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NESA · HSC Mathematics Advanced · Mixed

HSC Mathematics Advanced practice paper

15 minutes · 10 marks · 4 questions

  1. 1

    Differentiate y = x²eˣ.

    2 marks
  2. 2

    Evaluate ∫₁ᵉ (1/x) dx.

    2 marks
  3. 3

    The population P of a town grows so that dP/dt = 0.02P. If P = 5000 when t = 0 years, find the population after 10 years, to the nearest whole number.

    3 marks
  4. 4

    Find the area of the region enclosed between the curve y = x² and the line y = 2x.

    3 marks

Mark scheme

1. dy/dx = 2xeˣ + x²eˣ = xeˣ(x + 2). 1 mark for applying the product rule; 1 mark for the simplified result.

2. [ln x]₁ᵉ = ln e − ln 1 = 1. 1 mark for ln x as the primitive; 1 mark for the value 1.

3. P = 5000e^0.02t; P(10) = 5000e^0.2 ≈ 6107. 1 mark for the exponential form; 1 mark for substituting t = 10; 1 mark for 6107.

4. Intersections at x = 0 and x = 2; ∫₀² (2x − x²) dx = 4/3 square units. 1 mark for the intersections; 1 mark for the correct integral; 1 mark for 4/3.

Worked solution · Question 4

Find where the curve and line meet: x² = 2x ⇒ x² − 2x = 0 ⇒ x(x − 2) = 0, so x = 0 or x = 2.

On 0 ≤ x ≤ 2 the line lies above the curve (test x = 1: 2 > 1), so the area is ∫₀² (2x − x²) dx.

= [x² − x³/3]₀² = (4 − 8/3) − 0 = 4/3.

The enclosed area is 4/3 square units.

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8/10 marks (80%) · based on 4 questions

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Calculus 5/5
Exponential growth fix first 1/3

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