Math III - 1st Indicative Requirement (2010)

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Mathematics III

1st Indicative Exam Requirement

Chapter 1 Relations and Functions I. Determine the domain of the following functions

1. d(y) = y + 3

2. g(k) = 2k2 + 4k - 6

3. b(n) = 82 −n

4. 292)( xtm −+−=

5. 42

5)(

+

−=

x

xxu

6. 1

1)(

−+=

rrra

7. cc

cy3

2)(

2+

=

8. 3

)(+

=x

xxf

9. 82)( 2−+= vvvt

10. t

ttn

+=

1)(

II. Find the domain of the following graphs

11. 12. 13.

14. 15. 16.

III. Determine if the followings are functions or not, explain your reasoning

17. {(4, –5), (0, –9), (1, 0), (7, 0)} 18. {(–2, –3), (6, –8), (4, 2), (6, –5), (2, –5)} 19. {(5, –12), (–1, –2(\), (8, –5), (4, –2), (3, –5)} 20. {(5, 2), (–2, 15), (–7, 15), (1, 5), (4, 15), (–7, 2)}

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21. 22. 23. 24.

25. 26. 27. 28.

IV. Perform the following operations with functions

45)( += xxf xxg −= 4)(

29. ( )( )xgf +

30. ( )( )xgf −

31. ( )( )xgf ⋅

32. ( )xg

f

33. ( )( )xfg o

34. ( )( )3gf +

432)( 2+−= xxxf 3)( += xxg

35. ( )( )xgf +

36. ( )( )xgf −

37. ( )( )xgf ⋅

38. ( )xg

f

39. ( )( )xgf o

40. ( )( )3gf +

Chapter 2 The Linear Function V. Using points A(–10, 6), B(–5, 8) find:

41. Slope 42. y and x-intercepts 43. Graph

44. Point-slope form equation 45. Slope-intercept form equation 46. General form equation

VI. Using points A(–4, –2), B(4, 0) find:

47. Slope 48. y and x-intercepts 49. Graph

50. Point-slope form equation 51. Slope-intercept form equation 52. General form equation

VII. Solve the following problems

53. Write the slope-intercept form of an equation of the line that passes through point (3, 1) and is parallel to the graph of 2x + y – 5 = 0 and draw its graph.

54. Write the slope-intercept form of an equation of the line that passes through point (2, 4) and is perpendicular to the graph of x – 6y – 2 = 0 and draw its graph.

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55. Write the general form of an equation of the line that passes through point (–1, –2) and is parallel to the graph of 3x – y – 5 = 0 and draw its graph.

56. Write the general form of an equation of the line that passes through point (–4, 1) and is perpendicular to the graph of 4x + 7y – 6 = 0 and draw its graph.

57. Flourishing Flowers charges $125 plus $60 for each standard floral arrangement to deliver and set up flowers for a banquet. (a) Write an equation in slope-intercept form that shows the cost for flowers for any number of arrangements. (b) Find the cost of providing 20 floral arrangements.

58. A person weighing 150 pounds burns about 320 Calories per hour walking at a moderate pace. Suppose that the same person burns an average of 1500 Calories per day through basic activities. The total Calories burned by that person can be represented by a linear equation. Find the particular equation in the general form.

59. Renata González wants to increase the energy efficiency of her house by adding to the insulation previously installed. The better material protects against heat loss, the higher its R-value, or resistance to heat flow. A 1in coat has an R-value of 3.2, while 3in has 9.6 as well. (a) Write an equation in slope-intercept form that shows the total R-value in the attic if she adds x number of inches of additional insulation. (b) Estimate the total R-value in the attic if she adds 6 inches of insulation.

60. The value of an automobile that has 6 years of use is $42,000, but 3 years ago, it was 96,000. If the value of the automobile varies linearly with the time: (a) Find an equation relating the value of the automobile at t years (b) How much does the value of the car depreciates per year? (c) Determine the value of the car when it was new (d) When does the automobile will not have commercial value?

