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S-01 The Number System 1. Put T (true) or F (false) beside each statement. (a) -3 is not an integer (b) 6 is a rational number (c) 7 is a rational number 9 (d) (e) 7 is a natural number 9 (f) (g) 7 is a real number 9 3 is irrational (h) a natural number can’t be negative 4 is irrational (i) an integer is always rational (j) 0.2777654 is irrational (k) -7 is in (l) 150% is an integer 2. Place each number from the box in its correct place in the diagram below: -5 0.2 7 2 3 19 1 7 3 0 6 R Q Z N 3. Give an example of: (a) an irrational number which lies between 1 and 2 (c) a number which is in but not in (b) a number which is in but not in (d) a rational number which lies between 4. Two natural numbers are chosen at random. Put true or false beside each statement: (a) If the two numbers are added the answer must be a natural number. (b) If one number is divided by the other, the answer must be a natural number. (c) If one number is subtracted from the other, the answer must be a natural number. 1 2 and 3 2 S-02 Accuracy and Estimation When you’re doing a calculation on you r calculator and the answer is not exact the rule is to give answers general correct to 3 significant figures unless, of course, the question gives other instructions such as “correct to 4 decimal places”. 1. Use the 3 significant figures rule for these: (a) 11.32 (b) 2.3 4.286 11.77 (c) 232 382 (c) 7.382 (d) (e) 37.6 5.33 (f) 2.083 4. Give the answer correct to the specified level of accuracy. “dp” means “decimal places” and “sf” means “significant figures”. (a) Find the area. [2sf] (d) 17.7% of 10.89 (b) 1 2 (g) 120 49 (h) A motor-cyclist travels 18.5km in 11 minutes. Find the average speed (i) in km per minute (ii) in kmh-1 (i) Find the volume of a cuboid which is 4.6cm by 6.9cm by 8.2cm. (j) (3 3) 1.85m 83cm (e) Find the area of a rectangular field that 237m by 124m. (f) 77.5 3 3 (cube root!) [6dp] (c) A school fund-raising event raises 226,432 Japanese yen. There are 487 students in the school. Find the average amount raised per student. [nearest whole number] (d) 1.277 4389 (e) 71.7 2 24.3 (f) 0.2755 [4sf] [2sf] [3sf] 4 One situation where the 3 significant figures rule doesn’t make much sense is with MONEY. Often it is reasonable to give 2 decimal places : €12.56. 2. (a) When would it not make much sense to give 2 decimal places for money? Give 2 decimal places here : (b) Find 3.7% of $583.25 (c) €130 17 (d) Petrol costs €1.19 a litre. Find the cost of 8.55 litres. 3. Estimate these without a calculator. Give your estimate correct to 1 significant figure. (a) 9.7 4.6 (b) 12.6 1.95 (g) There are about 270 million people in the USA. Estimate the number who are female and left-handed and aged over 70 years old. [1sf] (h) 1 (1 0.9856)3 [nearest thousand] 5. Find the percentage error (correct to 2 significant figures). true value approx value 68.544 107.4 2 69 110 1.41 22 7 50 49 % error S-03 Standard Form : a 10k (where 1 a 10 and k 1. (a) One of the statements in (b) – (g) below is wildly wrong! Which one? In (b) to (g) put the number used in the statement into the form a 10k , where 1 a 10 and k . (b) A sheet of normal A4 paper weighs about 0.000005 tonnes. (c) The total population of China and India is about 2.1 billion people. 3. These can be done purely by brain-work or purely by calculator : p 4 106 q 6 105 r 3 108 Find the value of these, giving answers in the form a 10k , where 1 a 10 and k : (a) pr (b) q p (c) p (d) r 20 th (d) On your 17 birthday you are about 536,000,000 seconds old. (e) There are about 500,000,000,000 atoms in a slice of toast. (e) q 2 (f) p + r (f) A normal sized novel might contain about 120,000 words. (g) pqr (h) (g) If you throw four normal 6-sided dice, your chance of getting four 6s is about 0.00077. r q (i) 6% of q (j) qr + p 2. Put these numbers in the form a 10 , where 1 a 10 and k : k 4. Given f 5.5 105 g 1.6 107 : (a) 427000 (b) 0.000042 (c) 108,000,000,000,000 (a) Which is larger, 7f or (b) Solve for x : g ? 