IMAT2202 2017 Exam PDF

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Exam Paper for IMAT2202 2017...

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Sheet 1 of 7 EXAMINATION PAPER Code: IMAT2202 Session - 2016/2017 Faculty of Technology Module Code – IMAT2202 Module Title – Further Mathematical Methods Date - Wednesday 3rd May 2017 Time Allowed – 2 hours Start 14:00 Finish 16:00 _________________________________________________________ Instruction and information for candidates: Restricted Open book: During this test you are allowed to refer to your formula/information sheet. This can be no more than 4 sides of A4 and must be handed in with your test paper at the end of the test. Tables provided: Laplace and Z transform tables are provided at the end of this exam paper. Answer all 4 questions. Each question is worth 25 marks and there are 100 marks in total.

Total marks achievable = 100 Programmable calculators are permitted during this examination provided they are ‘reset’ using the reset button found on the underneath of some calculators, ‘cancelled’ (by battery removal) or otherwise checked and proved not to carry textual information, or formulae required by the examination, other than normal scientific/statistical functions CANDIDATES MUST NOT REMOVE THIS EXAMINATION PAPER FROM THE EXAMINATION ROOM BUT MUST HAND IT TO THE INVIGILATOR AT THE END OF THE EXAMINATION

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Sheet 2 of 7 Code: IMAT2202

Q1. a) (i) Find all the first and second order derivatives of the function, f(x,y) = xyey (5 marks) (ii) Show that f (x, y) has only one critical point, and this is a saddle point. (3 marks) b) An open fish tank has length one and a half times the width. The tank costs £0.30/cm2 for the base glass and £0.60/cm2 for the high-strength glass of the sides There is a budget constraint on the glass cost of £150 per tank (i)The height of the tank is given by h cm, and the width by w cm. Show that the cost of the glass is given by 0.45w2 +3wh (3 marks) (ii) Using the method of Lagrange multipliers, find the dimensions of the tank for maximum volume under the budget constraint and the value of this volume (8 marks) Use the method of Lagrange multipliers in part (c) c) (i) The 3 real numbers x, y and z are subject to the linear constraint x2 + y2 + z2 = k,

k a real constant.

Find general formulas for x, y and z in terms of k which will maximise their product . (4 marks)

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Sheet 3 of 7 Code: IMAT2202 (ii) A cuboid is designed with a fixed space diagonal of 5 3 cm. Find the dimensions of the cuboid for maximum volume (2 marks) Q2. a) By reversing the order of the variables, find 4 2

∫∫ 0

3

e y dy dx

x

Express your answer in terms of e (8 marks) b) The force F = x2i + xy j + y2 k is applied over the curve r(t) = ati +btj + k

from t = 0 to t = 1 a, b, c > 0, all constants.

The work done by the force is (i)

80 3

Find an equation linking a and b (6 marks)

(ii)

The magnitude of r (0.5) is

5

Find the values of a and b (5 marks) c) f (x, y, z) = xy + yz + z2 (i)

The vector field V = grad(f)

Find V. (3 marks)

(ii)

Show that V is a conservative field (3 marks)

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Sheet 4 of 7 Code: IMAT2202 Q3. a) A function f(t) is defined on the closed interval (a,b). It satisfies the Dirichlet condidions and is piecewise continuous (i)

Explain the ‘Dirichlet conditions’ in terms of this function

(ii)

Define what is meant by ‘piecewise continuous’

(iii)

Define the Laplace Transform of function f (4 marks)

b) Use the definition and integration by parts twice to prove that 3 the Laplace Transform of t2 is given by 2/s (6 marks) c) Determine the Laplace Transform of the following functions. (i)

g(t) = 3e-t sinh(2t) + e-3tcosh(2t) (2 marks)

ii)

h(t) = (t + 2) cos(3t) (4 marks)

You may use the tables attached at the end of this paper d) Use Laplace Transforms to solve the differential equation:

d 2y dy 6 + + 13 y = 0, dt 2 dt

y (0) = 3,

dy (0) = 7 dt (9 marks)

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Sheet 5 of 7 Code: IMAT2202 Q4. a) Assuming that the sum exists, and z is a complex variable, Define the Z transform F(z) of a sequence f(k), where k is a Natural number (2 marks) b) Determine, using your definition, the Z transform of the sequence: f(k) = ak, where a is a constant (3 marks) c)

F(z) = 10z (z2 – 3z + 2)-1 Find f(k), the inverse Z transform of F(z) (5 marks)

d) Find the transfer function which represents the system modelled by the difference equation: y(k+2) + 3y(k+1) –y(k) = x(k) (4 marks) Assume zero initial conditions e) Solve the second-order difference equation by means of Z. transforms y(k+2) - 5y(k+1) + 6y(k) = 0, y(0) = 0, y(1) = 2 (11 marks)

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Sheet 6 of 7 Code: IMAT2202

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Sheet 7 of 7 Code: IMAT2202

Table of basic Z transforms...