Mathematics Core Flashcards

What is the value of ∫₀² (3x² - 2x) dx?

Answer: 4

Integrating: [x³ - x²]₀² = (8 - 4) - (0 - 0) = 4.

The binomial expansion of (1 + 2x)⁵ has a term in x³. What is its coefficient?

Answer: 80

Using ⁵C₃ × (2x)³ = 10 × 8x³ = 80x³. The binomial coefficient ⁵C₃ = 10 and (2x)³ = 8x³.

A circle has equation x² + y² - 6x + 4y - 12 = 0. What is its radius?

Answer: 5

Completing the square: (x-3)² + (y+2)² = 9 + 4 + 12 = 25. So the radius is √25 = 5.

What is the exact value of sin(π/3)?

Answer: √3/2

π/3 radians = 60°. From the standard triangle, sin(60°) = √3/2. This is a key exact trigonometric value required for A-Level.

The vector a = 3i + 4j. What is the unit vector in the direction of a?

Answer: (3i + 4j)/5

|a| = √(9 + 16) = √25 = 5. The unit vector is a/|a| = (3i + 4j)/5 = 0.6i + 0.8j.

Solve 2sin(x) - 1 = 0 for 0 ≤ x ≤ 2π. How many solutions are there?

Answer: 2

sin(x) = 1/2 gives x = π/6 and x = 5π/6 in the range [0, 2π]. Sine is positive in the first and second quadrants, giving exactly 2 solutions.