5 fx = g (d) 0.00000000000000155 1 in the form a 10k , where fg 1 a 10 and k . In these, perform the calculation then make sure the value of a (in 1 a 10 ) is given correct to three significant figures: (d) True or false: g + 100f = 71000000 (c) Write (e) 1.270 (f) 0.2275 0.00986 ) (e) Calculate f 20 giving your answer in the form a 10k , where 1 a 10 and k . 1 (f) Write as a normal decimal: g S-04 Arithmetic Sequence 1. Write down the next two terms in these arithmetic sequences: (a) (b) (c) (d) (e) 7 , 11 , 15 , ... 3.8 , 5.1, 6.4 , ... 20 , 18 , 16 , ... 33 , 19 , 5 , ... 1 1 1 , , ,... 6 3 2 6. Find the sum of the first 10 terms of each arithmetic sequence: (a) (b) (c) (d) 7. Here you can see in the square brackets how many terms of the arithmetic sequence to sum. 2. The first four terms of some sequences are given below. Which ones are not arithmetic sequences? (a) (b) (c) (d) (e) 1 , 2 , 4 , 8 , ... 2 , 4 , 6 , 8 , ... 1 , 3 , 6 , 10 , ... 5 , 1 , –3 , –7 40 , 20 , 10 , 5 3. In this question you are given the first two terms of arithmetic sequences. That’s enough to be able to work out the term shown in square brackets. (a) 12 , 19 , ... (b) 4.6 , 5.2 , ... (c) 30 , 26 , ... (d) 1 , 3 , ... (e) 36 , 34.5 , ... (f) 100 , 97 , ... (g) 22 , 22.8 , ... u 20 u13 u10 u 40 u 25 u 22 u 25 4. The three sequences below are all arithmetic. Find the values of the letters a to g: 4 , a , 16 , b c , 21 , d , e , 27 –3 , f , 10 , g 5. In each arithmetic sequence below, find the least value of n such that u n 100 . (a) (b) (c) 17 , 25 , 33 , ... –25 , –18 , –11 , ... 24 , 33 , 42 , ... 4 , 7 , 10 , ... 2 , 2.3 , 2.6 , ... 8 , 6.5 , 5 , ... 5 , 10 , 15 , ... (a) 13 , 16 , 19 , ... (b) 1.2 , 1.6 , 2 , ... (c) –20 , –14 , –8 , ... (d) 1 3 1 , , , ... 4 8 2 (e) 2 , 4 , 6 , ... S14 S30 S15 S25 S100 8. The nth term of an arithmetic sequence is given by the formula: u n 5 3n (a) Find u1 , u 2 and u 3 . (b) Find S16 . 9. 13 + 17 + 21 + ..... + 49 + 53 (a) How many terms are there in this sum? (b) Now find the sum. 10. Find: (a) 11 + 18 + ..... + 81 + 88 + 95 (b) –3 + 1 + 5 + ..... + 45 + 49 11. A child’s parents put $100 into a bank account for her on her first birthday, then $120 on her second birthday, and so on, increasing the amount by $20 each year. How much will they have paid into her account the day after her 18th birthday? 12. The second term of an arithmetic sequence is 32 and the fourth term is 26. Work out: (a) the common difference (b) the first term (c) S18 . S-05 1. 2. 3. Geometric Sequences Give the next two terms of each geometric sequence: (a) 3 , 6 , 12 , 24 , … (b) 1 , 3 , 9 , 27 , … (c) 0.2 , 1 , 5 , 25 , … (d) 80 , 40 , 20 , 10 , … (e) 2 , –6 , 18 , –54 , … 5. 20 , 30 , 45 , … (b) 2 , 5 , 12.5 , … (c) 54 , 36 , 24 , … (d) 48 , 36 , 27 , … 1.5 , 3 , 6 , 12 , … [ u10 ] (b) 0.2 , 0.6 , 1.8 , … [ u12 ] (c) 4 , –12 , 36 , … [ u8 ] 6. Give answers correct to 4 significant figures: 4. (b) 0.2 , 0.6 , 1.8 , … S9 (c) 3 , 6 , 12 , … S8 (d) 64 , 96 , 144 S10 (e) 4 , 5 , 6.25 , … S20 (f) 4 , –6 , 9 , … S8 S15 Give these correct to 4 decimal places (a) (d) 4,6,9,… [ u15 ] (e) 12 , 4 , 4 [ u15 ] (f) 1 , 0.5 , 0.25 , … (g) 10 , 8 , 6.4 ,… S10 (g) 8 , 6 , 4.5 , … For each geometric sequence find the term given in square brackets: 3 (a) 1 , 2 , 4 , … Give these correct to 3 significant figures: For each geometric sequence below find (i) the common ratio, and (ii) the next term: (a) Find the sum of the required number of terms of each geometric sequence: , 1 9 ,… S6 (i) 1 , 1 3 , 1 9 ,… S18 Find the final value of the investments, correct to the nearest whole number. Interest is compounded annually. €5000 $25000 ₤1000000 $1 Interest (per annum) 2% 1.5% 5% 5% Time FINAL VALUE 10 yrs 4 yrs 18 mths 100 yrs 7. If 10000 pesos are put in an account for 3 years earning 7.5% interest per annum compounded annually, how much interest will be earned (correct to the nearest peso)? 8. A child’s parents put $100 into a bank account for her on her first birthday, then $120 on her second birthday, and so on, increasing the amount by 20% each year. How much (correct to the nearest dollar) will they have paid into her account the day after her 18th birthday? 