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ENG1065/5/LSA/20/21 (1 hand-out) 1 a) 1 7 3 Given matrices π¨ = (β3 β5) , π© = (β5 β4 2 6 i) ii) 0 5 2 4 β1) and πͺ = (3 β1 2 ): 4 1 4 β3 Calculate ππ¨ + ππ© β πͺπ¨. [4 marks] β5 Calculate |πͺ| and also the product πͺπ« where π« = ( 11 13 What then are the inverse πͺβπ and πππ πͺ? 22 8 β19 2 ). β18 β11 [5 marks] b) Consider the following system of equations: βπ₯1 + 2π₯2 + 3π₯3 = 7 2π₯1 + 3π₯2 β π₯3 = β10 3π₯1 β 2π₯2 + 2π₯3 = 22 i) Write this system of equations in matrix form and, if possible, solve it by Cramerβs rule. [5 marks] ii) If the third equation is replaced by β11π₯1 + π₯2 + 18π₯3 = 65 can the system be solved using Cramerβs rule and if not, why not? Can you determine if any solutions exist, and if so, what they are? [8 marks] 2. a) Find all the second order partial derivatives of the function π(π₯, π¦) = π₯ 3 π¦ cos(π₯π¦ 2 ) β ln(π₯ 2 π¦ 3 ) [6 marks] b) Find all the stationary points of the function π(π₯, π¦) = 2π₯ 3 + 6π₯π¦ 2 β 3π¦ 3 β 150π₯ and classify them. [9 marks] c) A lamina occupies the region of the first quadrant of π₯ β π¦ plane bounded on all sides by the line π¦ = 2π₯ , the curve π¦ = 6 β π₯ and the line π₯ = 1. Sketch the region, then find the mass and the π₯ coordinate π₯Μ of the centre of mass of the lamina, if the mass per unit area is ππ (π₯, π¦) = π₯ 2 + π¦. [12 marks] Continued next page ENG1065/5/LSA/20/21 (1 hand-out) 3. a) The function π(π₯, π¦, π§) is given by π(π₯, π¦, π§) = 2π₯ 2 π¦π§ 2 β 3π¦π§ + 4π₯ 3 + π₯π¦ 2 . Find the expression for βπ. Hence find the directional derivative of the function π(π₯, π¦, π§) at P(β1,1,2), in the direction of the vector π = 3πΜ + 4πΜ. Find maximum rate of change of π(π₯, π¦, π§) at P(β1,1,2). In which direction is this? [6 marks] b) Find the divergence and curl of the vector π where Μ, π = (π₯ 2 π¦ 3 π§ + π¦) πΜ + (π₯π§ β 3π¦) πΜ + (2π₯π¦π§ β 4π₯) π and evaluate them at the point P (1,-1, -1). Show that for any vector π, β β (β Γ π) = 0 c) 4. a) [6 marks] [3 marks] Solve the following first-order differential equations, subject to any initial conditions if given: i) π‘ 2 + 1 ππ₯ = 3π‘ + 1 π₯ ππ‘ [4 marks] ii) π‘π₯ ππ₯ = 2π‘ 2 + 3π₯ 2 , ππ‘ π₯(1) = β2 [5 marks] iii) ππ₯ + π₯ tan π‘ = sin 2π‘ , ππ‘ π₯(0) = 1 [5 marks] b) Solve the differential equation π2π₯ ππ₯ β 3 β 4π₯ = π‘ + 2π βπ‘ ππ‘ 2 ππ‘ subject to the initial conditions π₯(0) = 35 , 16 ππ₯ 47 (0) = ππ‘ 20 [12 marks] Continued next page ENG1065/5/LSA/20/21 (1 hand-out) 5. a) b) 8 4 0 Find all the eigenvalues and eigenvectors of the matrix π΄ = (β5 2 6). 5 4 2 [12 marks] Write the system of linear equations below in matrix form, and convert this to an eigenvalue problem. Hence solve the system of equations. ππ¦1 = 8π¦1 + 4π¦2 ππ‘ ππ¦2 = β5π¦1 + 2π¦2 + 6π¦3 ππ‘ ππ¦3 = 5π¦1 + 4π¦2 + 2π¦3 ππ‘ subject to the initial conditions π¦1 (0) = 8, π¦2 (0) = β2, π¦3 (0) = 6. 6. [6 marks] The time-dependent temperature π(π₯, π‘) in a one-dimensional rod is described by the following equation: ππ π 2π =4 2 ππ‘ ππ₯ The temperature at either end of the rod is given by π(0, π‘) = 0 and π(5, π‘) = 0. At this stage the initial temperature of the rod is as yet unspecified. a) Assuming trial solutions of the form πππ₯ ππ (π₯, π‘) = ππ (π‘) sin ( ), π i) π = 1,2,3 β¦ . . What is the maximum value that π can take in the trial solutions to ensure the boundary conditions are satisfied? ii) Using this maximum value of π, find the general solution to the partial differential equation. b) [9 marks] If the initial temperature in the bar is specified as π(π₯, 0) = 3 sin 2ππ₯ + 4 sin π π₯ 5 determine the complete solution to the problem. c) [3 marks] [4 marks] To what physically would the boundary conditions below correspond, and what form do you think you would have to assume for the trial solution in this case? [3 marks] ππ (0, π‘) = 0, ππ₯ ππ (5, π‘) = 0 ππ₯ Continued next page ENG1065/5/LSA/20/21 (1 hand-out) 7. a) b) c) A biased coin which lands on heads 40% of the time is tossed five times. i) What is the probability of tossing exactly 2 heads? [3 marks] ii) What is the probability of tossing at least 2 heads? [3 marks] People arrive at the queue for a cash machine according to a Poisson distribution with mean arrival rate of 2 per minute. i) What is the probability that exactly 4 people arrive in a given minute? [2 marks] ii) What is the probability that more than 2 people arrive in a given minute? [3 marks] iii) What is the probability that exactly 7 people arrive in a 5 minute period? [3 marks] The weights of Choci chocolate bars are distributed normally with a mean 200 g and a standard deviation of 0.5 g. i) What is the probability that the weight of a randomly selected chocolate bar will be more than 201 g? [2 marks] ii) What is the probability that a randomly selected bar will have a weight of less than 198g? [3 marks] iii) If to comply with weights and measures legislation, 99% of the bars must weigh more than 199 g , and the process for producing the bars means that /the standard deviation is fixed at 0.5 g, to what minimum value should the mean be set? [4 marks] Internal Examiner: Dr N Rockliff

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