MATH 148, SPRING 2016
COMMON EXAM II (PART 1) - VERSION A
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Question
1–13
Points Awarded
Points
52
1
PART I: Multiple Choice (4 points each)
1. Find the vector ~x that has magnitude r = 12 and forms an angle of θ = 300◦ measured counterclockwise from the
positive x-axis.
√
(a) h6, 6 3i
√
(b) h6 3, 6i
√
(c) h6, −6 3i
√
(d) h6 3, −6i
√
(e) h−6 3, −6i
2. Suppose that A is a 2 × 2 matrix. The eigenvalues of A have negative real part if and only if
(a) trA < 0 and det A < 0
(b) trA > 0 and det A < 0
(c) trA > 0 and det A > 0
(d) trA < 0 and det A > 0
(e) None of these.
2
3. Suppose that a population is divided into two age classes with Leslie matrix
3 5
L=
.
0.8 0
Find the growth parameter for the population and the stable age distribution.
5
6
1
Growth parameter: 4, Stable age distribution:
6
5
Growth parameter: 1, Stable age distribution:
6
1
Growth parameter: 1, Stable age distribution:
6
None of these.
(a) Growth parameter: 4, Stable age distribution:
≈ 83.33%
(b)
≈ 16.67%
(c)
(d)
(e)
≈ 83.33%
≈ 16.67%
4. Find the inverse (if it exists) of the matrix
1
A=
4
(a) A
−1
(b) A−1
(c) A−1
(d) A−1
1/4 −1
=
−3/4 2
−1/4 3/4
=
1
−2
−2 −3/4
=
−1 −1/4
−2 3/4
=
1 −1/4
(e) The matrix is singular, so A−1 does not exist.
3
3
.
8
5. Suppose that
dy
= y(2 − y)(y − 5).
dx
Use the stability of the equilibria to determine which of the following statements is true?
(a) If y(0) = 1, then lim y(x) = 5.
x→∞
(b) None of these.
(c) If y(0) = 3, then lim y(x) = 0.
x→∞
(d) If y(0) = 3, then lim y(x) = 2.
x→∞
(e) If y(0) = 1, then lim y(x) = 0.
x→∞
6. Consider the Leslie matrix

0
L = 0.3
0
1.4
0
0.7

2.3
0 .
0
Determine the average number of female offspring produced by a one-year-old female.
(a) 2.3
(b) 1.4
(c) 0.7
(d) 0.3
(e) 0
4
7. Find an equation of the plane through the point (2, 4, −1) and orthogonal (perpendicular) to the vector h2, 3, 4i.
(a) 2x + 4y − z = 12
(b) 2x + 3y + 4z = 20
(c) 2x + 3y + 4z = 12
(d) 2x + 4y − z = 20
(e) None of these.
8. Suppose that A is a 2 × 5 matrix and B is a 4 × 2 matrix. Which of the following statements is true?
(a) AB is a 2 × 2 matrix and BA is a 4 × 5 matrix
(b) AB is undefined and BA is a 4 × 5 matrix
(c) AB is a 2 × 2 matrix and BA is undefined
(d) AB is undefined and BA is a 5 × 4 matrix
(e) Both AB and BA are undefined.
5
9. Solve the underdetermined system
x + 7z
=
10
2x + y + 16z
=
22.
(a) There is no solution.
(b) (x, y, z) = (10, 2, 0)
(c) None of these.
(d) (x, y, z) = (3, 0, 0)
(e) {(10 − 7α, 2 − 2α, α)|α ∈ R}
10. Find ALL values of x such that the vectors ~v = hx, x, −1i and w
~ = h1, x, 6i are orthogonal (perpendicular).
(a) x = −3
(b) x = −2, 3
(c) x = 2, 3
(d) x = −2, −3
(e) None of these.
6
11. Find an equation of the line through the point (3, 2) and orthogonal (perpendicular) to the vector h−2, 1i.
3
(a) y = − x − 2
2
(b) y = 2x + 4
(c) y = 2x − 4
3
(d) y = − x + 2
2
(e) None of these.
−→
12. Let A = (1, 2, 3) and B = (−1, 6, 4). Find a unit vector in the direction of AB.
4
1
2
(a) − √ , √ , √
21
21
21
4
1
2
(b) − √ , − √ , √
21
21
21
1
2
4
(c) − √ , − √ , √
13
13
13
4
1
2
(d) − √ , √ , √
13
13
13
(e) None of these.
7
13. Consider the triangle with vertices A = (2, −2, 5), B = (1, 1, 4), and C = (3, 1, 3). Find the cosine of the angle at
vertex B.
(a) None of these.
3
(b) √
11
1
(c) √
55
1
(d) √
11
3
(e) √
55
8