9. Use a GDC method. [ u 20 ] Use your GDC to solve these. (Trial (a) In the geometric sequence 6 , 15 , 37.5 , … what is the first term that is greater than 200? (b) In the geometric sequence 14 , 7 , 3.5 , .. what is the least value of n such that u n 0.01 ? (c) In the geometric sequence 5 , 6 , 7.2 , … what is the least value of n such that u n 50 ? 1 3 Amount invested [ u12 ] and error is an acceptable method, but show some working!) (h) 1 , $5000 was put in an account earning r% interest per annum compounded annually. At the end of 5 years the value of the investment was $5600. What is the value of r correct to 2 decimal places? S-06 Pairs of Linear Equations Answer on this sheet 1. These are easy to solve without a GDC: (a) a + b = 11 a – b = 3 (b) a + b = 11 2a – b = 1 (c) x + 2y = 15 3x – 2y = 29 (d) p + q = 17 p – 2q = -1 (e) f – 3g = 7 2f + g = 21 (f) 2y + z = 11 y + z = 8 (g) m + 3n = 14 2m – n = 0 (h) x + y = 21 2x + 3y = 50 (i) 3a – 5b = 1 b = a - 3 2. Using the graph on the right, write down solutions to these pairs of linear equations. Estimate answers correct to 1 decimal place where necessary. 3. (a) 2x + 3y = 11 4x – y = 8 (b) x + y = –4 x – 2y = 6 (c) 2x + 3y = 11 x – 2y = 6 (d) a – 2b = 6 4a – b = 8 Use your GDC to write down solutions to these pairs of linear equations. Give answers correct to 2 decimal places where necessary. (a) 7x + 11y = 30 x – 5y = 11 (b) 3x + 7y = 20 5x – 3y = 30 (c) 3x + 5y = 16 2x + 9y = 39 (d) y = 11x – 5 (e) 9x – 5y = 20 5x + 6y = 100 7x – 4y = 10 S-07 1. Quadratic Equations Write down the solutions to these quadratic equations: 3. (a) x 2 4 These need re-arranging then factoring then solving: (a) x 2 2x 24 (b) (x 2)(x 5) 0 (c) (x 3)(x 4) 0 (b) x 2 x 20 (d) (x 6)(x 6) 0 (e) (x 1)(x 4) 0 (c) x 2 x 6 (f) x(x 8) 0 (g) (x 3)2 0 (h) (x 2)(2x 5) 0 (d) x(x 3) 10 (i) (3x 2)(4x 7) 0 2. These need factoring before you can solve them: 4. (a) x 2 8x 12 0 (b) x 2 4x 3 0 (c) x 5x 4 0 (d) x 2 x 12 0 (e) x 2 7x 10 0 Use your GDC or “the formula” or any other method to solve these: (a) x 2 4x 13 0 (b) x 2 x 100 (c) 3x 2 x 12 0 2 (d) x(x 5) 22 (e) (x 3)2 10 (f) 20 2x 3x 2 0 (g) x 2 (f) x 2 4x 21 0 5. (g) x 2 2x 15 0 (h) x 2 4x 0 (i) x 2 x 30 0 (a) 50 3x 4 Write down a quadratic equation whose roots are x = 2 and x = 4. (b) What happens if you try to solve x 2 4x 7 0 ? (c) Use any method to solve x(2x 5) (x 1)(x 4) . (d) Write down three roots of the equation. (x 1)(x 4)(x 2) 0 . S-08a Word Problems Involving Equations S-08b Word Problems Involving Sequences 1. A coffee and three donuts will cost you $2.50, while three coffees and five donuts will cost you $5.30. If a coffee costs c cents and a donut costs d cents, make two linear equations and solve them by any method. How much would you pay for two coffees and seven donuts? 2. A rectangle whose area is 80cm2 is 4cm longer than it is wide – see diagram. x x+4 1. $20000 is invested at 4% interest. Below, find the value of the investment (i) if it is simple interest, and (ii) if it is compounded annually. Answer correct to the nearest dollar. (a) after 1 year (c) after 20 years. (b) after 5 years 2. The human population of the earth is currently about 6 billion. Find (in billions, correct to one decimal place) what the population would be in 50 years time if it grew at 80cm2 Solve the quadratic equation suggested by the diagram and find the perimeter of the rectangle. 3. If you subtract twice the son’s age from the mother’s age the answer is 12. (a) 1% per annum (c) 5% per annum. (b) 3% per annum 3. 7 sections of the spiral below have been drawn. From the center outwards the lengths are 1 , 1 , 2 , 2 , 3, 3 and 4 units. Find the length of the spiral after 20 sections have been drawn. If you double the mother’s age and add it to the son’s age the answer is 99. Form two linear equations and find the age of each. 4. When two consecutive odd integers are multiplied together the answer is 4623. (a) If the smaller of the two integers is n, what is the larger? (b) Form a quadratic equation in n and solve it. 5. A box contains x 5-cent coins and y 2-cent coins. There are 33 coins altogether and their total value is €1.02. Form a pair of linear equations and solve them. 4. A woman invested $1000 at 3% simple interest on 1 Jan 1995, then another $1000 at 3% simple interest on 1 Jan 1996, and so on. Find the total value of her investment on 31 Dec 2005. 