Additional Mathematics Flashcards

Find the range of values of k for which the equation x² + kx + 9 = 0 has no real roots.

Answer: -6 < k < 6

For no real roots, the discriminant must be negative: b² - 4ac < 0. k² - 4(1)(9) < 0, k² - 36 < 0, k² < 36, -6 < k < 6.

What is the exact value of cos 60°?

Answer: 1/2

cos 60° = 1/2 is a standard trigonometric value derived from the equilateral triangle (60-60-60) bisected to form a 30-60-90 triangle.

What is the equation of the tangent to the curve y = x² at the point (2, 4)?

Answer: y = 4x - 4

dy/dx = 2x. At x = 2, gradient = 4. Using y - y₁ = m(x - x₁): y - 4 = 4(x - 2), y = 4x - 4.

Simplify: (2ⁿ⁺¹ × 4ⁿ) / 8ⁿ

Answer: 2

Express all as powers of 2: (2ⁿ⁺¹ × 2²ⁿ) / 2³ⁿ = 2ⁿ⁺¹⁺²ⁿ / 2³ⁿ = 2³ⁿ⁺¹ / 2³ⁿ = 2³ⁿ⁺¹⁻³ⁿ = 2¹ = 2.

What is the area under the curve y = x² between x = 0 and x = 3?

Answer: 9

Area = ∫₀³ x² dx = [x³/3]₀³ = 27/3 - 0 = 9.

If the roots of 2x² - 5x + 1 = 0 are α and β, what is α + β?

Answer: 5/2

For ax² + bx + c = 0, the sum of roots = -b/a. Here, α + β = -(-5)/2 = 5/2.