Australian National University Student Number: u Mathematical Sciences Institute EXAMINATION: Semester 1 — Mid Exam, 2022 MATH1014: Mathematics and Applications 2, Semester 1 Exam Duration: 120 minutes. Reading Time: 15 minutes. Materials Permitted In The Exam Venue: Unmarked English-to-foreign-language dictionary (no approval required). One A4 page with hand written notes on both sides. Materials To Be Supplied To Students: None Instructions To Students: You must justify your answers and show your work. Please be neat. Q1 10 Q2 10 Q3 10 Q4 10 Q5 10 Q6 20 Q7 10 Total / 80 Question 1 10 marks (a) Determine the angle between the following vectors in R3 : v1 = 2 1 3 , v2 = 4 1 3 . (1) (5 marks) (b) Consider the plane inside R3 with normal vector n = 1 1 1 which passes through the origin, and the line L() = 2 0 4 + 3 1 2 , where ∈ R . Find the point of intersection, or determine that they do not intersect. (5 marks) Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 2 of 8 Question 2 10 marks Recall that the space 2×2(R) of 2 × 2 matrices with real number entries forms a vector space. Determine whether or not the set = {[ 1 0 0 1 ] , [ 1 0 0 1 ] , [ 0 1 0 0 ] , [ 0 0 1 0 ]} (2) is a basis for 2×2(R) . Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 3 of 8 Question 3 10 marks The sets 1 = {1, , 2} and 2 = {1 + , 1 , 2} are both bases for the vector space P2 of polynomials with degree at most 2 . Construct the change of coordinates map 2←1 from the basis 1 to the basis 2 , and use it to write the coordinate vector [p()]2 , where p() = 32 + 2 4 . Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 4 of 8 Question 4 10 marks Consider the following parameter-dependent 3 × 3 matrix, where the parameter ∈ R : () = 2 8 4 1 2 0 0 3 . (3) (a) Determine the rank of the matrix . (6 marks) (b) Determine the dimension of the null space of , i.e. calculate dimnul() . (2 marks) (c) Is the matrix invertible Justify your answer. (2 marks) Your answers to the above questions may depend on the value of the parameter . Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 5 of 8 Question 5 10 marks Determine whether the following series converge or diverge. (a) ∞∑ =1 1√ 3 + 1 , (5 marks) (b) ∞∑ =1 (!)25 (2)! (5 marks) Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 6 of 8 Question 6 20 marks Consider the power series ln 2 + ∞∑ =1 2 Answer the following questions: (a) (i) Find the domain of convergence. (4 marks) (ii) For which does the series converge conditionally (2 marks) (b) For the values of determined in part (a), define () = ln 2 + ∞∑ =1 2 . Check that ′() = 12 . (7 marks) (c) Apply the fundamental theorem of calculus () (0) = ∫ 0 ′() to check that ln 23 = ∞∑ =1 ( 1) 2 . (7 marks) Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 7 of 8 Question 7 10 marks Sketch the polar curves = 2 cos and = 2 sin and find the area of the region where the interiors of the two polar curves overlap. Midsemester Exam, Semester 1, 2022 MATH1014 Mathematics and Applications 2, Page 8 of 8