5. A car is estimated to lose 20% of its value for each year of its age. Find (correct to the nearest hundred dollars) the value of a car which costs $35000 new, after: (a) 1 year (b) 2 year (c) 5 years (d) 10 years S-09 Sets 1 : Notation 1. U = {a, b, c, d, e, f, g, h} 2. U = {x x and 2 x 12 } A = {a, d, f, h} B = {b, c, e, g} C = {a, c, f} D = {b, c, d, e} List the elements of: (a) B C W = {multiples of 3} X = {factors of 24} Y = {prime numbers} List the elements of W, X and Y. Check your answers are correct before continuing! Now list the elements of: (b) C D (c) C (d) C B (e) ( B C ) (f) D (g) D C (h) D A (i) B (f) n(U ) 10 (j) B D (g) {3, 6} (W X ) (k) ( B D) Find: (a) W (b) X Y (c) (W X ) (d) X Y (e) W X Y True or false: (h) Find: (l) n(U ) (m) n(C D) (n) n( A B ) True or false: n(W Y ) 3. U = { , , , , , } A = { , , , } B = { , } C = { , } D = { , } F = { , , , } (o) A B E = { , , } (p) g B (a) Which sets are subsets of A? (q) f C (b) C is a subset of two other sets (apart from (r) (B C) D U). Which ones? Solve these: (c) A X U . Which sets could X stand for (s) Construct a set E such that: (apart from U)? n( E ) = 3 E D = a E (t) How many subsets of A having exactly two elements can you find? (u) Find B (C D) and ( B C ) D . (d) Find n( F ( B D)) . (e) True or false: (i) A D F (ii) C D (f) List the elements of B E . (g) B Y . Which sets could X stand for? S-10 Sets 2 :Venn Diagrams 1. Represent the sets below with a Venn Diagram, putting each element in its correct place in the diagram. (a) (b) U A B C = = = = {1 , 2 , 3 , 4 , 5 , 6} {4} {1 , 5 , 6} {2 , 5} U W X Y = = = = {x x and 2 x 12 } {multiples of 3} {factors of 24} {prime numbers} 2. Make copies of this Venn Diagram for each part of the question below. 5. Students at Polyglot High School all study at least one language other than English. The three languages on offer are Spanish, German and Russian. An interview with a class of 22 students produced the following information: 3 students study all three languages nobody studies Russian only, but 2 students study German only and 4 study Spanish only 5 students study Spanish and German (but not Russian) 3 students study Spanish and Russian (but not German). By making a Venn Diagram find: (a) the number of students who study German and Russian (but not Spanish), Shade the area corresponding to: (b) the number of students who study German. (a) P Q (b) Q (c) ( P Q ) (d) P Q 3. Use the Venn Diagram above to list the elements of: (a) X Y (b) Y Z (c) X Y Z (d) ( X Y ) Z (e) ( X Z ) (f) Y X 4. Two sets A and B have the following properties: n( A B ) 4 n( B ) 9 n( A) 7 n(U ) 20 By making a Venn Diagram or otherwise find n( A). 6. By a curious coincidence students at Polyglot High School can only practice three sports: athletics, basketball and cricket. In the same class of 22 students the following facts were found: 2 students practiced no sport the number of students who only practiced athletics was equal to the number who only practiced cricket. Call this number x. 4 practiced all three sports 5 students practiced exactly two sports 3 students only practiced basketball. By making a Venn Diagram find: (a) the value of x, (b) the number of students who practiced exactly one sport. S-11 Logic 1 : Propositions and Basic Symbols ( ) 1. Tick the propositions: 5. N stands for a positive integer in this question. (a) 2 + 2 = 4 (b) 10 - 7 a : N is even (c) Italian carrots have blue ears. b : N is greater than 10 (d) Their arre know speling misstakes hear. c : N is less than 15 (f) Go away! d : N is a multiple of 4 e : N is 14 f : N is less than 6 (g) 3y – 7 2. Assign T (true) or F (false) to each proposition: Assign T or F to each proposition: (a) 2 is the only even prime number (a) ea (b) ad (b) The Mathematical Studies course involves two examination papers. (c) cf (d) e f (e) e d (f) (f d) N is 4 (c) –20 –2 (d) Indonesia has a larger population than Japan. Write in symbols: (g) N is an even number greater than 10. 