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# test 2 2011.pdf

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Western University

Mathematics

Mathematics 1229A/B

Vicki Olds

Fall

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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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