Chapter 3 Inequalities VIII. Solve and draw the graph of each of the following inequalities

61. 158 −≥− x

62. xx 534 <+

63. )13(211)1(3 +−<++ zz

64. 0)6(35 ≥−− xx

65. 11235 ≤+< x

66. 1025 ≤− xy

67. 752<3 ≤+− x

68. 933 +> xy

69. 32 <− xy

70. The area of a triangular garden can be no more than 120 square feet. The base of the triangle is 16

feet. What is the height of the triangle? 71. The value of a new television is $4600. If its value depreciates linearly 6% per year, between what

years of use its value is between $3220 and $2392?

Chapter 4 Quadratic Function IX. Given the following quadratic functions determine:

43)( 2+−= xxxf

72. Where does the graph open? (Concavity) 73. Vertex 74. Vertex form equation 75. Symmetry axis 76. Maximum/minimum value of f(x) 77. Nature of the roots or zeros of the function 78. Zeros of the function (x-intercepts) 79. y-intercept 80. Domain and range 81. Graph

xx 1052 2=+


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822++−= xxy


102. Find the expression of the quadratic function that passes through points (3, 8), (–1, –12), (–4, –6).

103. The manager of a symphony computes that the symphony will earn PP 110040 2+− per concert if

they charge P dollars for tickets. What ticket price should the symphony charge in order to maximize its profits?

104. A theater operator predicts that the theater can make xx 1604 2+− dollars per show if tickets are

priced at x dollars. What are the maximum profits the theater can earn?

105. David threw a baseball into the air. The function of the height of the baseball in feet is 21680 tth −= ,

where t represents the time in seconds after the ball was thrown. What is the maximum height reached? At what time?

X. Simplify the following imaginary or complex numbers

106. )5)(2)(3( iii −

107. 11

i

108. 429i

109. )412()87( ii −−+−

110. )37()410( ii +−−

111. )32)(67( ii −−

112. i

i

3

68 −

113. i

i

24

3

+

Chapter 5 Polynomial Functions XI. Determine if the binomial is a factor or not of the given polynomial function.

114. 15262)( 2++= xxxf 4+x

115. 103113)( 234++−−= xxxxxf 5−x

XII. Divide or evaluate the following polynomials. Use synthetic division and reminder theorem.

116. 15192)( 23+−−= xxxxf ÷ 3−x

117. 4263)( 34+−+= xxxxf ÷ 2+x

118. 2053243)( 234−−−+= xxxxxf )4(−f

119. 15132)( 2++= xxxf )5(−f

XIII. Find all the zeros of each function. Hence, give the factors.

120. 10136)( 23−+−= xxxxg

121. 863)( 23−−+= xxxxq

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122. 8021)( 24+−= xxxf

123. Determine the value of k for which x + 3 is a factor of 66 23++− kxxx

124. Write a polynomial function of least degree that has 3, -1, and 2 as zeros and f (1) = 8.

125. One of the zeros of the function 2425)( 23−−+= xxxxf is –3. Find the addition of the other two.

Chapter 6 Rational Functions XIV. Determine what they ask you for

49

284)(

2−

−=

x

xxf

126. Domain 127. Vertical asymptote 128. Horizontal asymptote 129. Coordinates of the

removable discontinuity

3613

16)(

2

2

+−

−=

xx

xxf



158

152)(

2

2

++

−−=

xx

xxxf



Chapter 8 Irrational Functions XV. Find the domain and range of the following irrational functions

138. yc =+ 342

139. 2113 +−= xy

140. yx =− 215

141. yx

=+

−52

13

Let 524)( −+= xxf

142. Find f(15) 143. Determine the value of x if f(x) = 4

Chapter 9 Variations of proportionality XVI. Solve the following problems

144. When a body falls freely form the rest, the travelled distance is directly proportional to the square of time fall. If a body falls 44.1m in three seconds, calculate the distance at 3.6 seconds.

145. 8 men can perform a work in 9 hours. How long does it takes to 12 men? 146. The volume of a gas enclosed in a container varies inversely proportional to pressure, if the temperature

is constant. If the volume of the gas is 405 in3 and the pressure is 16psi, find its volume when the pressure is 20psi.

Math III - 1st Indicative Requirement (2010)

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