3. p : I am happy (h) N is less than 6 or greater than 10. q : I am working (i) N is greater than or equal to 15. r I am at my computer (j) N is an odd number and less than 6. : Write a short sentence corresponding to: (a) p (b) qr (c) p q (d) (e) p q r (p) (f) q q (g) qp (k) If N is greater than 10, then N is not less than 6. (l) N lies between 11 and 14 (inclusive). (m) N is an even number, but not a multiple of 4, and is less than or equal to 10. Assign T or F to each proposition for the given N value: (n) b c [N = 16] (o) a f [N = 11] (p) (d c) f [N = 4] (q) (d c) f [N = 40] (r) (d c) f [N = 14] S-12 Logic 2 : Truth Tables ( ) Complete these truth tables. 1. p T T F F q T F T F 2. p T T F F p q T F T F p T T F F p T T F F 5. p T T F F q T F T F q T F T F pq p T T T T F F F F q T T F F T T F F r pq T F T F T F T F r (p q) r p q 7. Complete these truth tables. What do you notice about the final column of each? 3. 4. 6. q T F T F q pq T F T F p q pq p q q T F T F p T T F F pq pq p T T F F (p q) 8. Construct the truth table for (p q) q : p p (p q) p Compare this table with the table in Q2 and comment: (p q) (pq) q pq (p q) q S-13 Logic 3 : Tautology, Contradiction and Implication ( ) 1. Complete this truth table for the proposition (p q) p . Use your truth table to explain why this proposition is a tautology. p T T F F q T F T F pq 5. “If I study logic, then my mind improves.” (p q) p Give the following related implications: INVERSE: CONVERSE: CONTRAPOSITIVE: 2. Complete this truth table for the proposition (p q) q . Use your truth table to explain why this proposition is a contradiction. p T T F F q T F T F p pq p (p q) 3. Complete this truth table for the proposition (p q) p . Use your truth table to explain why this proposition is a tautology. p T T F F q T F T F p pq (p q) p 6. Complete this truth table and comment on the result: p q T T F F T F T F : N is even q : N is a multiple of 5 r : N is odd s : N is a multiple of 2 t : (N+1) is odd u : N is a multiple of 10 Assign a truth value to each of these: (a) pt (b) s r (c) qr (d) (q r) u (e) s u (f) u q q (p q) q (p q) q p 7. Use the truth tables below to show that (p q) and p q are logically equivalent: p T T F F 4. N stands for a positive integer in this question. p pq p T T F F q T F T F q T F T F pq p q (p q) p q S-14 Probability 1 : Sample Spaces It’s usually easiest to give probabilities as FRACTIONS. 1. In a bag there 6 red counters, 4 green counters, 3 yellow counters and 2 blue counters. If a counter is chosen at random, find the probability that it is: (a) yellow (b) not yellow (c) green or blue (d) not red or yellow. In fact the counter is yellow. It is not put back in the bag. If another counter is now chosen, find the probability that it is: (e) green or yellow 2. (f) not blue. 4. If a student is chosen at random, find the probability that the student is: (a) a girl (b) aged 16 (c) a 17-year-old girl (d) a 16-year-old or a boy. (e) A girl is chosen at random. Find the probability that she is not 18 years old. 5. A coin has its two faces marked ‘1’ and ‘2’. A die has its six faces marked from ‘1’ to ‘6’ in the usual way. The sample space can be shown like this: V I E T N A M 1 2 3 4 5 6 If one of the seven letters above is chosen at random, find the probability that it is: 1 2 If the coin and the die are thrown together, find the probability that the two numbers seen: (a) a vowel (b) in the word MEXICO. (a) add up to 4 (b) add up to 5 or more 3. The natural numbers from 1 to 15 are written on counters. If a counter is chosen at random, find the probability that its number is: (a) even (c) are the same. (d) Find also the probability that the number on the die is larger than the number on the coin. (b) a factor of 12 6. Questions involving throwing two dice can be solved with a larger version of the grid in Q5: (c) a multiple of 3 (d) an even multiple of 3 1 2 3 4 5 6 (e) an even number or a multiple of 3 (f) less than 6 (g) greater than 20. 4. The ages and sexes of participants in a school trip are summarised below: aged 16 aged 17 aged 18 boys 9 11 4 girls 10 12 2 So, for example, there were nine 16-year old boys on the trip. 1 2 3 4 5 6 If two dice are thrown, find the probability that: (a) the scores add up to 9, (b) the scores add up to 4 or less, (c) the score on each die is the same, (d) the product of the scores is 18 or more, (e) the scores differ by 2 or less. S-15 Probability 2 : Independent Events and Tree Diagrams 1. In each question below the two events are independent. (a) I throw a coin and a die (dice). Find the probability that I get a Tail and a 5. 