math 2200 midterm review - ms. simms...

Math 2200 Midterm Review

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. Which of the following numbers occurs in the sequence –12, –8, –4, 0, 4, . . .?

A 6 C 24

B –3 D 15

____ 2. The common difference in the arithmetic sequence , , , , , . . . is

A

C 9

B 3 D

____ 3. In the formula for the general term of an arithmetic sequence tn = –7 + (n – 1) (–2.5), the common difference is

A 17.5 C –4.5

B –7 D –2.5

____ 4. Which of the given formulas for the general term of the sequence –9, –1, 7, 15, 23, . . . is correct?

A C

B D

____ 5. What is the 18th term of the sequence –22, –21.2, –20.4, –19.6, –18.8, …?

A –6.8 C –8.4

B 0.8 D –35.6

____ 6. The sum of the series (–5) + (–7) + (–9) + + (–19) is

A –96 C –192

B –304 D 26

____ 7. The sum of an arithmetic series where , , and n = 19 is

A C

B

D

____ 8. On the first day of the month, Michael places 5¢ in a jar. The next day, he places 7¢ in the jar. The third day, he

places 9¢ in the jar, and so on for 24 days. What amount will be in the jar at the end of this period of time?

A $6.72 C $6.96

B $6.36 D $6.12

____ 9. The common ratio for the geometric sequence 8, 1, 0.125, 0.015625, . . . is

A

C 8

B –8 D

____ 10. The population of a community was 82 000 at the beginning of 2000. Assuming a rate of growth of 1.6% per year

since 2000, what will the population be at the beginning of 2025?

A 123 894 C 121 943

B 2 082 800 D 120 023

____ 11. In the formula for the general term of a geometric sequence , the common ratio is

A

C 4

B 5 D

1

2

5

6

7

6

3

2

11

6

5

12

551 1045

2

165

21045

A

B

12

24

____ 12. The first three terms of the sequence given by are

A 11, 121, 1331 C , ,

B , ,

D 11, ,

____ 13. The sum of the geometric series 13 + 6.5 + 3.25 + + 0.203125 is

A

C

B

D

____ 14. What is the sum of the infinite geometric series

15 + + + + ?

A C

B

D

____ 15. Which of the following best describes the series –50 + ( ) + ( ) + ( ) + ?

A The series is convergent and has a sum of .

B The series is divergent and has a sum of .

C The series is divergent and has no sum.

D The series is convergent and has no sum.

____ 16. The first three terms of the sequence defined by are

A 0.5, 0.8, 1.1 C 0.2, –0.1, –0.4

B –0.3, 0.2, 0.7 D –0.3, –0.8, –1.3

____ 17. What is the reference angle for 15in standard position?

A 255 C 345

B 30 D 15

____ 18. What are the three other angles in standard position that have a reference angle of 54?

A 99144234 C 144234324

B 108162216 D 126234306

____ 19. What is the exact sine of A?

A 1/ C 2/

B 1/3 D ½

____ 20. Which set of angles has the same terminal arm as 40?

A 80, 120, 160 C 200, 380, 560

B 130, 220, 310 D 400, 760, 1120

____ 21. The point (40, –9) is on the terminal arm of A. Which is the set of exact primary trigonometric ratios for the angle?

A , ,

B , ,

C , ,

D , ,

11

4

11

16

11

64

1111

4

11

16

11

3

11

9

559

64

1651

64

1677

64

819

32

120 135

8

135

17135

45 81

2

729

20

500500

3 3

27.1º

A

16 cm

34 cm

Diagram not drawn to scale.

B

C

a

A

aB C

bc


____ 22. What is the exact value for ?

A

C 1

B D

____ 23. Solve to the nearest tenth of a unit for the unknown side in the ratio

.

A 24 C 6.6

B 21.8 D 24.6

____ 24. Determine, to the nearest tenth of a centimetre, the two possible lengths of a.

A 72.8 cm and 26.3 cm C 72.8 cm and 55.8 cm

B 34.3 cm and 26.3 cm D 55.8 cm and 34.3 cm

____ 25. If , c = 10.3 cm, and b = 10.5 cm, and ABC is acute, what is the measure of , to the nearest tenth of

a degree?

