331

PART I: Solutions to Odd-Numbered Exercises and Practice Tests

Chapter 6 Practice Test !

For Exercises 1 and 2, use the Law of Sines to find the remaining sides and angles of the triangle. 1. A= 40°, B = 12°, b= 100

2. C= 150°, a=5, c=20

3. Find the area of the triangle: a = 3, b = 6, C = 130°. 4. Determine the number of solutions to the triangle: a = 10, b = 35, A = 22.5°. For Exercises 5 and 6, use the Law of Cosines to find the remaining sides and angles off the triangle. 5. a=49, b=53, c=38

6. C=29°, a= 100, b=300

7. Use Heron’s Formula to find the area of the triangle: a = 4.1, b = 6,8, c = 5.5. 8. A ship travels 40 miles due east, then adjusts its course 12° southward. After traveling 70 miles in that direction, how far is the ship from its point of departure? 9. w=4u-7vwhereu=3i+jandv=-i+2j. Findw. 10. Find a unit vector in the direction of v = 5i - 3j. 11. Find the dot product and the angle between u = 6i + 5j and v = 2i - 3j. 12. v is a vector of magnitude 4 making an angle of 30° with the positive x-axis. Find v in component form. 13. Find the projection of u onto v given u = (3, -1} and v = (-2, 4}. 14. Give the trigonometric form of z = 5 - 5i. 15. Give the standard form of z = 6(cos 225° + i sin 225°) 16. Multiply [7 cos 23° + i sin 23o)][4(cos 7° + i sin 7°)]. 9 cos + 17. Divide -~- i sin ........... 3(cos ~" + i sin or) ’ 18. Find (2 + 2i)8. 19. Find the cube roots of 8 cos 3 + i sin ~). 2t). Find all the solutions to x4 + i = 0.

42~

Chapter 8

PART I: Solutions to Odd-Numbered Exercises and Practice Tests

Practice Test /

1. Write the matrix in reduced row-echelon form.

For Exercises 2-4, use matrices to solve the system,of equations. 2. 3x+5y= 3 2x- y= -11

3. 2x+3y=-3

x + 3z = -5 2x+y =0 3xq-y- z= 3

3x+2y= .8 x+ y= 1

E9

6. GivenA = -4 7. Find f(A):

f(x) = xz 7x + 8, A = 8. True or false: (A +B)(A + 3B) = A~ + 4AB + 3B2 where A and B are matrices. (Assume that A2, AB, and B2 exist.) For Exercises 9-10, find the inverse of the matrix, if it exists.

11. Use an inverse matrix to solve the systems. (a) x+2y=4 3x+5y= 1

(b) x-bay= 3 3x,+5y=-2

For Exercises 12-13, find(the determinant of the matrix. 3 9 2

426

PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

Use a graphing utility to find the determinant of the matrix. 1 -2 / : 5 -1 0 6

.

64 15. Evaluate ~

5 0

3 1 2

0 4 7

0

0

0

6 ~ .

16. Use a determinant to find the area of the triangle with vertices (0, 7), (5, 0), and (3, 9). 17. Use a determinant to find the equation of the line through (2, 7) and (- 1, 4). For Exercises 18-20, use Cramer’s Rule to find the indicated value. 18. Find x. 6x-7y= 4 2x+5y= 11

19. Find z. 3x x-y

20. Find y.

+ z= 1 y+4z=3 =2

721.4x - 29.1y = 33.77 45.9x + 105.6y = 19.85

472

PART I: Solutions to Odd-Numbered Exercises and Practice Tests

Chapter 9 Practice Test /

2n 1. Write out the first five terms of the sequence an - (n + 2)!’ ’

)

2. Write an expression for the nth term of the sequence , 9, 27, 81, 2,~3 ..... 6

3. Find the sum E(2i - 1). i=1

4. Write out the first five terms of the arithmetic sequence where a1 = 23 and d = -2. 5. Find an for the arithmetic sequence with a1 = 12, d = 3, and n = 50. Find the sum of the first 200 positive integers. 7. Write out the first five terms of the geometric sequencewith a1 = 7 and r = 2. ¯ ~ /2\n 8" Evaluate 9. Evaluate ~,~ (0.03)n. rt=0