3. The probability that a student works hard on any day is 0.6. If she works hard, the probability that she passes the test next day is 0.8. If she doesn’t work hard then that probability is only 0.5. Complete the tree diagram below, and find the probability that she passes the test. (b) I throw a die twice in a row. Find the probability that the first is a 2 and the second is 5 or 6. pass work fail (c) I throw a die twice in a row. Find the probability that both times it shows 3 or more. pass not work (d) I pick a day of the week at random, twice in a row. Find the probability that both are Friday. (e) I pick a day of the week at random, twice in a row. Find the probability that the first is Saturday and the second is not Saturday. fail 4. An unbiased coin is thrown three times. Complete the tree diagram below, and find the probability of getting (a) two heads and one tail (b) all three throws the same. (f) I pick a day of the week at random, twice in a row. Find the probability that both days contain the letter “r”. 2. In a bag are 6 green balls and 4 yellow balls. A ball is picked at random and its color is noted. It is replaced, and the process is repeated. (a) Complete the tree diagram below: G G 0.6 Y H T 5. The experiment in Q2 is repeated except this time the first ball is not replaced. (a) Think carefully about the entry 5 below 9 and complete the tree diagram. (b) Find the probability that both balls are the same color. G 5 9 G G Y 6 10 Y Y (b) Write down the probabilities that: G (i) both balls are green (ii) the two balls are the same color (iii) the two balls are different colors. Y Y S-16 Probability 3 : Mutually Exclusive Events and Venn Diagrams 1. Market research has been done on a group of 50 people asking them whether they were in interested in Archery, Billiards, Cooking or Diving. The numbers expressing an interest are shown in the Venn Diagram below. (a) If a person is chosen at random, find the probability that (s)he expresses an interest in: (i) Archery, (ii) Archery, but not Cooking, (iii) Billiards or Diving, (iv) exactly one of A, B, C and D, (v) three or more of A, B, C and D. 3. Here are three events involving throwing two normal 6-sided dice: A : there is at least one 3 B : the total is 10 or more C : there are no 5s or 6s Which pairs of events are mutually exclusive? 4. A group of students were surveyed about whether they studied Biology and/or Chemistry. In the Venn Diagrams of Q1 and Q2 the numbers of people were shown. In the Venn Diagram below the probabilities have been shown. (f) An interest in billiards is mutually exclusive with which other interest? (g) Name another pair of mutually exclusive interests. Find the probability that a student chosen at random: (a) studies Chemistry (b) does not study Biology. 5. In a small international school the breakdown of students by gender and grade is as follows: 2. A group of 30 American international business travelers were asked if they traveled regularly to Asia or Europe. 12 said ‘yes’ to Asia 19 said ‘yes’ to Europe 4 said ‘no’ to both Asia and Europe. (a) Thinking particularly about A E complete the Venn Diagram below. (b) A traveler is chosen at random, Find the probability (s)he says ‘yes’ to Asia and ‘no’ to Europe. male gr 9 - 12 25 20 gr 6 - 8 female 30 25 (a) With F representing Female and S representing Grade Six to Eight, complete the Venn Diagram below, which shows probabilities. (b) Find p(F S) . 6. A and B are two events. It is known that: p(A) = 0.6 ; p(B) = 0.3 ; p(A B) = 0.75. Find: (a) p(A B) (b) p(A B) . S-17 Probability 4 : Conditional Probability 1. In a small international school the breakdown of students by sex and grade is as follows: gr 9 - 12 gr 6 - 8 boy 25 20 girl 40 15 4. An unbiased 6-sided die is thrown twice in succession. (a) Find the probability that the total is 7 (not a conditional probability problem!). (b) Find p (total is 7 | the first die did not show 6). (a) Find the probability that a student chosen at random is in Grade 6 – 8 (not a conditional probability problem!). (b) Find the probability that a girl chosen at random is in Grade 6 – 8. HINT : completely (c) Comment on you answers to (a) and (b). 