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
   A family in Nairobi borrows 240 000 shillings to buy equipment. The loan is repaid in equal monthly instalments over 15 years, at a nominal annual interest rate of 4.8 %, compounded monthly.
   1. (a)Find the monthly repayment.3
   2. (b)Find the total interest paid over the 15 years.2
2. Question 26 marks
   Let log₁₀2 = p and log₁₀3 = q.
   1. (a)Express log₁₀24 in terms of p and q.3
   2. (b)Solve the equation 2 ln x = ln 9 + ln 4.3
3. Question 36 marks
   A ball is dropped from a height of 2.4 m onto a hard floor. After each bounce it rises to 62 % of the height from which it fell.
   1. (a)Find the height reached after the fifth bounce.1
   2. (b)Find the total vertical distance travelled by the ball before it comes to rest.4
   3. (c)State the condition on the common ratio r for the sum of an infinite geometric sequence to exist.1
4. Question 47 marks
   Consider the matrices A = ( 3 4 1 2) and B = ( 2 0 −1 5)
   1. (a)Find AB.2
   2. (b)Find det A.1
   3. (c)Find A⁻¹.2
   4. (d)Hence use a matrix method to solve the system 3x + y = 11, 4x + 2y = 16.2
5. Question 57 marks
   Let z₁= 3 + 4 i and z₂= 1 − 2 i.
   1. (a)Find z₁z₂, giving your answer in the form a + bi.2
   2. (b)Find | z₁| and arg ( z₁), giving the argument in radians.3
   3. (c)Hence write z₁ in the form r e^iθ.2
6. Question 67 marks
   Let f ( x) = 2 x − 5 and g ( x) = x²+ 1.
   1. (a)Find ( f ∘ g) ( 3).2
   2. (b)Find an expression for ( g ∘ f) ( x).2
   3. (c)Find f⁻¹( x).2
   4. (d)Verify that f ( f⁻¹( 7)) = 7.1
7. Question 77 marks
   A sector AOB of a circle has centre O, radius 9 cm and angle AOB = 2 π 5 radians.
   1. (a)Find the length of the arc AB.2
   2. (b)Find the area of the sector AOB.2
   3. (c)Find the area of triangle AOB.1
   4. (d)Hence find the area of the shaded segment between the chord [AB] and the arc AB.2
8. Question 87 marks
   Let a = 3 i − j + 2 k and b = i + 4 j − 2 k.
   1. (a)Find | a |.2
   2. (b)Find a·b.2
   3. (c)Hence find the angle between a and b.3
9. Question 97 marks
   The graph below shows the direct road links between five villages A, B, C, D and E.
   
   1. (a)Write down the degree of each vertex.2
   2. (b)State, with a reason, whether this graph has an Eulerian circuit, an Eulerian trail, or neither.2
   3. (c)Let M be the adjacency matrix of the graph. Find the number of distinct walks of length 3 from A to E, and explain how you obtained it.3
10. Question 107 marks
    A cable is to be laid so that all five villages are connected. The weighted graph below shows the cost, in thousands of dollars, of laying a cable along each possible link.
    
    1. (a)Use Kruskal’s algorithm to find a minimum spanning tree, listing the edges in the order in which they are chosen, and stating clearly any edge that is rejected.4
    2. (b)Write down the total cost of the minimum spanning tree.1
    3. (c)Explain why a minimum spanning tree for a graph with n vertices must contain exactly n − 1 edges.2
11. Question 117 marks
    Calls arrive at a helpdesk in Warsaw at an average rate of 3.4 calls per 10-minute interval. The number of calls, X, in a 10-minute interval is modelled by a Poisson distribution.
    1. (a)State one assumption required for a Poisson model to be appropriate here.1
    2. (b)Find P(X = 5).2
    3. (c)Find the probability that at least two calls arrive in a 10-minute interval.2
    4. (d)Find the probability that exactly 8 calls arrive in a 20-minute interval.2
12. Question 127 marks
    A random sample of 12 rechargeable batteries is tested. The sample mean lifetime is 41.2 hours and the sample standard deviation is 3.6 hours. The lifetimes are known to be normally distributed, but the population standard deviation is unknown.
    1. (a)State, with a reason, which distribution should be used to construct a confidence interval for the population mean.1
    2. (b)Find a 95 % confidence interval for the mean lifetime of the batteries.4
    3. (c)Interpret your confidence interval in context.2
13. Question 137 marks
    A random variable X has E(X) = 24 and Var(X) = 9.
    1. (a)Find E(3X − 5).2
    2. (b)Find Var(3X − 5).2
    3. X₁, X₂and X₃ are three independent random variables, each with the same distribution as X.
    4. (c)Find E ( X₁+ X₂+ X₃) and Var ( X₁+ X₂+ X₃).3
14. Question 147 marks
    1. (a)Differentiate y = x²e^3x with respect to x.3
    2. (b)Hence find the gradient of the curve y = x²e^3x at the point where x = 1.2
    3. (c)Differentiate y = ln ( x²+ 1) with respect to x.2
15. Question 158 marks
    1. (a)Find ∫ sin ( 3 x) dx.2
    2. (b)Use the substitution u = x²+ 4 to find ∫ 6 x ( x²+ 4)²dx.3
    3. Consider the curve y = x²− 4 for 0 ≤ x ≤ 3.
    4. (c)Find the total area of the regions enclosed between the curve and the x-axis.3
16. Question 168 marks
    Consider the differential equation dy dx = x + y, with the initial condition y = 1 when x = 0.
    1. (a)Write down Euler’s formula for approximating the solution, using a step length h.1
    2. (b)Use Euler’s method with a step length of 0.2 to complete a table of values of x and y from x = 0 to x = 1.4
    3. (c)Hence write down an approximate value of y when x = 1.1
    4. The exact solution of the differential equation is y = 2 e^x− x − 1.
    5. (d)Find the percentage error in your approximation of y when x = 1.2