Pakuranga College. Year 10 Mathematics Examination. Time: 2 hours

Transcription

1 NAME: SOLUTIONS TEACHER: Pakuranga College Year 10 Mathematics 2014 Examination Time: 2 hours Sections Page number Result 1 Number 2 /47 2 Algebra 4 /62 3 Graphs 6 /55 4 Measurement 9 /43 5 Trigonometry 12 /46 6 Angles 15 /47 7 Statistics 18 /64 8 Probability 21 /59 Answer ALL questions in the spaces provided in this booklet. Show ALL working. Page 1

2 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 1 Number Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================= QUESTION ONE (d) 2 3 (4 + 2) + 16 Tina s teacher tells the class that they are not allowed to use their phones as calculators! Tina = 6 for working did not remember to bring her calculator to class. for solution Show how these questions could be solved without a calculator (show working). (a) Find 20% of = 160 for appropriate working for answer (b) = = 23 or (c) 62 + = 77 1 for new denominator for correct working for correct answer 62 + x = 76 for some working x = 14 for correct answer (e) Find the lowest common multiple of 6 and 8. 6,12,18,24,30,36, for multiples of 6 8,16,24,32,40, for multiples of 8 LCM(6,8) = 24 for answer (f) = some working = for answer Or 2,080,000,000 QUESTION TWO Some students weighed their phones. Here are the weights (in g). Write them in order from smallest to largest: 110.7, 112.0, , 111.3, , , 110.7, 111.3, Page 2

3 QUESTION THREE Complete the rounding table Number Rounded to Nearest d.p. 3 s.f for each correct answer up to total of 9. QUESTION FOUR A new smart phone has a recommended retail price of $1049. (a) Shady Sam says he can get it for 65% of the recommended price. What is Shady Sam s price? = $ working (b) Techfilla Company sells the phone at its recommended price but then holds a 30% off everything sale. What is the sale price of the phone? = $ working (c) CheapSellaz holds a 20% off sale and lists the phone s sales price as $782. What is their non-sale price for the phone? = $ working QUESTION FIVE A phone cost a retailer $530 to get into the store. 72% profit is added to get the GST exclusive selling price, then 15% GST is added. (a) What will the GST inclusive selling price be? = $ working (b) If the price in (i) is then discounted 30%, what will the new selling price be? = = $ working appropriate rounding (c) What is the percentage decrease between the original GST exclusive price and the discount price in (ii)? ( ) = 19.5% working answer (d) Another phone has a GST inclusive price of $870. What is the GST exclusive price? (GST is 15%) = = $ working rounded Page 3

4 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 2 Algebra Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================== QUESTION ONE QUESTION TWO If the first two scales are in perfect balance, what needs to be added (in place of the question mark) Solve the following equations: to balance the third set? (a) 10 + = (b) 53 = (c) 2n + 5 = 29 2n = 24 working n = 12 (d) 6n 4 = 3n + 8 Correct answer is 4 6n = 3n n = 12 n = 4 (e) 5(n 3) = 35 working 5n 15 = 35 5n = 50 n = 10 working (f) (n 4)(n + 3) = 0 n = 4 or n = 3 for each answer Total of two Page 4

5 QUESTION THREE Simplify the following expressions: (a) p p p p = p 4 (b) 4n n = 3n (c) 5n + 4p 3n + p = 2n + 5p (f) (x + 4)(x 2) = x 2 2x + 4x 8 correct exp. = x 2 + 2x 8 simplified (g) (p 6) 2 + 6p = p 2 12p p exp. = p 2 6p + 36 simp. (d) 7n 8n (e) (3n 4 ) 2 (f) 2y 5 + y 3 = 6y + 5y = 11y 15 = 56n 2 = 9n 8 working answer QUESTION FIVE To cater an afternoon tea, it was decided to provide 3 biscuits per person and supply an extra 10 biscuits in case of greedy people! The following formula was used: b = 3n + 10 a) Explain what b and n stand for n is the number of people at afternoon tea b is the number of biscuits required for n people 1 each (g) 14n4 x 35nx = 2n3 5 QUESTION FOUR Expand the following, simplify if necessary. (a) 5(b + c) = 5b + 5c (b) 12(n + 4) = 12n + 48 (c) p(5p + 1) = 5p 2 + p (d) n(6 n) + 2(n + 3) = 6n n 2 + 2n + 6 (one for each bracket) = n 2 + 8n + 6 (e) 4(y + 3) 3(y 1) = 4y y + 3 one for each bracket = y + 15 b) How many biscuits will be needed for 20 people? b = subst. b = biscuits will be required for 20 people. context c) How many people were there at the last party if they provided 175 biscuits? 175 = 3n + 10 subst. n = = 55 working and answer 3 55 people were at the last party. context QUESTION SIX Fully factorise the following expressions (a) 5p + 10 = 5(p + 2) one for each part (b) 42n 12 = 6(7n 2) one for each (c) x 2 6x + 8 = (x 2)(x 4) one for each Page 5

