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Test Two This is a self-diagnostic test. Every pair of questions relates to a worksheet in a series available in the MUMS the WORD series. For example question 5 relates to worksheet 2.5 Arithmetic with Surds. If you score 100% on this test and test 1 then we feel you are adequately prepared for your introductory mathematics course. For those of you who had trouble with a few of the questions, we recommend working through the appproriate worksheets and associated computer aided learning packages in this series. 1. (a) Find the highest common factor of 6uvw and 18uv, where u, v, and w are prime numbers. (b) Factorize 3xy + 6y + 3y 2 2. (a) Solve for x, x 4 = 3 (b) Solve for y, 6(y + 2) = 30 3. (a) Simplify 2 3 + x+2 x (b) Solve for x in the equation 2x+2 3x+5 4. (a) Draw x > −3 on the number line. = 2 (b) Rewrite x > 3 without the absolute value signs. 5. (a) Simplify (1 + √ 2)(1 − √ 2) (b) Rationalize the denominator: 6. (a) Factorize x 2 + 6x + 8 (b) Simplify x+3 x 2 −9 7. (a) Is x − 3 a factor of x 3 − 27? 1 2+ √ 2 (b) Factorize x 3 + x 2 + x − 14 as far as you can with integer coefficients. 8. (a) If log 2 x 8 (b) Simplify 3 log e y = 3, what is x? 9. (a) In diagram A below, what is θ? (b) In diagram B below, what is sin θ, cos θ, and tan θ? What is θ? ❅ ❅❅❅❅❅ 45 ◦ θ Diagram A ❆ ❆❆❆❆❆ √ 2 3 θ Diagram B 10. (a) If I drew y = 5x + 3 on graph paper, what would it look like? (b) What is the slope and the y-intercept of the line 3y = 5x + 2? Page 2
Worksheet 2.1 Factors of Algebraic Expressions Section 1 Finding Common Terms In worksheet 1.2 we talked about factors of whole numbers. Remember, if a × b = ab then a is a factor of ab and b is a factor of ab. In a similar way we can look at the factors of an algebraic expression. So, for instance, 3uv has factors 1, 3, u, v and combinations of these like 3u, 3v, uv and of course 3uv. Other examples:- 1. 2ab = 2 × a × b has factors 1, 2, a, b, 2a, 2b, ab and 2ab. 2. uvz has factors u, v, z, uv, uz, vz, 1, uvz. 3. xy has factors 1, x, y and xy. In simplifying algebraic expressions we need to find common factors. Given two terms in an algebraic expression the common factors are those things which divide both of them. Example 1 : (a) 3uv and 6u have common factors 1, 3, u and 3u. (b) 2xy and 4xyz have common factors 1, 2, x, y, 2x, 2y, 2xy and xy. The highest common factor is, as was the case with numbers, the biggest or largest factor that divides two expressions. So the highest common factor of 3uv and 6u (from example 1(a)) is 3u; the highest common factor of 2xy and 4xyz (from example 1(b)) is 2xy. As with whole numbers we can also find the smallest algebraic expression that is a multiple of two expressions. This is called the lowest common multiple. Example 2 : 3uv and 6u have as their lowest common multiple 6uv since both 3uv and 6u divide into 6uv and do not both divide into a smaller expression. To find the lowest common multiple we take the highest common factor and then multiply it by whatever is missing from each expression. From the example above 3uv and 6u have highest common factor 3u. Now 3uv = 3u × v and 6u = 3u × 2 so the lowest common multiple is 3u × v × 2 = 6uv. Page 3
Page 1: Table of Contents Module 1 1.1 Orde
Page 5 and 6: By factorizing we can make a slight
Page 7 and 8: Example 3 : 4a + 2ab = 2a 2a(2 + b)
Page 9 and 10: 1. What are the factors of 18? 2. W
Page 11 and 12: In each of the above calculations,
Page 13 and 14: Solution 1 Solution 2 Putting in J
Page 15 and 16: Exercises: 1. Solve the following e
Page 17 and 18: Worksheet 2.3 Algebraic Fractions S
Page 19 and 20: y a fraction is the same operation
Page 21 and 22: Section 3 Solving Equations Sometim
Page 23 and 24: 2. Solve for x in terms of y: (a) 3
Page 25 and 26: Worksheet 2.4 Introduction to Inequ
Page 27 and 28: Example 7 : The number of matches i
Page 29 and 30: Example 6 : Exercises: Example 7 :
Page 31 and 32: Example 3 : this is the same as Exa
Page 33 and 34: Exercises for Worksheet 2.4 1. (a)
Page 35 and 36: Note: After expansion of this expre
Page 37 and 38: Example 2 : Example 3 : √ √ 7 +
Page 39 and 40: Example 4 : Example 5 : Example 6 :
Page 41 and 42: 1. (a) √ 2 × √ 2 (b) ( √ 5)
Page 43 and 44: Exercises: 1. Factorize the followi
Page 45 and 46: Exercises: 1. Expand the following,
Page 47 and 48: Section 4 Factorizing Quadratics Be
Page 49 and 50: Example Factorize 6x 2 − x − 12
Page 51 and 52: the quadratic formula. Note that a
Page 53 and 54:
Example 3 : Solve (x + 3) 2 = x + 5
Page 55 and 56:
Section 7 Multiplication and Divisi
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Worksheet 2.7 Logarithms and Expone
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Note that for all of the above prop
Page 61 and 62:
exactly in decimal form, but it is
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1. Evaluate (a) log 10 1000 (b) log
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Example 4 : If CD and AB are parall
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The triangle ABC in the above examp
Page 69 and 70:
Section 3 Pythagoras’ Theorem All
Page 71 and 72:
Section 4 Introductory Trigonometry
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Exercises: 1. Find the following ra
Page 75 and 76:
Worksheet 2.9 Introduction to the C
Page 77 and 78:
Exercises: 1. Complete the table of
Page 79 and 80:
Example 1 : y = 2x + 3. The gradien
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Example 8 : The lines y = 10x + 3 y
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which doesn’t make sense as we ca
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2. (a) Plot the points X(−1, 2) a
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Worksheet 2.1 Section 1 1. (a) 6 (b
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2. (a) -1 (b) 46,47 (c) 22,24 Works
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Exercises 2.4 1. (a) i. ii. iii. iv
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2. (a) 13 √ 3 (b) −12 √ 5 (c)
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2. (a) 5x2 +13x+9 (x+2)(x+3) (b) 2(
Page 97 and 98:
Exercises 2.7 1. (a) 3 (b) 0 (c) 3
Page 99 and 100:
2. (a) Section 2 1. (a) & (d) Secti
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