H2 Mathematics Flashcards

Find dy/dx if y = 3x⁴ - 5x² + 2x - 7.

Answer: 12x³ - 10x + 2

Differentiating term by term: d/dx(3x⁴) = 12x³, d/dx(-5x²) = -10x, d/dx(2x) = 2, d/dx(-7) = 0. Result: 12x³ - 10x + 2.

Evaluate ∫(2x + 3) dx

Answer: x² + 3x + C

∫(2x + 3) dx = 2x²/2 + 3x + C = x² + 3x + C. Remember to include the constant of integration C.

What is the sum to infinity of a geometric series with first term 4 and common ratio 1/3?

Answer: 6

S∞ = a/(1-r) = 4/(1 - 1/3) = 4/(2/3) = 4 × 3/2 = 6. Valid since |r| = 1/3 < 1.

Solve: e^(2x) = 5

Answer: x = ln5/2

Taking natural log of both sides: 2x = ln5, so x = ln5/2.

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

Answer: y = -x/4 + 9/2

dy/dx = 2x. At x=2: gradient of tangent = 4. Gradient of normal = -1/4. Equation: y - 4 = -1/4(x - 2) → y = -x/4 + 1/2 + 4 = -x/4 + 9/2.

Find the range of values of k for which x² - 6x + k = 0 has no real roots.

Answer: k > 9

For no real roots: discriminant 9.