6 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 3 Graphs Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================== SKILLS QUESTIONS The first point and last point of each missing section has already been plotted. QUESTION ONE Section 1: (3, 1), (7, -2), (5, -3), (9, -7), (2, -7), (3, -10) Section 2: (-5, -3), (-7, -2), (-3, 1), (-6, 2), (0, 7) total of 4 one mark off for each mistake. Part of a dot-to-dot graph picture is shown above, but two sections are missing. Complete them by plotting the points listed and joining them in the order they are given. Page 6

7 QUESTION TWO Give the next two terms in each of these patterns (a) 6, 10, 14, 18, 22, 26 (b) 11, 8, 5, 2, -1, -4 (c) 4, 6, 10, 16, 24, 34, 46 (d) What pattern number would require 244 matches to make? 244 = 3P + 1 subst. 3P = 243 rearranged P = 81 It would be pattern 81 that requires 244 matches to make. context. (e) If the rule in part (b) was plotted on a graph, what would its y intercept be? y intercept would be 1. (d) n + 4, 2n + 1, 3n 2, 4n 5, 5n-8,6n-11 QUESTION THREE Ellen is already excited about Christmas and makes a Christmas tree pattern out of matches. QUESTION FOUR Give the gradients of the lines shown above (a) Complete the table for pattern numbers and numbers of matches. Pattern (P) Matches (M) (b) Write a rule linking the number of matches to the pattern number. M = 3P+1 for 3P and for +1 (c) How many matches would be required to make the tree that is Pattern number 23? M = subst. M = 70 It would take 70 matches to make pattern 23 Page 7 (a) Gradient = 2 (b) Gradient = 1 3 (c) Gradient = 0

8 QUESTION FIVE At 2pm one day, Petra left her house to walk and visit Anna. Anna left her house to go on a walk. Kelly stayed home. The graph shows the three girls movements. Anna could have walked away from her home in the opposite direction than Petra walked towards it. Hence they may not meet. partial marks okay for incomplete explanation. (a) Give the equations of each girl s line. Petra: y=-4x+12 grad y-int Anna: y=2x (accept y=2x+0) Kelly: y=6 (accept y=0x+6) grad no y-int no grad y-int (b) How fast does Petra walk? 4kmh -1 4 units accept km/h (c) How far away from Anna does Kelly live? How is this shown on the graph? 6km This is because the graph shows the distance each person is away from Anna s house and Kelly started 6km away while at home. (d) How fast does Anna walk? 2kmh -1 (e) Explain why Anna and Petra do not necessarily meet each other. The graph shows the distance the girls are away from Anna s home, not each other. So Page 8

9 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 4 Measurement Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ======================================================================== QUESTION ONE QUESTION TWO Cherie doesn t like her bath too hot. She took the Circle the most sensible measurement temperature of the bath water before and after adding some cold to it (thermometer reads in (a) Width of a bathroom sink might be: degrees Celsius). What were the temperature readings? 47 cm (b) A bath soap might weigh: 90 g (c) Height of the bathroom door might be: 190 cm (d) The area of a face cloth might be: 625 cm 2 First temperature: 60 o C Second temperature: 42 o C Page 9

10 QUESTION THREE QUESTION FIVE Cherie is considering several different toothbrush holders. Calculate the volume of each one. (a) Rectangular prism (cuboid) V = = 288cm 3 working units (a) The dimensions of Cherie s bath towel are given above. What is the area of Cherie s towel? = 6000cm 2 (b) A towel weighs 500g per square metre. What does Cherie s towel weigh? = 300g working answer QUESTION FOUR Give the conversions for these metric units: (a) 49 cm = 0.49m (b) 1.02 kg = 1020g (c) 154 mm = 15.4cm (d) 24 ml = 0.024L (b) Cylinder V = 18 π 7 2 = 882π = cm 3 working radius used answer units Page 10

