Mathematical Reasoning Flashcards

A geometric sequence has first term 2 and common ratio 3. What is the sum of the first 5 terms?

Answer: 242

S₅ = 2(3⁵ − 1)/(3 − 1) = 2 × 242/2 = 242.

What is the remainder when 2x³ − 3x² + x − 5 is divided by (x − 2)?

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.

A cone has radius 6 cm and slant height 10 cm. What is its total surface area?

Answer: 156π cm²

Total surface area = πrl + πr² = π(6)(10) + π(36) = 60π + 36π = 96π cm².

If sin θ = 3/5 and θ is in the first quadrant, what is tan θ?

Answer: 3/4

cos θ = 4/5 (from Pythagorean theorem), so tan θ = sin θ/cos θ = (3/5)/(4/5) = 3/4.

What is the area enclosed between y = x² and y = 2x?

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.

Expand and simplify (x + 2)⁴ using the binomial theorem. What is the coefficient of x²?

Answer: 24

C(4,2) × x² × 2² = 6 × 4 × x² = 24x², so the coefficient is 24.

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?

Answer: 68%

By the empirical rule, approximately 68% of data falls within one standard deviation of the mean in a normal distribution.