A 57 C 30.5

B 123.0 D 149.5

____ 26. Which strategy would be best to solve for x in the triangle shown?

A cosine law C sine law

B primary trigonometric ratios D none of the above

____ 27. What are the x-intercepts of ?

A 6 and 9 C –6 and 9

B –6 and –9 D 6 and –9

____ 28. What is the quadratic function in vertex form for the parabola shown below?

A C

B D

1 2 3 4 5 6 7 8 9 10 11–1–2–3–4–5–6–7–8–9–10–11 x

1

2

3

4

5

6

7

8

9

10

11

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

y

x

30 cm

21º

7º


____ 29. Which graph represents the quadratic function ?

A C

B D

____ 30. What are the domain and range of ?

A Domain:

Range:

C Domain:

Range:

B Domain:

Range:

D Domain:

Range:

____ 31. The vertex of a parabola is located at . If the parabola has a y-intercept of 231, which quadratic function

represents the parabola?

A C

B D

____ 32. What information can be determined from the quadratic function ?

A the vertex is at (–2, –9) and the graph opens upward

B the vertex is at (–9, –2) and the graph opens downward

C the vertex is at (–2, –9) and the graph opens downward

D the vertex is at (–9, –2) and the graph opens upward

____ 33. Identify the characteristics of this graph.

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

A vertex: (–2, –5)

axis of symmetry:

y-intercept: 10.5

x-intercepts: –3 and –7

opens downward

C vertex: (–2, –5)

axis of symmetry:

y-intercept: 10.5


opens upward

B vertex: (–5, –2)

axis of symmetry:

y-intercept: 10.5


opens upward

D vertex: (–5, –2)

axis of symmetry:

y-intercept: 10.5

x-intercepts: 3 and 7

opens downward

____ 34. What are the coordinates of the vertex of the quadratic function ?

A (–6, –1) C (–1, –6)

B (8, –2) D (8, –6)

____ 35. What is the function written in standard form?

A C

B D

____ 36. If the points and are on the graph of the quadratic function , what are the

values of b and c?

A and C and

B and D and

____ 37. State whether the function has a maximum or minimum value and identify the coordinates of

the vertex.

A maximum at C minimum at

B maximum at D minimum at

2 4 6 8 10 12 14 16 18 20 22–2–4–6–8–10–12–14–16–18–20–22 x

2

4

6

8

10

12

14

16

18

20

22

–2

–4

–6

–8

–10

–12

–14

–16

–18

–20

–22

y

____ 38. How many x-intercepts does the graph of the quadratic function have?

A unknown C 1

B 2 D 0

____ 39. What is the x-intercept of the quadratic function graphed here?

A 0 C 4.8

B –2 D 2

____ 40. What is/are the root(s) of the quadratic function ?

A 2 and 5 C –0.45

B –2 and –5 D 2

____ 41. Solve .

A and C and

B and D and

1 2 3 4 5 6–1–2–3–4–5–6 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

1 2 3 4 5 6–1–2–3–4–5–6 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

____ 42. A rectangle has dimensions and , where x is in centimetres. If the area of the rectangle is 72 cm2,

what is the value of x, to the nearest tenth of a centimetre?

A C B D

____ 43. The value of k that makes the expression a perfect square trinomial is

A 1296 C 0

B 144 D 72

____ 44. Solve .

A 1 + and 1 C

B –1 + and –1 D

____ 45. For a science experiment, a projectile is launched. Its path is given by , where h is the

height of the projectile above the ground and d is the horizontal distance of the projectile from the launch pad, both

in metres. How far away from the launch pad is the projectile when it begins to fall, to the nearest tenth of a metre?

A 255.8 m C 0.3 m

B 7.7 m D 15.7 m

Matching

Match the correct term to its description below.

A arithmetic series D divergent series

B geometric sequence E geometric series

C arithmetic sequence F infinite geometric series

____ 1. a geometric series that has no last term

____ 2. the sum of the terms of a sequence in which the terms have a common ratio

____ 3. the sum of the terms of a sequence in which the terms have a common difference

____ 4. a sequence where there is a common difference between consecutive terms

____ 5. a sequence where there is a common ratio between consecutive terms

Match the correct term to its description below.

A reference angle D ambiguous case

B cosine law E sine law

C angle in standard position F terminal arm

____ 6. a problem with two or more solutions

____ 7. a law used when two sides and an opposite angle are given

____ 8. the acute angle between the terminal arm and the x-axis of an angle in standard position

____ 9. an angle with the initial arm on the positive x-axis

x + 10

5x –4

43 43 2 11

43 43 42

____ 10. a law used when two sides and a contained angle are given

Match the correct term or statement to its description below.

A D

B E

C –2 and 1 F 2

____ 11. the number of roots of the equation

____ 12. the factored form of

____ 13. the quadratic formula

____ 14. the zeros of

____ 15. the discriminant

____ 16. an equation in standard form that represents

Short Answer

For each arithmetic sequence, determine

a) the value of t1 and d

b) an explicit formula for the general term

c) t20

1. 3a, 3a – 2b, 3a – 4b, 3a – 6b, …

2. The starting wage at a bookstore is $8.50 per hour with a yearly increase of $0.75 per hour.

a) Write the general term of the sequence representing the hourly rate earned in each year.

b) Use your expression from part a) to determine the hourly rate after 6 years.

c) How many years will someone need to work at the store to earn $15.25 per hour?

For each arithmetic series, determine

a) an explicit formula for the general term

b) a formula for the general sum

c)

d)

3. = 2, d = 3, n = 4

Determine the sum of each arithmetic series.

4.

5. Find the value of given and . Be sure to show all of your work.

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

6. A bouncy ball bounces to its height when it is dropped on a hard surface. Suppose the ball is dropped from 20 m.

a) What height will the ball bounce back up to after the sixth bounce?

b) What is the total distance the ball travels if it bounces indefinitely?

7. The point A(–3, –5) is on the terminal arm of an angle . Determine exact expressions for the primary trigonometric

ratios for the angle.

8. In acute , o = 7 cm, p = 9 cm, and . Solve . What type of triangle is this?

9. A drive belt wraps around three pulleys, A, B, and C, as shown.

What is the measure of ?

10. Express the quadratic function in vertex form.

11. For what values of k does the equation have

a) one real root?

b) two distinct roots?

c) no real roots?

12. Solve the quadratic function by completing the square. Round roots to the nearest hundredth, if

necessary.

13. Use the quadratic formula to find the roots of the equation x2 + 4x – 21 = 0. Express your answers as exact roots.

14. Determine the number of real roots for the equation 3x2 = 8x – 4. Then, find the roots of the equation by

a) using the quadratic formula

b) factoring

15. Write the equation of this parabola.

(–2, 0) (3, 0)

(0.5, –6.25)

1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x

1

2

3

4

5

6

7

8

–1

–2

–3

–4

–5

–6

–7

–8

y

Problem

1. Consider an arithmetic sequence with and .

a) Determine the values of t1 and d.

b) Determine an explicit formula for the general term of the sequence.

c) Determine the value of each term.

i) t12

ii) t20

iii) t26

d) Determine the term number of each term.

i)

ii) 9

iii)

2. A toy car is rolling down an inclined track and picking up speed as it goes. The car travels 4 cm in the first second,

8 cm in the second second, 12 cm in the next second, and so on. What is the total distance travelled by the car in the

first 30 s?

3. The value of an antique increases in value at a rate of 2.5% every year. In 2000, the antique was purchased for

$5000.

a) Determine an explicit formula to represent the value of the antique since the year 2000.

b) Use your formula to write the first three terms of the sequence.

c) What was the value of the antique in 2008?

d) In which year will the value of the antique be $11 866?

4. a) Without using a calculator, determine two angles between 0° and 360° that have a sine ratio of .

b) Use a calculator and a diagram to verify your answers to part a).

5. Two wires are connected to a tower at the same point on the tower. Wire 1 makes an angle of 45° with the ground

and wire 2 makes an angle of 60° with the ground.

a) Represent this situation with a diagram.

b) Which wire is longer? Explain.

c) If the point where the two wires connect to the tower is 35 m above the ground, determine exact expressions for

the lengths of the two wires.

d) Determine the length of each wire, to the nearest tenth of a metre.

e) How do your answers to parts b) and d) compare?

6. a) Write the function y = –(x – 2)2 + 9 in standard form.

b) Sketch the graph of the function. Use your answer to part a) to identify the y-intercept.

1 2 3 4 5 6 7 8–1–2 x

1

2

3

4

5

6

7

8

9

10

11

–1

–2

–3

–4

–5

–6

–7

y

7. A ball is launched from a 50-m cliff vertically upward on Jupiter. The table shows the height, h, in metres, of the ball

above the surface at time, t, in seconds, after it was launched.

t h(t)

0 50

1 88

2 101

3 88

4 50

a) State the domain and range of the function modelled by the data.

b) Estimate the vertex of the function.

c) What is the approximate equation, in vertex form, that models this situation?

8. Consider the function y = –2(x – 12)2 + 18.

a) What is the axis of symmetry of the graph of the function.

b) What is the vertex of the graph of the function?

c) State the domain and range of the function.

9. A store can increase its profit by increasing the price of the sweaters it sells. The relation between the income, R, and

the dollar increase in the price per sweater, d, can be modelled by the equation .

a) What is the maximum possible income?

b) What would the income be if the price per sweater were increased by $10?

10. Supermarket cashiers try to memorize current sale prices while they work. A survey shows that, on average, the

percent, P, of prices memorized after t hours is modelled by the relation .

a) What is the greatest percent of prices memorized?

b) How long does it take to memorize this greatest percent?

11. The path of a ball can be modelled by the function , where h is the height of the ball and d is the

horizontal distance travelled, both in metres. What total horizontal distance does the ball travel?

12. A ball is thrown straight up from a bridge over a river and falls into the water. The height, h, in metres, of the ball

above the water t seconds after being thrown is approximately modelled by the relation .

a) What is the maximum height of the ball above the water?

b) How long does it take for the ball to reach the maximum height?

c) After how many seconds does the ball hit the water?

d) How high is the bridge above the river?