5. The Venn Diagram below shows the probabilities of students in a certain region studying Economics and Geography. ignore the ‘boy’ column. NOTE : expressed Question as (b) follows could have been E G 0.1 0.3 0.15 : Find the probability that a student chosen at random is in Grade 6 – 8, given that the student is a girl. (c) Find the probability that a student chosen at random is in Grade 9 – 12, given that the student is a boy. (d) Find the probability that a student chosen at random is a girl, given that the student is in Grade 9 – 12. 2. A 12-sided die (numbered from 1 to 12) is thrown . Find: (a) p(score is 9) (b) p(score is 9 | it’s a multiple of 3) (c) p(score is 7 or more | it’s 10 or less) Find: (a) p(E) (b) p(E | G) (c) p(G | E) (d) p(G | E ) 6. Use the Venn Diagram below to find: (a) p(A | B) (b) p(B | A) (c) Are A and B independent? Explain! A A 6 x 6 grid as a sample space will help with these. Two unbiased 6-sided dice are thrown. Find: B 0.49 0.21 0.09 (d) p(score is even | it’s not a multiple of 3) 3. 0.45 0.21 7. For events C and D it is known that p(C) = 0.25, p(D) = 0.35 and p( C D ) = 0.5. (a) Find p( C D ) . (a) p (total is 10 or more | there is at least one 6) (b) Find p(C | D) (b) p (total is 10 or more | dice show different scores) (c) p (total is 10 or more | there are no 6s) (d) p (total is 10 or more | there are no 1s or 2s) (e) p (there is at least one 6 | total is 10 or more) 8. X and Y are mutually exclusive events. p(X) = 0.4 and p(Y) = 0.3. Find: (a) p(X Y) (b) p(X Y) S-18 Functions f (x) 3x 2 x 1 h(x) 2 1. g(x) 10 3x 5. Find the domain and range of each function: (a) 6 Find the values of: 3 (a) f (6) (b) g(1) (c) h(5) (d) f ( 1) (e) g(1) (f) h(7) (g) f (2.5) 2 (h) g 3 (i) f (2) - f (3) (j) f (g(2)) 0 5 (b) 10 1 -1 1 2. Using the functions from Q1, solve: (a) f (x) 7 (b) g(x) 8 (c) h(x) 5 -1 6. The graph below shows y f (x) and y g(x) . (a) Find (i) f (6) (ii) g(1.5) (iii) g(7) 3. f :x 2x 1 h:x g:x x(x 3) (b) Solve (i) f (x) 4 sin(5x) (ii) g(x) 5 Find the values of: (a) h(6) (b) f (4) (c) g(4) (d) f (0) (e) f ( 1) (f) h(36) (g) g(3) (h) h(18) (i) f (10) (j) g(0.5) (c) Write down: (i) the domain of f (x) (ii) the range of g(x) (d) Write down three integers n for which g(n) f (n) 7 y=g(x) 4. Using the functions from Q3 solve these. Your GDC could be useful! (a) f (x) 7 (b) g(x) 28 y=f(x) (c) f (x) g(x) 0 7 S-19 Linear Functions Answer each question on a separate sheet of graph paper. 1. Construct axes with 4 x 4 using a scale of 2cm to 1 unit, and with 20 y 20 using a scale of 1cm to 2 units. Now construct the following lines accurately: (a) y 2x 3 (b) 2x 3y 12 (c) x 4y 6 0 (d) Write down the coordinates of the point of intersection of lines (a) and (b) correct to 1 decimal place. 2. For a small car hire company charges €22 per day plus 13 cents for each kilometer driven. (a) Find the total cost of hiring a car for a day if you travel (i) 50km (ii) 100km (iii)300km. (b) Use your answers to (a) to construct a graph showing the cost of hiring a car for any distance up to 300km. Put ‘distance’ on the x-axis using a scale of 1cm to 20km, and ‘cost’ on the y-axis using a scale of 1cm to 4€. (c) Using your graph find how many kilometers you traveled if your bill was €47. Note: show lines on your graph to indicate clearly that you have used your graph! 3. If you change euros for dollars at the Royal Island Principal Office of Fair Finance bank, they calculate as follows: deduct 5 euros as the bank’s charge convert the rest to dollars at the rate 1€ to 1.2$. 18 So if you change €20 you will receive 15 1.2 = $18, and the effective rate you have received is = 20 0.9€ per dollar. (a) Complete the table: € given $ received 10 20 18 100 (b) Using your own scale construct a graph to show what you will receive in dollars for any amount up to 100 euros. (c) Using your graph find how many dollars you will receive for 65 euros, and calculate the effective rate. (d) Using your graph find out how many euros you need to change to receive $100. 