University of Bristol Mathematics Exam Practice Paper PDF

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 mar...

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. w1∗ w2 w1 − 3w2 ii. w2 (b) Write down a quadratic equation with real coefficients that has w1 as a root. 2. The graph of the function y = x2 − 2x − 3 is sketched below. Find the area of the shaded region. 3. Given that i and j represent two perpendicular unit vectors, (a) Find the magnitude of the vector −8i + 15j (b) Find the angle θ between the vector −8i + 15j and the vector j, giving your answer in degrees to 2 decimal places. 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 = 0.2P dt (a) Find an equation for P in terms of t. (b) Hence estimate, to the nearest hundred, how many puffins there will be on Lundy island in the year 2030. (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. 5. Let w = 5e3πi/4 and z = 4eπi/2. (a) State the value of |wz|. w (b) State the value of arg z The complex number v satisfies v 2 = 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  6. The matrices A and B satisfy the equation AB = I − 2A   5 −4 where I is the identity matrix and B =. −3 0 Find A. 7. It is given that x and y are related by the ordinary differential equation dy 3y =8− dx 1+x and that when x = 0, y = −14. Find the value of y when x = 1.    2 −1 8. Find a vector that is perpendicular to both −4 and −3. 5 1 9. A complex number z has modulus 1 and argument θ. (a) Show that 1 zn + = 2 cos (nθ) zn for any integer n. 1 (b) Hence show that cos5 θ = (cos 5θ + 5 cos 3θ + 10 cos θ) 16 Z p 10. (a) Find 9x5 x3 + 2 dx Z ln 4 (b) Find 2ue2u du, giving your answer in the form a ln 2 + b, where a and b are constants to be found. ln 2 11. Consider the simultaneous equations 3x + 4y = 6 −x + 4y = 19 (a) Write these equations as a matrix equation Ax = b.   6 4 (b) Find the determinant of the matrix. 19 4 (c) Use Cramer’s rule to solve the simultaneous equations. You must show detailed reasoning. 3  √ 12. The sketch below shows the curves y = A − 2 x and y = x2 − B for x in the range 0 ≤ x ≤ a, where A, B, and a are all positive constants. The sketch is not drawn to scale. Both curves meet the x-axis at x = a. (a) Find the area of the finite region bounded by the y-axis and the two curves, giving your answer in terms of A. The area of this region is 504. (b) Find the value of A. (c) Hence find the values of B and a. 13. The points A(4, 5, −1), B(4, −5, 8), C(6, 1, −1), and D(6, 11, −10) are the four corners of a parallelogram. Find the area of the parallelogram ABCD. 14. (a) Sketch an Argand diagram showing the locus of points satisfying the equation |z + 2| = 2. (b) Given that there is a unique complex number w that satisfies both |w + 2| = 2 and 3π arg(w − k) = , 4 where k is a positive real number, i. Find the value of k. ii. Express w in the form r (cos θ + i sin θ), giving r and θ to 2 significant figures. 4 15. A particle moves around a fixed point O with damped harmonic motion. The displacement in metres of the particle from O at time t seconds is denoted by x, where x satisfies the equation d2 x dx +6 + 9x = 0 dt2 dt (a) Find the general solution to this differential equation. When t = 0 the particle is a distance of 15 m from O and is moving with velocity 5 ms−1. (b) Find an expression for x in terms of t. (c) Briefly explain what happens to the particle as t increases. A second particle has displacement u from O at time t given by the equation d2 u du +6 + 9u = e−3t dt2 dt (d) Find the general solution to this equation, expressing u in terms of t. End of exam. 5