Sit this paper under timed conditions before you look at the marking key. Show your working — method marks are given for it. There is a printable PDF below if you would rather write on paper.

1. Question 16 marks
   1. (a)Simplify ( 3 x² y³)³9 x⁴ y², giving your answer in the form a x^m yⁿ.3
   2. (b)Prove that the sum of any three consecutive integers is always divisible by 3.3
   3. (no calculator)
2. Question 26 marks
   Consider the binomial expansion of ( 2 + 3 x)⁵.
   1. (a)Find the first three terms of the expansion, in ascending powers of x.3
   2. (b)Find the coefficient of x³ in the expansion.3
   3. (no calculator)
3. Question 36 marks
   The quadratic equation k x²+ ( k + 2) x + 3 = 0, where k ≠ 0, has two equal real roots.
   1. (a)Show that Δ = k²− 8 k + 4.3
   2. (b)Hence find the exact values of k.3
   3. (no calculator)
4. Question 46 marks
   Let θ be an acute angle with sin θ = 3 5.
   1. (a)Find the exact value of cos θ, and hence find the exact value of sin 2θ.4
   2. (b)Find the exact value of cos 2θ.2
   3. (no calculator)
5. Question 56 marks
   Consider the curve y = x³− 4 x.
   1. (a)Find dy dx.2
   2. (b)Find the equation of the tangent to the curve at the point where x = 2, giving your answer in the form y = mx + c.4
   3. (no calculator) Section B [50 marks] Answer all questions. (no calculator)
6. Question 616 marks
   Consider the function f ( x) = x²− 6 x + 11, defined for x. ∈ ℝ
   1. (a)Write f(x) in the form ( x − h)²+ k, where h and k are integers.3
   2. (b)Write down the coordinates of the vertex of the graph of f, and the equation of its axis of symmetry.2
   3. (c)Write down the range of f.1
   4. (d)Explain why f does not have an inverse function when its domain is, and state the largest value ℝ of a for which f has an inverse on the domain x ≥ a.3
   5. The domain of f is now restricted to x ≥ 3.
   6. (e)Find an expression for f⁻¹( x).4
   7. (f)Write down the domain of f⁻¹.1
   8. The graph of f, with domain, is translated 2 units in the positive x-direction and 5 units in the ℝ negative y-direction to give the graph of a function g.
   9. (g)Find an expression for g(x), giving your answer in the form x²+ bx + c.2
   10. (no calculator)
7. Question 717 marks
   Consider the function f ( x) = x³− 3 x².
   1. (a)Find f ′ ( x), and hence find the coordinates of the two stationary points on the graph of f.5
   2. (b)Find f ″ ( x), and use it to determine the nature of each stationary point.3
   3. (c)Find the coordinates of the points where the graph of f meets the x-axis.2
   4. (d)Sketch the graph of f for −1 ≤ x ≤ 4, showing the stationary points and the x-intercepts.3
   5. (e)Find the exact area of the region enclosed by the graph of f and the x-axis between x = 0 and x = 3.4
   6. (no calculator)
8. Question 817 marks
   Consider the function f ( x) = 3 sin ( 2 x) + 1, for 0 ≤ x ≤ 2π.
   1. (a)Write down the amplitude, the period, and the range of f.3
   2. (b)Solve the equation f ( x) = 5 2 for 0 ≤ x ≤ 2π, giving your answers as exact multiples of π.6
   3. (c)Sketch the graph of f for 0 ≤ x ≤ 2π, showing clearly the coordinates of the maximum and minimum points.4
   4. (d)State the number of solutions of the equation f(x) = 4 for 0 ≤ x ≤ 2π, and justify your answer.4