Number Sets, Measurements, and Laws of Algebra

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3 Number Sets, Measurements, and Laws of Algebra

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7 Sequences and Series

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9 Descriptive Statistics

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11 Sets and Venn Diagrams

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15 Area of a triangle: Law of Sines: 1 2 ab sinc a sin A = b sin B = c sinc sin A a = sin B b = sinc c Law of Cosines: a 2 = b 2 + c 2-2bc cos A cos A = b2 + c 2 - a 2 2bc 1. The quadrilateral ABCD has AB = 10 cm, AD = 12 cm and CD = 7 cm. The size of angle ABC is 100 and the size of angle ACB is 50. a) Find the length of AC in centimetres. b) Find the size of angle ADC.

16 2. A greenhouse ABCDPQ is constructed on a rectangular concrete base ABCD and is made of glass. Its shape is a right prism, with cross section, ABQ, an isosceles triangle. The length of BC is 50 m, the length of AB is 10 m and the size of angle QBA is 35. a) Write down the size of angle AQB. b) Calculate the length of AQ. c) Calculate the length of AC. d) Show that the length of CQ is m, correct to 4 significant figures. e) Find the size of the angle AQC. f) Calculate the total area of the glass needed to construct i) the two rectangular faces of the greenhouse; ii) the two triangular faces of the greenhouse. g) The cost of one square metre of glass used to construct the greenhouse is 4.80 USD. Calculate the cost of glass to make the greenhouse. Give your answer correct to the nearest 100 USD.

17 3. A cross-country running course consists of a beach section and a forest section. Competitors run from to, then from to and from back to. The running course from to is along the beach, while the course from, through and back to, is through the forest. The course is shown on the following diagram. Angle is. a) It takes Sarah minutes and seconds to run from to at a speed of. Using distance = speed time, show that the distance from to is metres correct to 3 significant figures. b) The distance from to is metres. Running this part of the course takes Sarah minutes and seconds. Calculate the speed, in, that Sarah runs from to.

18 c) The distance from to is metres. Running this part of the course takes Sarah minutes and seconds. Calculate the distance, in metres, from to. d) The distance from to is metres. Running this part of the course takes Sarah minutes and seconds. Calculate the total distance, in metres, of the cross-country running course. e) The distance from to is metres. Running this part of the course takes Sarah minutes and seconds. Find the size of angle. f) The distance from to is metres. Running this part of the course takes Sarah minutes and seconds. Calculate the area of the cross-country course bounded by the lines, and.

19 4. In triangle,, and. i. diagram not to scale a) Find the length of. b) is the point on such that. Find the length of. c) is the point on such that. Find the area of triangle.

20 5. ABC is a triangular field on horizontal ground. The lengths of AB and AC are 70 m and 50 m respectively. The size of angle BCA is 78. a) Find the size of angle. b) Find the area of the triangular field. c) is the midpoint of. Find the length of.

21 Logic 1. Police in a town are investigating the theft of mobile phones one evening from three cafés, Alan s Diner, Sarah s Snackbar and Pete s Eats. They interviewed two suspects, Matthew and Anna about that evening. Matthew said: I visited Pete s Eats and visited Alan s Diner and I did not visit Sarah s Snackbar Let p, q and r be the statements: p : I visited Alan s Diner q : I visited Sarah s Snackbar r : I visited Pete s Eats (a) Write down Matthew s statement in symbolic logic form. What Anna said was lost by the police, but in symbolic form it was (q r) p (b) Write down, in words, what Anna said. 2. In a particular school, students must choose at least one of three optional subjects: art, psychology or history. Consider the following propositions a: I choose art, p: I choose psychology, h: I choose history. (a) Write, in words, the compound proposition h ( p a).

22 (b) Complete the truth table for a p. a p a a p T T T F F T F F (c) State whether a p is a tautology, a contradiction or neither. Justify your answer. 3. Consider two propositions p and q. (a) Complete the truth table below. p q q p q p p q T T T F F T F F (b) Decide whether the compound proposition (p q) ( p q) is a tautology. State the reason for your decision.

23 4. (a) Complete the truth table shown below. p q p q p (p q) (p ( p q)) p T T F F T F T F (b) State whether the compound proposition (p ( p q)) p is a contradiction, a tautology or neither. Consider the following propositions. p: Feng finishes his homework q: Feng goes to the football match (c) Write in symbolic form the following proposition. If Feng does not go to the football match then Feng finishes his homework. (d) Given the basic implication p Þ q above, write in words, its (i) inverse (ii) converse (iii) contrapositive.

24 5. (a) Complete the following truth table. p q... p q T T F T F T F T F F F T Consider the propositions p: Cristina understands logic q: Cristina will do well on the logic test. (b) Write down the following compound proposition in symbolic form. If Cristina understands logic then she will do well on the logic test (c) Write down in words the contrapositive of the proposition given in part (b). 6. Consider the two propositions p and q. p: The sun is shining q: I will go swimming Write in words the compound propositions (a) p q; (b) p q.

25 The truth table for these compound propositions is given below. p q p q p p q T T T T T F F F F T T T F F T T (c) Complete the column for p. (d) State the relationship between the compound propositions p q and p q. 7. Consider the following propositions (a) Write in words (q r) p p: The number is a multiple of five. q: The number is even. r: The number ends in zero. (b) Consider the statement If the number is a multiple of five, and is not even then it will not end in zero. (i) (ii) Write this statement in symbolic form. Write the contrapositive of this statement in symbolic form.

26 8. Consider the statement p: If a quadrilateral is a square then the four sides of the quadrilateral are equal. (a) (b) Write down the inverse of statement p in words. Write down the converse of statement p in words. (c) Determine whether the converse of statement p is always true. Give an example to justify your answer.