Sample Question Paper for 9210-100 Graduate Diploma in Engineering Engineering mathematics Duration: three hours You should have the following for this examination • one answer book • drawing instruments • non-programmable calculator The following data are attached • percentage points of the chi-squared distribution • percentage points of the t-distribution • table of the standard Normal probability distribution General instructions • This paper consists of nine questions in three sections A, B and C. • Answer five questions, at least one from each section. • An electronic calculator may be used but candidates must show sufficient steps to justify their answers. • Drawings should be clear, in good proportion and in pencil. • All questions carry equal marks. The maximum marks for each section within a question are shown. © The City and Guilds of London Institute Section A 1 a Function u(x, y) is such that w 2u w 2u wx 2 wy 2 0. (4 marks) If x s 2 t 2 and y 2 st , find b w 2u w 2 u . ws 2 wt 2 The cost C ( x, y ) of making a single unit of product in a production facility is given by C ( x, y ) 8 x 3 12 xy 3 y 2 where x and y are the material and labor costs respectively, required to make a single unit. c i Locate the stationary points of C (4 marks) ii Determine their nature. (4 marks) A plate with the measurements shown in Figure Q1c is to be constructed so that the outer perimeter has a length of 30 cm. Use the Lagrange multiplier method to find lengths for x, y and z in order that the area of the plate is the maximum possible. x y y z z z z Figure Q1c 2 (8 marks) 2 a F and a Fscalar function M it is given that For a vector For afunction vector function .(MF ) (M ).F M.F 3 where r xi yj zk i Show i that S . r ii Using above equations show that ii the Using thetwo above t .(r r ) 2 ³³∫∫ 5r 2 (2 marks)(2 marks) (6 marks)(6 marks) 2 2 ((r 2 r))..nndsds, , iii Hence Hence evaluate iii evaluate Hence evaluate ss (6 marks)(6 marks) where Swhere is the surface a cylinder S is the on surface on a x2 + y2 = Ǡ2, 0 ч z ч Ǡ . b If EbandIfH are respectively the electrical and magnetic field vectors in a charge-free, current-free electro-magnetic field in free it isthen known theythat satisfy current-free electro-magnetic fieldspace, in freethen space, it isthat known theythe satisfy the equations, equations, .E 0 .H 0 xE xH 1 wE c wt 1 wH c wt where c is the velocity of light in free space. Show that wH wt i xi ii x ii xE 1 w2E and c wt 2 1 w2E c 2 wt 2 (3 marks)(3 marks) (3 marks)(3 marks) 2 3 See next page a i State the Cauchy-Riemann equations for an analytic function f(z) (2 marks) where u ( x, y ) jv ( x, y ) , z f ( z) ii Given that u ( x, y ) x jy x sin( x ) cosh(Dy ) y cos( x) sinh( y ) (2 marks) x cos( x) sinh( y ) y sin( x ) cosh( y ) v ( x, y ) Find D such that f ( z) u ( x, y ) jv ( x, y ) is an analytic function in the z-plane. iii By assigning the values of D obtained in part ii above to u ( x, y ) , show that f ( z ) can be expressed as f ( z ) [Note: cosh( y ) b (4 marks) z sin( z ) . cos( jy ) and j sinh( y ) sin( jy ) ] C is the closed curve obtained by joining points ( -1, 0) and (1, 0) along the x- axis with the semi circle z 1 and with y > 0. Given z x jy and with integration carried out in the counter-clock wise sense, z sin( z ) dz . 4 1) f x sin( x ) dx . ii Hence evaluate ∫³ f ( x 4 1) i evaluate ³∫c ( z (6 marks) C 8 3 (6 marks) 4 Calculate the Laplace Transform of sin 2 t . (2 marks) b Calculate the inverse Laplace Transform of (6 marks) 4 a 3 2 (S 1)(S 2 9) c Determine the Laplace Transform L{u (t )} of the unit step function u (t ) with (2 (2marks) marks) u (t ) 1 , t t 0 and u (t ) 0 , t 0. d In the flight of a helicopter, the pitch angle T is controlled by adjusting the rotor angle G , where T and G satisfy the differential equation, with T 5 a b d 2T dT 0.4 6G 2 dt dt dT 0 and 0 , at t 0 . Also taking G dt i obtain the equation satisfied by I , where I ii Hence determine T as a function of t . L{T (t )} . (5 marks) (5 marks) Solve using the Z Transform method, the difference equation yn 2 4 yn 0 , y0 1 , y1 0 (6 marks) Obtain the half range Fourier sine series expansion for (6 marks) 4 x (S x) f ( x) c u (t ) , as defined in part c, S2 , 0d x dS . The temperature T ( x, t ) of a rod of length S at a point distance x from one end at time t, satisfies the differential equation with T (0, t ) i dT dt d 2T , 0 d x d S , t t 0, dx 2 0 , and T (S , t ) 0 , for all t t 0 . c2 Show by the use of the variables separable method that the solution for ƍ is T ( x, t ) f ¦e c 2 n 2t (4 marks) Bn sin( nx ) n 1 ii By taking T ( x,0) f ( x ) as defined in part b find the solution for T ( x, t ) . 