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Unformatted text preview: The University of Melbourne Department of Mathematics and Statistics 620—157 Accelerated Mathematics 1 Semester 1, 2010 Exam duration: Three hours Reading time: 15 minutes This paper has 4 pages. The total number of marks allocated is 90. Authorized Materials: No materials are authorized. Calculators and mathematical tables are not permitted. Candidates are reminded that no written or printed materials related to the subject may be brought into the examination. If you have any such materials in your possession, you should immediately surrender it to an invigilator. Instructions to Invigilators: One 14 page script book is to be given to each student initially. Students may retain this examination paper. No written or printed materials related to the subject may be brought into the examination. N0 mathematical tables or calculators may be used. Instructions to Students: This examination consists of 13 questions. All questions may be attempted, The number of marks for each question is indicated on the examination paper. Use of calculators is not allowed. Paper to be held by Baillieu Library; This paper may be reproduced and lodged with the Baillieu Library. Page J 0f4 Please turn over . . . 5. . The system of equations (k -|— 1):r — y + 32 = k 43: + Icy — z = —1 2x + z = k — 2 has reduced row echelon form % sis—1 For which values of 16 does the system of equations have (a) no solution; (b) one solution; (c) infinitely many solutions [4 marks] Consider the following two planes in R3. x+2y+2=5, x—y+z=~1. (a) Find the angle between the two planes. [3 marks] (b) Find the line of intersection between the two planes7 expressing it in vector form. [3 marks] . Solve the following equation for the 3 X 3 matrix B 1 —1 3 1 1 1 4 1 —1 B = 1 —1 1 2 O 1 1 0 0 [5 marks] . Use mathematical induction to prove that the following statement is true for all positive integers mi 2.1 + 3.2 + 4.22 + + (n + 1).?“1 = 72.2". [5 marks] (a) Find all solutions of z6 — 9223 — 8 = 0 expressing your answers in the form 2 2 :10 + iy E (C. [4 marks] (b) Prove 7r 1 1 —— = 2 arctan ~— — arctan E [3 marks] Page 2 0f4 Please turn over . A . 6. Let 1 2 8 3 3 0 1 0 2 —1 0 1 _ 1—1—1—3 2 5 _ 0 1 3 2 0 —2 A‘ 2 1 7 0 5 5 andB‘ 0 0 0 0 1 1 2 4 16 6 —1—7 0 0 0 0 0 0 You are given that the matrix B is the reduced row echelon form of the matrix A. Using this information, or otherwise, answer the following: (a) Write down a basis for the column space of A. (b) Are the vectors v1 2 (1,1,2,2), '02 = (2, —1, 1,4), 123 = (8, ~1,7, 16) linearly independent? if not, express 113 as a linear combination of U1 and 122. (c) Write down a basis for the row space of A. ((1) Find a basis for the solution space of A. (e) Write down the dimension of the solution space of A. ( f) Verify the Rank—Nullity theorem for the matrix A. (g) Do the columns of A span R4? Explain your answer. [11 marks] 7. Find all solutions in Z; of the following linear system. $1+x2+x3+a34+$5 -— 0 201 + 932 + 964 — 1 x1+x2+ass+rcs = 1 Express your answer as a finite list. [6 marks] 8. (a) Use the method of least squares to find the line of best fit y = a + by: to the data (1,2), (2,1), (—1,3), (3,»1), (0,0) and plot the data points together with your straight line. [6 marks] (b) Find a non—zero vector in 732 2 space of polynomials of degree 3 2, orthogonal to p(a:) = :1: with respect to the inner product [2 marks] Page 3 0f4 Please turn over . . . 9. Let T : R3 —> R3 be orthogonal projection onto the plane :6 + y + z = O. (a) Show that the closest point to (1, 0, 0) on the plane x+y +2 2 0 is (g, —%, ——%) 1 2 — 1 —1 (b) Show that the standard matrix for T is A = — —1 2 —1 —1 — 1 2 (c) Find the matrix M for T with respect to the basis B = {(1,—1,0), (0,1, —1), (1, 1.1)}. (d) Find a 3 X 3 invertible matrix P such that M = P“1AP. [2+3+3+1 marks] 10. Let V = {10le a 1R |p(:r) = p(a: + 1)7 all derivatives p’(:r), p”(:z:), exist } (a) Show that V is a vector space. (b) Show that Tp(x) = p’(:c) defines a linear transformation from V to itself. (c) Calculate kerT. ( d) Calculate im T. [2+3+1+2 marks] 11. Consider the matrix C _ —1 ——1 F 1 3/2 ” (a Find all eigenvalues and corresponding eigenvectors for the matrix C. ) (b) Find an invertible matrix P and a diagonal matrix D such that C : PDP‘l. (c) Use your results from (a) to find a formula for C” valid for each integer n 2 1. ) (d Describe the limiting behaviour of C" as n —> oo. [4+2+1+2 marks] 12. (a) Write the cartesian equation for the tangent plane of the hyperboloid 562 + y2 — 3.22 = 1 at the point (2,0,1). [5 marks] (b) Find the intersection of the hyperboloid 3:2 -|— y2 — 322 z 1 with its tangent plane at the point (2,0,1). [2 marks] 13. Approximate the length of the hypotenuse of a right—angle triangle with perpendic— ular sides of lengths 3.01cm and 3.990m. [5 marks] END OF EXAMINATION PAPER Page 4 of] ...
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