13. Two whole numbers differ by 3. The sum of their squares is 89. What are the numbers?

14. A rectangle, 4 cm longer than it is wide, has a diagonal 20 cm long. What are the dimensions of the rectangle?

20

x + 4

x

Math 2200 Midterm Review

Answer Section

MULTIPLE CHOICE

1. ANS: C PTS: 1 DIF: Easy OBJ: Section 1.1

NAT: RF 9 TOP: Arithmetic Sequences KEY: nth term

2. ANS: D PTS: 1 DIF: Average OBJ: Section 1.1

NAT: RF 9 TOP: Arithmetic Sequences KEY: common difference | fraction

3. ANS: D PTS: 1 DIF: Easy OBJ: Section 1.1

NAT: RF 9 TOP: Arithmetic Sequences KEY: common difference | general term

4. ANS: B PTS: 1 DIF: Average OBJ: Section 1.1

NAT: RF 9 TOP: Arithmetic Sequences KEY: general term | arithmetic sequence

5. ANS: C PTS: 1 DIF: Average OBJ: Section 1.1

NAT: RF 9 TOP: Arithmetic Sequences KEY: terms | arithmetic sequence

6. ANS: A PTS: 1 DIF: Average OBJ: Section 1.2

NAT: RF 9 TOP: Arithmetic Series

KEY: sum | number of terms | arithmetic series


NAT: RF 9 TOP: Arithmetic Series KEY: sum | arithmetic series

8. ANS: A PTS: 1 DIF: Easy OBJ: Section 1.2

NAT: RF 9 TOP: Arithmetic Series KEY: sum | arithmetic series


NAT: RF 10 TOP: Geometric Sequences KEY: common ratio | geometric sequence

10. ANS: C PTS: 1 DIF: Difficult OBJ: Section 1.3

NAT: RF 10 TOP: Geometric Sequences KEY: explicit formula | terms


NAT: RF 10 TOP: Geometric Sequences

KEY: common ratio | explicit formula | geometric sequence

12. ANS: B PTS: 1 DIF: Easy OBJ: Section 1.3

NAT: RF 10 TOP: Geometric Sequences

KEY: terms | explicit formula | geometric sequence


NAT: RF 10 TOP: Geometric Series

KEY: sum | number of terms | geometric series


NAT: RF 10 TOP: Infinite Geometric Series KEY: sum | infinite geometric series


NAT: RF 10 TOP: Infinite Geometric Series

KEY: convergent series | infinite geometric series | sum


NAT: RF 9 TOP: Arithmetic Sequences

KEY: terms | explicit formula | arithmetic sequence


NAT: T 1 TOP: Angles in Standard Position KEY: reference angle | < 180°


NAT: T 1 TOP: Angles in Standard Position KEY: reference angle


NAT: T 1 TOP: Angles in Standard Position KEY: special angles | sine


NAT: T 1 TOP: Angles in Standard Position KEY: co-terminal angles


NAT: T 1 TOP: Trigonometric Ratios of Any Angle

KEY: point on terminal arm | cosine | sine | tangent


NAT: T 1 TOP: Trigonometric Ratios of Any Angle

KEY: tangent | reference angle | related angles


NAT: T 3 TOP: The Sine Law KEY: sine law | side length

24. ANS: B PTS: 1 DIF: Difficult OBJ: Section 2.3

NAT: T 3 TOP: The Sine Law

KEY: sine law | side length | ambiguous case


NAT: T 3 TOP: The Sine Law KEY: sine law | angle measure


NAT: T 3 TOP: The Sine Law KEY: sine law | solution method


NAT: RF 3 TOP: Investigating Quadratic Functions in Vertex Form

KEY: intercept



KEY: vertex form

29. ANS: A PTS: 1 DIF: Difficult OBJ: Section 3.1


KEY: vertex form | graph



KEY: domain | range

31. ANS: B PTS: 1 DIF: Difficult OBJ: Section 3.1


KEY: vertex | y-intercept



KEY: vertex | direction of opening


NAT: RF 4 TOP: Investigating Quadratic Functions in Standard Form

KEY: vertex | axis of symmetry | y-intercept | x-intercept | direction of opening



KEY: vertex



KEY: standard form

36. ANS: D PTS: 1 DIF: Difficult OBJ: Section 3.2


KEY: standard form | determine equations from two points

37. ANS: D PTS: 1 DIF: Difficult OBJ: Section 3.3

NAT: RF 4 TOP: Completing the Square KEY: max/min


NAT: RF 5 TOP: Graphical Solutions of Quadratic Equations

KEY: x-intercepts



KEY: x-intercepts | one real root



KEY: two real roots


NAT: RF 5 TOP: Factoring Quadratic Equations KEY: solve factored trinomial


NAT: RF 5 TOP: Factoring Quadratic Equations KEY: area problem | solve trinomial


NAT: RF 5 TOP: Solving Quadratic Equations by Completing the Square

KEY: perfect square trinomial



KEY: square root

45. ANS: B PTS: 1 DIF: Difficult + OBJ: Section 4.4

NAT: RF 5 TOP: The Quadratic Formula KEY: vertex coordinates

MATCHING

1. ANS: F PTS: 1 DIF: Easy OBJ: Section 1.5

NAT: RF 10 TOP: Infinite Geometric Series KEY: infinite geometric series

2. ANS: E PTS: 1 DIF: Easy OBJ: Section 1.4

NAT: RF 10 TOP: Geometric Series KEY: geometric series


NAT: RF 9 TOP: Arithmetic Series KEY: arithmetic series


NAT: RF 9 TOP: Arithmetic Sequences KEY: arithmetic sequence


NAT: RF 10 TOP: Geometric Sequences KEY: geometric sequence


NAT: T 3 TOP: The Sine Law KEY: ambiguous case

7. ANS: E PTS: 1 DIF: Easy OBJ: Section 2.3

NAT: T 3 TOP: The Sine Law KEY: sine law


NAT: T 1 TOP: Angles in Standard Position KEY: reference angle


NAT: T 1 TOP: Angles in Standard Position KEY: standard angle


NAT: T 3 TOP: The Cosine Law KEY: cosine law

11. ANS: F PTS: 1 DIF: Easy OBJ: Section 4.1


KEY: number of roots

12. ANS: E PTS: 1 DIF: Average OBJ: Section 4.3

NAT: RF 5 TOP: Factoring Quadratic Equations KEY: factor trinomial


NAT: RF 5 TOP: The Quadratic Formula KEY: quadratic formula

14. ANS: C PTS: 1 DIF: Average OBJ: Section 4.1 | Section 4.2

NAT: RF 5

TOP: Solving Quadratic Equations by Completing the Square | Factoring Quadratic Equations

KEY: zeros


NAT: RF 5 TOP: The Quadratic Formula KEY: discriminant



KEY: vertex form | completing the square

SHORT ANSWER

1. ANS:

a) t1 = 3a, d = –2b

b)

c)

PTS: 1 DIF: Average OBJ: Section 1.1 NAT: RF 9

TOP: Arithmetic Sequences KEY: terms | explicit formula | arithmetic sequence

2. ANS:

a)

b)

The hourly rate after 6 years is $12.25.

c)

You would need to work at the bookstore for 10 years to earn $15.25 per hour.