5 (4 marks) See next page Section B x1 6 a i2 i1 K1 R2 M M V R1R x3 x2 M1 i3 K2 R3 Figure Q6a Figure Q6a Figure Q6c Figure Q6c Circular currents together with the resistances and voltage in the circuit shown s in Figure Q6a satisfy the equations, R1 (i1 i2 ) V R1 (i2 i1 ) R2 (i2 i3 ) R2 (i3 i2 ) R3i3 0 0 By taking R1 = 1 ƺ, R2 = 4 ƺ, R3 = 2 ƺ and V = 1 volt, write down i (2 marks) the system of equations as Ai b , where iT [i1 , i2 , i3 ] . and with the matrix A , having positive diagonal terms. ii b Use a matrix factorization method to evaluate the currents i1, i2 and i3. Starting ffrom point (1, 1) obtain the next iteration point in a search for (6 marks) (6 marks) the minimum point of the function 2( x 1) 2 ( y 1) 2 xy 2 by use of the steepest gradient method. c Two masses masses each each of of mass mass M coupled with two springs of spring constants Two K1 and K2 can move on a smooth trolley of mass M1 which also can move on a smooth horizontal table. If the trolley and the two masses have displacements x1, x2 and x3 as show n in Figure Q6c, then it is known displacements that the displacements satisfy the equations of motion. M1x1 K1 ( x2 x1 ) M x2 K1 ( x2 x1 ) K 2 ( x3 x2 ) M1 Ta Taking 2, M M x3 K 2 ( x3 x2 ) 1 , K1 K 2 1 and xi Z 2 xi , i = 1, 2, 3, it can be shown that the system of equations can be written as Ax Z2x , 6 (6 marks) marks) (6 1 §1 ¨ 2 ¨2 where A ¨ 1 2 ¨ 0 1 ¨ © xT [ x1 , · 0¸ ¸ 1¸ 1 ¸¸ ¹ x 2 , x3 ] and Z is the frequency of oscillations of the system. Given [ x ( 0) ]T [1 0 1] , perform two iterations to determine the maximum frequency of vibrations Z . 7 See next page Section C 7 a Liquid is poured into a cylindrical tank of uniform cross section A at a rate Q. The tank has an orifice at the base, causing the level of liquid in the tank x(t) to satisfy the differential equation dx dt 1 1 [Q D x 2 ] A where D is a constant. Taking Q = 0.3, A = 1, and D = 0.01, with x(0) = 0.5. Determine x i at t = 0.1, 0.2, by the use of the Euler Method, (6 marks) ii at t = 0.1, by the use of the second order Runge-Kutta (RK2) method. (6 marks) [For the equation dx dt f (t , x) , at (t0 , x0 ) , the RK2 method is given by k1 h f (t 0 , x 0 ), k2 h f (t 0 h , x 0 k1 ), t1 x1 b t0 h, x 0 0 . 5 * ( k1 k 2 )] The temperature u ( x, t ) in a rod at distance x along the rod and time t satisfies the differential equation wu wt w 2u , 1 d x d 1, t t 0 wx 2 with, U(X, 0) = 3( 1 - | x | ) 1 d x d 1 , and where the two ends of the rod are kept at u ( r1, t ) Find, u ( xi , t j ) for xi i 1 1 i, t j j, 3 27 0, r 1, r 2 j 1, 2, 3 by the use of a suitable scheme. 8 0, t t 0 (8 marks) 8 a A factory has utilized two machines A and B to produce pistons for engines with an intended diameter of 10.00 cm. A sample of pistons produced by each machine gives the results in Table Q8a. Machine Number Mean Standard of Diameter Deviation Items (cm) (cm) A 9 10.02 0.02 B 9 10.01 0.01 Table Q8a i Examine statistically whether there is a significant difference between the (6 marks) diameters of the pistons produced by the two machines ; [Assume that diameters of items produced both machines are normally distributed with equal variances.] ii Find the mean and standard deviation of the 18 items obtained by combining (2 marks) the two samples. iii The factory also produces cylinders with internal diameter 10.03 cm and (6 marks) standard deviation 0.02 cm. Calculate the percentage of pistons that can fit to the cylinders. Assume that diameters of cylinders and also of pistons are normally distributed and the mean and standard deviation of diameters of the pistons are those found in part ii. b The effort in person-months(E) required to complete a number of software development projects with Lines of Code in units of 1000 (KLOC) is shown below in Table Q8b. KLOC (x) 1.0 1.4 2.1 2.5 3.1 E (y) 0.6 2.0 2.4 2.8 3.2 Table Q8b 9 See next page Given that x 2.0 y 2.2 ¦ (x i x)( yi y ) ¦ (x i x) 2 2.828 y)2 4. 0 ¦(y i 3.14 i Find the correlation coefficient between x and y values. ii If the equation y (4 marks) a bx , has been obtained by the use of (2 marks) Least Squares Curve Fitting to the set of results for x and y, and given that a 0.043 , find the value of b. 9 a A machine in a production unit has to be shut down in the event of a mechanical failure or an electrical failure. These occur at an average of 1 mechanical failure and 2 electrical failures per month. Given that the failures are described by Poisson distributions, find the probabilities that in a month there will be, i no shut downs (3 marks) ii at least one shut down. (3 marks) iii If each shut down costs the company 10 000 units of currency, estimate the average monthly cost incurred by the shut downs. (4 marks) b Daily conditions regarding rainfall at a hydroelectric power station has been (4 marks) classified as dry, showery and heavy rain and the Table Q9b summarises the changes in conditions over a period of 50 days. Current day status Following day Status Dry Showery Heavy Rain Dry 14 4 2 Showery 6 12 2 Heavy Rain 2 6 2 Table Q9b Determine the probability transition matrix for a Markov chain model of the weather conditions. c Using the results from part (b) and given that the current conditions are showery, find the probabilities of having different conditions after i 1 day ii 2 days. (3 marks) (3 marks) 10 Data Attachments 9210-100

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