H2 Mathematics Flashcards

The mean of a sample of 8 values is 12. If one value of 20 is added, what is the new mean?

Answer: 12.89

Original total = 8×12 = 96. New total = 96+20 = 116. New mean = 116/9 ≈ 12.89.

Find the equation of the circle with centre (3, -1) and radius 5.

Answer: (x-3)² + (y+1)² = 25

The standard equation of a circle with centre (a,b) and radius r is (x-a)² + (y-b)² = r². Here: (x-3)² + (y+1)² = 25.

What is the Pearson correlation coefficient closest to for data showing a strong negative linear relationship?

Answer: -0.9

Pearson's r ranges from -1 to +1. A strong negative linear relationship gives r close to -1, such as -0.9.

Find the term independent of x in the expansion of (x + 1/x²)⁶.

Answer: 20

General term: ⁶Cᵣ × x^(6-r) × x^(-2r) = ⁶Cᵣ × x^(6-3r). For x⁰: 6-3r=0 → r=2. Term = ⁶C₂ = 15. Rechecking: ⁶C₂ = 15.

Evaluate lim(x→0) [sin(3x)/x].

Answer: 3

Using the standard limit lim(x→0)[sin(ax)/x] = a, or applying L'Hôpital's rule: differentiate numerator and denominator: 3cos(3x)/1 → 3 as x→0.

If the probability that a biased coin shows heads is 0.6, what is the probability of getting exactly 2 heads in 5 tosses?

Answer: 0.2304

P(X=2) = ⁵C₂ × (0.6)² × (0.4)³ = 10 × 0.36 × 0.064 = 10 × 0.02304 = 0.2304.