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# test 2 2011.pdf

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School
Western University
Department
Mathematics
Course
Mathematics 1229A/B
Professor
Vicki Olds
Semester
Fall

Description
Friday, November 4, 2011 CODE 111 Mathematics 1229A Page 1 Test 2 PART A (18 marks) NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE INDICATED ON THE SCANTRON SHEET. YOU SHOULD ALSO CIRCLE YOUR ANSWERS IN THIS BOOKLET. 1 1. Consider the following systems of equations in the variables x, y and z. mark (i) x2 − 2y + z = 5 (ii) 5 x + 3y = z x + y + z = 0 4x + 3z = 6 x y 4x − 5y = 7 2 − 3 = 9 (iii) x + 3y = 4z 5x − yz = 10 x + y + z = 3 Which of the above systems are linear in x, y and z? A: all of them B: (i) only C: (ii) only D: (iii) only E: none of them 1 2. Find the augmented matrix for the following system of linear equations. mark x1− 2x2+ x 3 = 0 4x1 = 5x 3 6 5 = x −1x 2 ▯ ▯ ▯ ▯ ▯ ▯ 1 −2 1 0 1 2 3 0 1 −2 1 0 A: 4 0 5 6 B: 4 −5 0 6 C: 4 0 −5 6 1 −1 0 −5 1 −1 0 5 −1 1 0 5 ▯ 1 −2 1 0 ▯ ▯ 1 −2 1 0▯ 4 0 −5 6 4 0 −5 0 D: 1 −1 0 5 E: 1 −1 0 0 1 3. Which of the following matrices is/are in row-reduced echelon form? mark     1 2 0 1 3 1 2 0 1 3 P =  0 0 1 2 4 Q =  0 1 0 2 4     0 0 0 0 0 0 0 1 0 5     0 2 0 1 3 1 1 0 0 0  1 1 0 2 4  0 0 1 0 0 R =   S =   0 0 1 0 5 0 0 0 1 1 A: P only B: P and R only C: R only D: P and S only E: S only ▯ ▯ a 1 0 1 4. Find all values of a and b so that the matrix is in row-reduced echelon form. mark 0 b 1 A: a = 1 and b = 0 only B: a = 0 and b = 1 only C: a = 1 and b = 0, or a = 0 and b D: a = b = 0 only E: a = 1 and b = 0, or a = 0 and b = 0 Mathematics 1229A CODE 111 Friday, November 4, 2011 Test 2 Page 2   1 5 3 1 5. Let A =  −1 −4 −1  and let B be the row-reduced echelon form of A. mark −2 −7 0 Then the ﬁrst row of B is A: [1 0 −7 ] B: [ 0 0 ] C: [ 0 7 ] D: [ 1 0 ] E: [ 0 1 ] 1 6. How many solutions does the system of linear equations represented by the following mark augmented matrix have?   1 −1 2 4 6 2  0 1 2 1 −1 −1     0 0 0 1 0 1  0 0 0 0 0 0 A: no solution B: exactly one solution C: a one parameter family of solutionD: a two parameter family of solutions E: a three parameter family of solutions 1 7. How many solutions does the system of linear equations represented by the following mark augmented matrix have?   3 −2 1 −2  1 −1 3 5    −1 1 1 −1 A: no solution B: exactly one solution C: a one parameter family of solutionD: a two parameter family of solutions E: a three parameter family of solutions In questions 8, 9 and 10 below, use the following augmented matrix, which represents a system of linear equations.   1 −2 0 5    0 1 3 −1  0 0 k − 1 k + 1 1 8. For what value(s) of k does the system of linear equations have exactly one solution? mark A: k = −1 only B: k = 1 only C: k = ±1 only D: all k ▯= ±1 E: no values 1 9. For what value(s) of k does the system of linear equations have inﬁnitely many solutions? mark A: k = −1 only B: k = 1 only C: k = ±1 only D: all k ▯= ±1 E: no values 1 10. For what value(s) of k does the system of linear equations have no solution? mark A: k = −1 only B: k = 1 only C: k = ±1 only D: all k ▯= ±1 E: no values 1 11. Let A be a 2 × 3 matrix and let B be a 3 × 2 matrix. Which one of the following is not mark deﬁned? A: AT + B B: (AB)T C: (AAT)2 D: (A + B) E: BA Friday, November 4, 2011 CODE 111 Mathematics 1229A Page 3 Test 2 ▯ ▯   1 0 −1 1 1 1 12. Let
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