + log₂(x − 2) = 3?

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).

A sequence is defined by aₙ = 3n − 1. What is the sum of the first 10 terms?

Answer: 155

Sum = Σ(3n−1) for n=1 to 10 = 3(55) − 10 = 165 − 10 = 155.

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?

Answer: 35°

The third angle is 70°, and the longest side is opposite this angle; by alternate interior angles, the angle equals 35°.

If f(x) = 2x² − 3x + 1, what is the minimum value of f(x)?

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.