4. In Tokyo two taxi companies charge as follows: Company A : 250 yen per kilometer traveled. Company B : a fixed charge of 500 yen, plus 180 yen per kilometer traveled. (a) Construct a graph showing the cost for each company for journeys up to 15km. (b) Using your graph find how much more expensive Company A is (compared to Company B) for a journey of 5.5km. (c) Someone asks you which company is cheaper. What would your answer be? S-20 Quadratic Functions 1. Using your GDC (or otherwise) sketch each quadratic function below. Mark the x- and yintercepts and the coordinates of the vertex: 3. Find the coordinates of the vertex of each quadratic in Q2 (a) to (c). (a) y x 4x 3 2 (a) (b) (c) 4. Find the x-intercepts of: -2 4 (a) y x 2 2x 15 (b) y x 2 6x 8 y 6 x x2 (b) 5. (a) The graph of y x 2 2x c passes through the point (3, 11). Find the value of c. -4 4 (b) The graph of y x 2 bx 12 passes through the point (2, –2). Find the value of b. 6. Each sketch below shows a quadratic graph of the form y x 2 bx c . Find the values of b and c. y (x 4)(x 2) (c) (a) -6 6 2. Find the equation of the line of symmetry: (a) y x 2 6x 10 (b) y x 2 2x 5 (c) y 10 4x x 2 (d) y 2x 2 3x (b) S-21 Exponential Functions 1. The points below all lie on the graph of the exponential function y 4x . Find the value of a , b , c , d and e: (3 , a) (-1 , b) (0.5 , c) (d , 16) (e , 1) 2. The population, P animals, of an endangered mammal on an island at time t years after 1990 appears to be modeled by the exponential function P 1800 0.96t (a) Write down the population in 1990. (b) Find the population (correct to the nearest 10) in 2005. (c) By GDC and/or trial and error find in which year the population will first drop below 500. (d) Find the approximate number of years it takes for the population to halve. 3. Each sketch graph below shows y k b x (k, b ). Find the values of k and b. 4. (a) For the year 2010 find the estimated: (i) gross domestic product (ii) population (iii)gross domestic product per capita G Don’t forget 3 significant figures! P (b) Repeat the calculations of (a) for the year 2025. (c) Use a GDC to find in which year the gross domestic product per capita is estimated to exceed $20000 for the first time. 5. During an epidemic the number of new cases (per day) of an illness reaches a peak (maximum) of 45 on June 1st. From then on the number of new cases per day, N, is predicted to decline according to the formula N 45 1.17 t where t is the number of days after June 1st. (a) If the prediction is correct, how many new cases (correct to the nearest integer) should be expected on: (i) June 7th (ii) June 20th ? (b) Use a GDC method to find the date on which the number of new cases should first drop below 0.5. (a) (c) Explain the significance of “below 0.5” in part (b). not to scale (b) 6. The temperature, T˚ Celsius, of a cooling object t minutes after the start of an experiment is given by the formula T 22 (78 20.2t ) (a) Find the value of T (correct to the nearest integer) when: 4. The gross domestic product (G billion dollars) of a small nation is estimated to be growing exponentially according to the formula G 45 1.04n where n is the number of years after 2000. The population (P million people) of the country appears to be increasing exponentially according to the formula P 3.2 1.015n (i) t = 0 (ii) t = 10 (iii) t = 30 (b) After how many minutes does the temperature reach 30˚C? Answer correct to 1 decimal place. (c) When t is very large, what value does T approach? Physically, what do you think this temperature represents? S-22 Trigonometric Functions S-23 Accurate Graphing S-24 Graph Sketching and GDC Skills Function Coordinates of key points (correct to 2 decimal places) Sketch – include an approximate y-scale Domain 4 x 4 1. x-intercepts: y 2 3x x y-intercept: minimum: -4 4 4 x 4 2. x-intercept: 8 y x2 x maximum: -4 4 2 x 4 3. x-intercepts: y x 3x x 2 3 2 y-intercept: -2 4 minimum: maximum: 5 x 3 4. y-intercept: y 1 2x 1 x 2 minimum: maximum: -5 3 equation of vertical asymptote: