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University of Bristol. Mathematics Exam. Practice paper. Instructions. • You have 24 hours to complete this exam. • Calculators are permitted.

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Download University of Bristol Mathematics Exam Practice paper and more Study notes Mathematics in PDF only on Docsity! University of Bristol Mathematics Exam Practice paper Instructions • You have 24 hours to complete this exam. • Calculators are permitted. • Show your working clearly. Answers without working will not be given full credit. • The marks for each question part are shown in brackets. 1 1. Let w1 = 3 + 4i and w2 = −1 + 2i. (a) Find the following, giving your answers in the form a+ bi. i. w∗ 1w2 [2] ii. w1 − 3w2 w2 [2] (b) Write down a quadratic equation with real coefficients that has w1 as a root. [2] 2. The graph of the function y = x2 − 2x− 3 is sketched below. Find the area of the shaded region. [4] 3. Given that i and j represent two perpendicular unit vectors, (a) Find the magnitude of the vector −8i + 15j [2] (b) Find the angle θ between the vector −8i + 15j and the vector j, giving your answer in degrees to 2 decimal places. [4] 4. In 2005 there were 20 puffins on Lundy island. Counting t = 0 as the year 2005, after t years the number of puffins on the island, P , is modelled by the equation dP dt = 0.2P (a) Find an equation for P in terms of t. [5] (b) Hence estimate, to the nearest hundred, how many puffins there will be on Lundy island in the year 2030. [2] (c) Is this model suitable for predicting how many puffins will be on the island in the year 2100? Give a reason for your answer. [1] 5. Let w = 5e3πi/4 and z = 4eπi/2. (a) State the value of |wz|. [1] (b) State the value of arg (w z ) [1] The complex number v satisfies v2 = z, with z as above. (c) Write down the possible values of v in the form reiθ with r > 0 and −π < θ ≤ π. [2] 2