10. Use mathematical induction to prove that 1 + 2 + 3 + 4 + ¯. ¯ + n 11. Use mathematical induction to prove that n! > 2n, n > 4. 12. Evaluate 13C4. Verify with a graphing utility. 13. Expand (x + 3)5. 14, Find the term involving x7 in (x - 2)12. 15. Evaluate 30P4. 16. How many ways can six people sit at a table with six chairs? 17. Twelve cars run in a race. How many different ways can they come in first, second, and third place? (Assume that there are no ties.) 18. Two six-sided dice are tossed. Find the probability that the total of the two dice is less than 5. 19. Two cards are selected at random form a deck of 52 playing cards without replacement. Find the probability that the first card is a King and the second card is a black ten. 20. A manufacturer has determined that for every 1000 units it produces, 3 will be faulty. What is the probability that an order of 50 units will have one or more faulty units?

530

PART I: Solutions to Odd-Numbered Exercises and Practice Tests

Chapter 10 Practice Test 1o Find the vertex,,~ focus and directrix of the parabola xa - 6x - 4y + 1 = 0. 2. Find an equation of the parabola with its vertex at (2, -5) and focus at (2, -6). 3. Find the center, foci, vertices, and eccentricity of the ellipse x~ + 4y~ - 2x + 32y + 61 = 0. 4. Find an equation of the ellipse with vertices (0, +6) and eccentricity e = ½. 5. Find the center, vertices, loci, and asymptotes of the hyperbola 16ya - x2 - 6x - 128y + 231 = 0. 6. Find an equation of the hyperbola with vertices at (+3, 2) and loci at (+5, 2). .

7. Rotate the axes to eliminate the xy-term. Sketch the graph Of the resulting equation, showing both sets of axes. 5xa + 2xy + 5ya- 10=0 ¯

8. Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hyperbola. (a) 6x2-2xy+ya=0 (b) xz+4xy+4y2-x-y+ 17=0 For Exercises 9 and 10, eliminate the parameter and write the corresponding rectangular equation. 9. x = 3 - 2 sin 0, y = 1 + 5 cos 0

10. x = eat, y = e4t

11. Convert the polar point (-,~, (3,rr)/4)to rectangular coordinates. 12. Convert the rectangular point (,/~, -1)to polar coordinates. 13. Convert the rectangular equation 4x - 3y = 12 to polar form. 14. Convert the polar equation r = .5 cos 0 to rectangular form. 15. Sketch the graph of r = 1 - cos 0. 16. Sketch the graph of r ~ 5 sin 20. 17. Sketch the graph of r -

3 6 - cos 0’

18. Find a polar equation of the parabola with its vertex at (6, ~’/2) and focus at (0, 0).

575

PART I: Solutions to Odd-Numbered Exercises and Practice Tests

Chapter 12 Practice Test 1. Use a graphing utility to complete the table and use the result to estimate the limit x

2.9

2.99

f(~)

3

3,1

?

3. Find the limit lim ex- a by ’,direct substitution.

and

2. Graph the function f(x) = estimate the limit lim

3.01

~ + 4 -2

x--~O

X.

x3" 1 4. Find the limit lira analytically.

5. Use a graphing utility to estimate the limit

x--~l X-- ~"

sin 5x lim 2x . x~0 6. Find the limit lim Ix + 21 x~-2 x+2"

7. Use the limit process to find the slope of the graph of f(x) = ~ at the point (4, 2).

8. Find the derivative of the function f(x) = 3x - 1. 9. Find the limits. " x2 (b) lim z x-~-~ x + 3

(a) ~,-~oo lira 3~

(c) lim [x___J_[ x~ 1 - x 1 -- ~/2

10. Write the first four terms of the sequence an - 2n2 +-------i and find the limit of the sequence. 25

11. Find the sum ~1"= (i2 nc i).

12. Write the sum ~-~ as a rational function s(n), and find lim s(n). i=1

t/---)oo

13. Find the area of the region bounded by f(x) = 1 - x2 over the interval 0 < x < 1.