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ENG1065/5/LSA/20/21 (1 hand-out) 1 a) 1 7 3 Given matrices 𝑨 = (−3 −5) , 𝑩 = (−5 −4 2 6 i) ii) 0 5 2 4 −1) and 𝑪 = (3 −1 2 ): 4 1 4 −3 Calculate 𝟐𝑨 + 𝟒𝑩 − 𝑪𝑨. [4 marks] −5 Calculate |𝑪| and also the product 𝑪𝑫 where 𝑫 = ( 11 13 What then are the inverse 𝑪−𝟏 and 𝒂𝒅𝒋 𝑪? 22 8 −19 2 ). −18 −11 [5 marks] b) Consider the following system of equations: −𝑥1 + 2𝑥2 + 3𝑥3 = 7 2𝑥1 + 3𝑥2 − 𝑥3 = −10 3𝑥1 − 2𝑥2 + 2𝑥3 = 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 −11𝑥1 + 𝑥2 + 18𝑥3 = 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(𝑥, 𝑦) = 𝑥 3 𝑦 cos(𝑥𝑦 2 ) − ln(𝑥 2 𝑦 3 ) [6 marks] b) Find all the stationary points of the function 𝑓(𝑥, 𝑦) = 2𝑥 3 + 6𝑥𝑦 2 − 3𝑦 3 − 150𝑥 and classify them. [9 marks] c) A lamina occupies the region of the first quadrant of 𝑥 − 𝑦 plane bounded on all sides by the line 𝑦 = 2𝑥 , the curve 𝑦 = 6 − 𝑥 and the line 𝑥 = 1. Sketch the region, then find the mass and the 𝑥 coordinate 𝑥̅ of the centre of mass of the lamina, if the mass per unit area is 𝑚𝑎 (𝑥, 𝑦) = 𝑥 2 + 𝑦. [12 marks] Continued next page ENG1065/5/LSA/20/21 (1 hand-out) 3. a) The function 𝑓(𝑥, 𝑦, 𝑧) is given by 𝑓(𝑥, 𝑦, 𝑧) = 2𝑥 2 𝑦𝑧 2 − 3𝑦𝑧 + 4𝑥 3 + 𝑥𝑦 2 . Find the expression for ∇𝑓. Hence find the directional derivative of the function 𝑓(𝑥, 𝑦, 𝑧) at P(−1,1,2), in the direction of the vector 𝒖 = 3𝒊̂ + 4𝒋̂. Find maximum rate of change of 𝑓(𝑥, 𝑦, 𝑧) at P(−1,1,2). In which direction is this? [6 marks] b) Find the divergence and curl of the vector 𝒗 where ̂, 𝒗 = (𝑥 2 𝑦 3 𝑧 + 𝑦) 𝒊̂ + (𝑥𝑧 − 3𝑦) 𝒋̂ + (2𝑥𝑦𝑧 − 4𝑥) 𝒌 and evaluate them at the point P (1,-1, -1). Show that for any vector 𝒗, ∇ ⋅ (∇ × 𝒗) = 0 c) 4. a) [6 marks] [3 marks] Solve the following first-order differential equations, subject to any initial conditions if given: i) 𝑡 2 + 1 𝑑𝑥 = 3𝑡 + 1 𝑥 𝑑𝑡 [4 marks] ii) 𝑡𝑥 𝑑𝑥 = 2𝑡 2 + 3𝑥 2 , 𝑑𝑡 𝑥(1) = −2 [5 marks] iii) 𝑑𝑥 + 𝑥 tan 𝑡 = sin 2𝑡 , 𝑑𝑡 𝑥(0) = 1 [5 marks] b) Solve the differential equation 𝑑2𝑥 𝑑𝑥 − 3 − 4𝑥 = 𝑡 + 2𝑒 −𝑡 𝑑𝑡 2 𝑑𝑡 subject to the initial conditions 𝑥(0) = 35 , 16 𝑑𝑥 47 (0) = 𝑑𝑡 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 = (−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 𝑑𝑡 𝑑𝑦2 = −5𝑦1 + 2𝑦2 + 6𝑦3 𝑑𝑡 𝑑𝑦3 = 5𝑦1 + 4𝑦2 + 2𝑦3 𝑑𝑡 subject to the initial conditions 𝑦1 (0) = 8, 𝑦2 (0) = −2, 𝑦3 (0) = 6. 6. [6 marks] The time-dependent temperature 𝑇(𝑥, 𝑡) in a one-dimensional rod is described by the following equation: 𝜕𝑇 𝜕 2𝑇 =4 2 𝜕𝑡 𝑑𝑥 The temperature at either end of the rod is given by 𝑇(0, 𝑡) = 0 and 𝑇(5, 𝑡) = 0. At this stage the initial temperature of the rod is as yet unspecified. a) Assuming trial solutions of the form 𝑛𝜋𝑥 𝑇𝑛 (𝑥, 𝑡) = 𝜃𝑛 (𝑡) sin ( ), 𝑎 i) 𝑛 = 1,2,3 … . . What is the maximum value that 𝑎 can take in the trial solutions to ensure the boundary conditions are satisfied? ii) Using this maximum value of 𝑎, find the general solution to the partial differential equation. b) [9 marks] If the initial temperature in the bar is specified as 𝑇(𝑥, 0) = 3 sin 2𝜋𝑥 + 4 sin 𝜋 𝑥 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] 𝜕𝑇 (0, 𝑡) = 0, 𝜕𝑥 𝜕𝑇 (5, 𝑡) = 0 𝜕𝑥 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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