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 15 marks
   Express 7 x + 1 ( x − 1) ( x + 3) in partial fractions. [5] (no calculator)
2. Question 26 marks
   1. (a)Using the compound angle identity for sin(A + B), show that sin 75 ° = √⁶⁺√²4.3
   2. (b)Hence, or otherwise, show that tan 105 ° = − ( 2 + √^3).3
   3. (no calculator)
3. Question 36 marks
   Consider the complex numbers z₁= √³⁺ⁱ and z₂= 1 − i.
   1. (a)Find z₁z₂, giving your answer in Cartesian form.2
   2. (b)Write z₁ and z₂ in modulus–argument form.2
   3. (c)Hence find z₁z₂ in modulus–argument form.2
   4. (no calculator)
4. Question 47 marks
   A drinks factory in Da Nang, Vietnam, uses three bottling machines. Machine A fills 50 % of all bottles, machine B fills 30 % and machine C fills 20 %. The proportions of bottles that are underfilled are 2 % for machine A, 4 % for machine B and 5 % for machine C. A bottle is chosen at random from the day’s production.
   1. (a)Show that the probability that the bottle is underfilled is 4 125.3
   2. (b)Given that the bottle is underfilled, find the probability that it was filled by machine B.4
   3. (no calculator)
5. Question 57 marks
   The curve C has equation x²+ xy + y²= 7. The point P(1, 2) lies on C.
   1. (a)Use implicit differentiation to show that dy dx = − 2 x + y x + 2 y.4
   2. (b)Find the equation of the tangent to C at P, giving your answer in the form ax + by = c, where a, b, c. ∈ ℤ2
   3. (c)Write down the gradient of the normal to C at P.1
   4. (no calculator)
6. Question 67 marks
   1. (a)Use l’Hôpital’s rule to find the limit of 1 − cos 3 x x² as x approaches 0.4
   2. (b)Find the limit of 5 x²+ 3 x e^2x− 1 as x approaches 0.3
   3. (no calculator)
7. Question 78 marks
   Consider the equation sec²x − tan x = 1, for 0 ≤ x ≤ 2π.
   1. (a)Using the identity 1 + tan²x = sec²x, show that the equation can be written as tan x ( tan x − 1) = 0.2
   2. (b)Hence solve the equation for 0 ≤ x ≤ 2π.4
   3. (c)Write down the domain and the range of the function f ( x) = arctan x.2
   4. (no calculator)
8. Question 88 marks
   1. (a)Use proof by contradiction to prove that √³ is irrational.5
   2. Consider the following statement: for every positive integer n, the number n²− n + 11 is prime.
   3. (b)Show, using a counterexample, that this statement is false. You must explain why your example is a counterexample.3
   4. (no calculator) Section B [56 marks] Answer all questions. (no calculator)
9. Question 918 marks
   The points A(2, −1, 3), B(4, 1, 2) and C(3, 1, 5) are the vertices of a triangle. Let u be the vector from A to B, and let v be the vector from A to C.
   1. (a)Find u and v, giving your answers in the form ai + bj + ck.2
   2. (b)Show that the triangle is isosceles, with | u | = | v | = 3.2
   3. (c)The angle between u and v is θ. Using the scalar product, find the exact value of cos θ.4
   4. The line L passes through the points A and B.
   5. (d)Write down a vector equation of L.2
   6. (e)Determine whether the point D(8, 5, 0) lies on L.3
   7. (f)Find the vector product u × v.3
   8. (g)Hence find the exact area of the triangle ABC.2
   9. (no calculator)
10. Question 1019 marks
    Consider the function f ( x) = x e^x.
    1. (a)Find f ′ ( x), and hence show that the graph of f has exactly one stationary point, at x = −1.3
    2. (b)Use integration by parts to show that ∫ x e^xdx = ( x − 1) e^x+ c.4
    3. (c)The region R is bounded by the graph of f, the x-axis, and the line x = 1. Find the exact area of R.3
    4. (d)Using integration by parts twice, show that ∫ x²e^xdx = ( x²− 2 x + 2) e^x+ c.4
    5. The region S is bounded by the curve y = x e x 2, the x-axis, and the line x = 1.
    6. (e)The region S is rotated through 2π about the x-axis. Use part
    7. (d)to find the exact volume of the solid generated.3
    8. (f)Use the substitution u = x², or otherwise, to find ∫ x e^x2dx.2
    9. (no calculator)
11. Question 1119 marks
    1. (a)Use mathematical induction to prove that ( cos θ + i sin θ)ⁿ= cos nθ + i sin nθ for all n ∈. ℤ⁺7
    2. Consider the complex number w = 1 + i.
    3. (b)Write w in modulus–argument form.2
    4. (c)Hence find w⁸.3
    5. (d)Find the smallest positive integer n for which wⁿ is a negative real number, and state the value of wⁿ for this n.4
    6. (e)Hence write down the four roots of the equation z⁴+ 4 = 0.3