TOP: Arithmetic Sequences KEY: explicit formula | terms

3. ANS:

a)

b)

c)

d)

PTS: 1 DIF: Easy OBJ: Section 1.2 NAT: RF 9

TOP: Arithmetic Series KEY: explicit formula | sum | terms | arithmetic series

4. ANS:

= 4k, d = 7k


TOP: Arithmetic Series KEY: sum | arithmetic series

5. ANS:


TOP: Geometric Series KEY: sum | geometric series

6. ANS:

a) Draw a diagram of the situation.

and

The ball will bounce to a height of approximately 1.76 m on the sixth bounce.

b) The distance the ball travels (both down and up) is

20 + 2t1 + 2t2 + 2t3 + = 20 + 2S.

The total distance the ball will travel is 100 m.

PTS: 1 DIF: Difficult OBJ: Section 1.5 NAT: RF 10

TOP: Infinite Geometric Series KEY: sum | infinite geometric series

7. ANS:

The angle is in the third quadrant, so only the tangent ratio will be positive.

From the given point, x = –3 and y = –5.

Therefore, .

PTS: 1 DIF: Average OBJ: Section 2.2 NAT: T 2

TOP: Trigonometric Ratios of Any Angle

KEY: primary trigonometric ratios | point on terminal arm

8. ANS:

Use the sine law.

Since ,

is an isosceles triangle.


TOP: The Sine Law KEY: sine law | solve a triangle

9. ANS:

is approximately 23°.


TOP: The Cosine Law KEY: cosine law | angle measure

10. ANS:


TOP: Completing the Square KEY: standard to vertex form

11. ANS:

a)

b)

c)


TOP: The Quadratic Formula KEY: number of roots

12. ANS:

To solve the equation, set it equal to 0 and solve for x.


TOP: Solving Quadratic Equations by Completing the Square KEY: completing the square

13. ANS:


TOP: The Quadratic Formula KEY: quadratic formula

14. ANS:

Rearrange the equation so all terms are on the same side:

Calculate the discriminant :

Since the discriminant is positive (greater than zero), the equation has 2 real roots.

a)

b)

PTS: 1 DIF: Average OBJ: Section 4.3 | Section 4.4

NAT: RF 5 TOP: Factoring Quadratic Equations | The Quadratic Formula

KEY: roots of quadratic equation | solve factored trinomial

15. ANS:

The x-intercepts are 2 and 3. These correspond to factors of and . The equation is of the form

y = a(x + 2)(x – 3).

Expand and simplify the right side of the equation:

Substitute the known point on the curve to determine the value of a:

The value of a is 1, so the equation is .


TOP: Graphical Solutions of Quadratic Equations KEY: quadratic function | parabola

PROBLEM

1. ANS:

a) Use tn = a + (n – 1)d to write and solve a system of equations.

and

Substitute into one of the original equations in the system.

b)

c) i) ii) iii)

d) i) ii) iii)


TOP: Arithmetic Sequences KEY: explicit formula | arithmetic sequence | terms

2. ANS:

and d = 4.

The toy car travels 1860 cm or 18.6 m.


TOP: Arithmetic Series KEY: sum | arithmetic series

3. ANS:

a) At a rate of increase of 2.5% per year, r = 1.025. The initial value of the antique in 2000 is $5000, so =

5000.

tn = 5000(1.025)n

b)

The first three terms of the sequence are $5125, $5253.13, and $5384.45.

c) The year 2008 represents t8.

d)

Using systematic trial, . So, n = 35. In 2035, the antique will be worth approximately $11

866.


TOP: Geometric Sequences KEY: explicit formula | geometric sequence | terms

4. ANS:

a) Since sin 30° = , the reference angle is 30°. The sine ratio is negative in the third and fourth quadrants.

