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ENG1065/5/LSA/20/21 (1 hand-out) 1 a) 1 7 3 Given matrices 𝑨 = (βˆ’3 βˆ’5) , 𝑩 = (βˆ’5 βˆ’4 2 6 i) ii) 0 5 2 4 βˆ’1) and π‘ͺ = (3 βˆ’1 2 ): 4 1 4 βˆ’3 Calculate πŸπ‘¨ + πŸ’π‘© βˆ’ π‘ͺ𝑨. [4 marks] βˆ’5 Calculate |π‘ͺ| and also the product π‘ͺ𝑫 where 𝑫 = ( 11 13 What then are the inverse π‘ͺβˆ’πŸ and 𝒂𝒅𝒋 π‘ͺ? 22 8 βˆ’19 2 ). βˆ’18 βˆ’11 [5 marks] b) Consider the following system of equations: βˆ’π‘₯1 + 2π‘₯2 + 3π‘₯3 = 7 2π‘₯1 + 3π‘₯2 βˆ’ π‘₯3 = βˆ’10 3π‘₯1 βˆ’ 2π‘₯2 + 2π‘₯3 = 22 i) Write this system of equations in matrix form and, if possible, solve it by Cramer’s rule. [5 marks] ii) If the third equation is replaced by βˆ’11π‘₯1 + π‘₯2 + 18π‘₯3 = 65 can the system be solved using Cramer’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 𝑓(π‘₯, 𝑦) = π‘₯ 3 𝑦 cos(π‘₯𝑦 2 ) βˆ’ ln(π‘₯ 2 𝑦 3 ) [6 marks] b) Find all the stationary points of the function 𝑓(π‘₯, 𝑦) = 2π‘₯ 3 + 6π‘₯𝑦 2 βˆ’ 3𝑦 3 βˆ’ 150π‘₯ and classify them. [9 marks] c) A lamina occupies the region of the first quadrant of π‘₯ βˆ’ 𝑦 plane bounded on all sides by the line 𝑦 = 2π‘₯ , the curve 𝑦 = 6 βˆ’ 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‘šπ‘Ž (π‘₯, 𝑦) = π‘₯ 2 + 𝑦. [12 marks] Continued next page ENG1065/5/LSA/20/21 (1 hand-out) 3. a) The function 𝑓(π‘₯, 𝑦, 𝑧) is given by 𝑓(π‘₯, 𝑦, 𝑧) = 2π‘₯ 2 𝑦𝑧 2 βˆ’ 3𝑦𝑧 + 4π‘₯ 3 + π‘₯𝑦 2 . Find the expression for βˆ‡π‘“. Hence find the directional derivative of the function 𝑓(π‘₯, 𝑦, 𝑧) at P(βˆ’1,1,2), in the direction of the vector 𝒖 = 3π’ŠΜ‚ + 4𝒋̂. Find maximum rate of change of 𝑓(π‘₯, 𝑦, 𝑧) at P(βˆ’1,1,2). In which direction is this? [6 marks] b) Find the divergence and curl of the vector 𝒗 where Μ‚, 𝒗 = (π‘₯ 2 𝑦 3 𝑧 + 𝑦) π’ŠΜ‚ + (π‘₯𝑧 βˆ’ 3𝑦) 𝒋̂ + (2π‘₯𝑦𝑧 βˆ’ 4π‘₯) π’Œ and evaluate them at the point P (1,-1, -1). Show that for any vector 𝒗, βˆ‡ β‹… (βˆ‡ Γ— 𝒗) = 0 c) 4. a) [6 marks] [3 marks] Solve the following first-order differential equations, subject to any initial conditions if given: i) 𝑑 2 + 1 𝑑π‘₯ = 3𝑑 + 1 π‘₯ 𝑑𝑑 [4 marks] ii) 𝑑π‘₯ 𝑑π‘₯ = 2𝑑 2 + 3π‘₯ 2 , 𝑑𝑑 π‘₯(1) = βˆ’2 [5 marks] iii) 𝑑π‘₯ + π‘₯ tan 𝑑 = sin 2𝑑 , 𝑑𝑑 π‘₯(0) = 1 [5 marks] b) Solve the differential equation 𝑑2π‘₯ 𝑑π‘₯ βˆ’ 3 βˆ’ 4π‘₯ = 𝑑 + 2𝑒 βˆ’π‘‘ 𝑑𝑑 2 𝑑𝑑 subject to the initial conditions π‘₯(0) = 35 , 16 𝑑π‘₯ 47 (0) = 𝑑𝑑 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 𝐴 = (βˆ’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. 𝑑𝑦1 = 8𝑦1 + 4𝑦2 𝑑𝑑 𝑑𝑦2 = βˆ’5𝑦1 + 2𝑦2 + 6𝑦3 𝑑𝑑 𝑑𝑦3 = 5𝑦1 + 4𝑦2 + 2𝑦3 𝑑𝑑 subject to the initial conditions 𝑦1 (0) = 8, 𝑦2 (0) = βˆ’2, 𝑦3 (0) = 6. 6. [6 marks] The time-dependent temperature 𝑇(π‘₯, 𝑑) in a one-dimensional rod is described by the following equation: πœ•π‘‡ πœ• 2𝑇 =4 2 πœ•π‘‘ 𝑑π‘₯ The temperature at either end of the rod is given by 𝑇(0, 𝑑) = 0 and 𝑇(5, 𝑑) = 0. At this stage the initial temperature of the rod is as yet unspecified. a) Assuming trial solutions of the form π‘›πœ‹π‘₯ 𝑇𝑛 (π‘₯, 𝑑) = πœƒπ‘› (𝑑) sin ( ), π‘Ž i) 𝑛 = 1,2,3 … . . What is the maximum value that π‘Ž can take in the trial solutions to ensure the boundary conditions are satisfied? ii) Using this maximum value of π‘Ž, find the general solution to the partial differential equation. b) [9 marks] If the initial temperature in the bar is specified as 𝑇(π‘₯, 0) = 3 sin 2πœ‹π‘₯ + 4 sin πœ‹ 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œ•π‘‡ (0, 𝑑) = 0, πœ•π‘₯ πœ•π‘‡ (5, 𝑑) = 0 πœ•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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