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A past examination paper from the math122 course at the university of cambridge. It covers various topics in mathematics including differential equations, poisson processes, and vector calculus. Students are required to solve problems related to borrowing and loan repayments, cooling bodies, traffic flow, population dynamics, and harmonic oscillators.
Typology: Exams
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TIME ALLOWED: Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
You may attempt all questions. All answers to Section A and the best THREE answers to
questions from Section B will be taken into account. Section A carries 55% of the available marks.
What is the interest rate charged by the Building Society and how much does he repay each year? What is the equilibrium solution, N , of this equation? If the house initially costs £92,040 calculate how long it will take to pay off the mortgage, and the total amount the borrower will repay. [8 marks]
At and. Obtain and solve the differential equation for y in terms of_._ From your results draw the phase diagram indicating which way around the curve the point moves. [8 marks]
write:
Briefly explain what this equation represents. Derive an equation for the rate of change of . [5 mark] A disease infects people in such a way that if they feel ill on one day, the probability that they feel ill on the next day is. If however they feel well on one day the probability they feel well the next day is. Consider this as a two-state process and write down the equation for the rate of change of in terms of and the probabilities of feeling well and ill on day respectively. Show that
Given that , solve this equation and find the long-term value of. [10 marks]
where is the local mouse population, and time t is measured in years. The cats prey on the mice and the mice population is thought to satisfy
Derive the equation for and hence find , given that at and. [7 marks] Sketch the graph of against , indicating the direction of and as time increases. Find the maximum of the mice population and the corresponding number of cats living in the neighbourhood. When , describe what happens to both populations. Is this realistic? [8 marks]
Find the equilibrium values of and determine their stability. Sketch the graph of against , and hence describe the way that varies for various initial populations. [7 Marks] Show, by the method of partial fractions, or otherwise, that
where A is a constant. Given that when , show that
[8 marks]
12. The lookout on a large ship travelling East at 20 km/h observes a speedboat some 500 metres due North of his position. It appears to be travelling in the direction which is 30° West from the direction to the South (third quadrant, 240°), at a speed of 40 km/h. By using vector methods, find the actual velocity and speed of the speedboat. [6 marks] At time t find the position vector of the speedboat with respect to the large ship. [3 marks] Also find the closest distance between the speedboat and the ship. [6 marks] [Hint: Use unit vectors i and j to represent East and North respectively.]
Find the potential energy for a particle that moves in a force field F (^) 2 rˆ r
. How much
work is done by the force field in moving the particle from a point on the circle r a 0 to a point on the circle r b 0? Does the work done depend on the path? Explain. [7 marks]