Look for reflections of the 30° angle in these quadrants.

third quadrant: 180° + 30° = 210°

fourth quadrant: 360° – 30° = 330°

b) Using a calculator, sin 210° = and sin 330° = .

PTS: 1 DIF: Easy OBJ: Section 2.2 NAT: T 2

TOP: Trigonometric Ratios of Any Angle KEY: sine | reference angle

5. ANS:

a)

b) Both wires are connected to the tower at the same height, which is the opposite side to the given angles. Each

wire length represents the hypotenuse of its respective triangle. The longer hypotenuse is the wire that forms the

smaller angle, as it will need to be longer to reach the tower.

c) Let x represent the length of wire 1 and y represent the length of wire 2.

Wire 1: Wire 2:

d) The length of wire 1 is 49.5 m, and the length of wire 2 is 40.4 m.

e) The values calculated in part d) support the answers in part b).

PTS: 1 DIF: Average OBJ: Section 2.1 | Section 2.2

NAT: T 1 | T 2 TOP: Angles in Standard Position | Trigonometric Ratios of Any Angle

KEY: special angles | sine

6. ANS:

a)

b) The y-intercept is 5.


TOP: Investigating Quadratic Functions in Vertex Form KEY: vertex to standard form | vertex

7. ANS:

a) domain:

range:

b) (2, 101)

c) Use the estimated vertex from part b) and a point, say (0, 50), from the table to determine the approximate

equation for the function.


TOP: Investigating Quadratic Functions in Vertex Form KEY: domain | range | vertex | vertex form

8. ANS:

a) x = 12

b) (12, 18)

c) Domain: Range:


(0, 5)

1 2 3 4 5 6 7 8–1–2 x

1

2

3

4

5

6

7

8

9

10

11

–1

–2

–3

–4

–5

–6

–7

y

TOP: Investigating Quadratic Functions in Vertex Form

KEY: axis of symmetry | domain | range | vertex

9. ANS:

a) The maximum profit occurs at the vertex (3.5, 4500) or $4500.

b) Substitute d = 10 into the equation:

The income would be $2387.50.


TOP: Investigating Quadratic Functions in Vertex Form KEY: vertex form | vertex

10. ANS:

a) Complete the square to find the vertex.

The maximum is 90 (the P-coordinate of the vertex). The greatest percent of prices memorized is 90%.

b) The maximum is at the point (1.5, 90). So, it takes 1.5 h to memorize 90% of the prices.

PTS: 1 DIF: Difficult + OBJ: Section 3.3 NAT: RF 4

TOP: Completing the Square KEY: x-intercepts | factors | vertex

11. ANS:

Graphical solution

Determine the zeros of the function (or roots of the equation) by setting h = 0 and then factoring the equation:

0.1 0.2 0.3 d

0.01

0.02

0.03

h

The ball travels 0.25 m or 25 cm horizontally.

PTS: 1 DIF: Difficult OBJ: Section 4.1 | Section 4.2

NAT: RF 5 TOP: Graphical Solutions of Quadratic Equations | Factoring Quadratic Equations

KEY: zeros | x-intercepts

12. ANS:

Graph the relation to visualize the situation.

From the graph:

a) The maximum height of the ball is 40 m.

b) It takes 1 s to reach the maximum height.

c) The t-intercept is approximately 3.8. The ball hits the ground after about 3.8 s.

From the equation:

d) When t = 0, h = 35. The bridge is 35 m above the river.


TOP: Graphical Solutions of Quadratic Equations

KEY: maximum | x-intercepts | parabolic motion

13. ANS:

Let x represent the first number and x + 3 represent the second.

Then,

Since the result is a whole number, the negative value is rejected and the numbers are 5 and 8.

(0, 35)

(1, 40)

(3.8, 0)

1 2 3 4 5 6–1–2 t

5

10

15

20

25

30

35

40

45

50

h


TOP: Factoring Quadratic Equations KEY: factor quadratic

14. ANS:

Use the Pythagorean theorem.

Factor the trinomial to find the zeros of the quadratic relation.

The zeros are 16 and 12.

Since a negative distance is not reasonable, the value for x is 12.

The dimensions of the rectangle are 12 cm by 16 cm.


TOP: Factoring Quadratic Equations KEY: roots of quadratic equation

math 2200 midterm review - ms. simms...

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