11 QUESTION SIX Reminder: Circle area formula is A = πr 2 Circumference formula is C = 2πr or πd A toilet roll has the following dimensions: width of 11 cm, diameter of roll = 10 cm, diameter of cardboard tube = 4 cm. (a) What is the volume of paper in the roll? V = (π ) (π ) V = V = 725.7cm 3 answer units (b) The roll has 200 sheets of toilet paper, each 12 cm long. If it was unrolled, what would the total area of the toilet paper be? (i) What is the area of one of the soap s trapezium shaped faces? A = = 48cm 2 (ii) What is the volume of the soap? V = = 120cm 3 (b) Cherie was also given this candle. It has a square base and is pyramid-shaped = 26400cm 2 Useful formulas for next questions: Volume of sphere V = 4 3 πr3 Volume of pyramid V = 1 base area height 3 Area of trapezium = (a+b) 2 QUESTION SEVEN h Cherie s friends know that she likes candles and soaps for her bathroom. (a) One friend gave her this soap, which is a trapezium prism. (i) What is the volume of the candle? V = = cm3 (ii) Sadly, the candle broke into pieces before Cherie could light it. She melted down the wax and created a new candle shaped like a cube. What will the dimensions of the new candle be? 3 side = = 7.69cm The candle will be a cube with each side measuring 7.69cm. context. Page 11

12 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 5 Trigonometry Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================== QUESTION TWO QUESTION ONE Use your calculator to find the values of n or A. Record your working. (a) = n 2 n 2 = 65 n = 8.1 (b) n = 12 2 n 2 = 80 n = 8.9 (c) n = sin 35 8 n = 4.6 (d) 9 n = cos 52 n = 0.07 A boat is sailing due East of a radio beacon. A plane is due North of the beacon. The plane and boat are 18 km apart and the boat is 10 km from the beacon. (e) 4 7 = tan A A = 29.7 (a) How far North of the beacon is the plane? n = 18 2 n 2 = n = 15.0km Page 12

13 (b) What is the angle between the boat s path and a path that would take it towards the plane? (The angle indicated on the diagram) cos A = 10 A = cos A = 56.3 QUESTION THREE A windsurfer sails a course marked by three buoys that form a right-angled triangle. QUESTION FOUR A kayak s sail is shaped like an isosceles triangle. If it is 1.8 m wide at the top and the equal sides are 3 m, calculate the height of the sail. The first leg of the course is 35 m. Calculate x and y, the lengths of the other two legs. a = 3 2 a 2 = a = 2.86m using pythag some additional working cos 72 = 35 x x = 35 cos 72 x = 113.3m tan 72 = y 35 y = 35 tan 72 y = 107.7m Page 13

14 QUESTION SIX The angle of elevation from a boat to a plane is 29 o. The relative positions of the boat, plane and a radio beacon on the horizontal are given in the second diagram. QUESTION FIVE A boat has two right-angled triangle-shaped sails. The mainsail is 8m wide and the smaller sail is 12 m high. (a) Calculate x, the height of the mainsail. x = 22 2 x 2 = x = 20.5m (b) Calculate A, the angle at the top of the mainsail. sin A = 8 22 a = sin a = 21.3 (c) Calculate y, a length on the smaller sail. cos 50 = 12 y y = 12 cos 50 y = 18.7m Calculate the height (altitude) at which the plane is flying. Horizontal distance to plane: h 2 = h 2 = 1168 h = 34.2km Height of plane: tan 29 = y 34.2 y = 34.2 tan 29 y = 18.9km The height of the plane is 18.9km Well set out solution Page 14

15 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 6 Angles Answer ALL questions in the spaces provided in this booklet. Show ALL working For Assessor s use only Curriculum Level ========================================================================== QUESTION ONE QUESTION TWO In the figure above (a) Draw a cross inside one acute angle. Any of the indicated (b) What size is angle ADC? (a) Size of angle? 66 o (accept 65-67) 90 o (c) The angle to the far right can be called ABC. Give another three letter name for this angle. CBA (d) Put a tick inside an obtuse angle. As indicated (e) What would the angles inside the shape ABCD add to? 360 o (b) Size of angle AOC? 155 o (c) Size of angle AOB? 40 o (d) Size of angle BOC? 115 o Page 15

16 QUESTION THREE Give the size of the marked angles. Give a geometric reason for each one if you can. QUESTION FOUR (a) A = 135 o because exterior angle of a triangle is equal to the sum of the two interior opposite angles. (b) This diagram shows an isosceles triangle situated between parallel lines. Calculate the size of angle E. You may need to first work out some of the angles marked a-d. Give a geometric reason and clearly identify each angle you calculate. a = 46 (angles on a line add to 180) b = 46 (base angles of an isosceles triangle are equal) c = 88 (angles in a triangle add to 180) d = 46 (cointerior angles add to 180) E = 46 (angles on a line add to 180) Any valid chain of reasoning with correct reasons is worth 10. A = 38 o because Co-interior angles on parallel lines add to 180 o B = 142 o because Vertically opposite angles are equal Page 16

17 QUESTION FIVE QUESTION SIX Calculate the size of angle A. You may need to calculate other angles in the diagram to do so. Label any angle that you use and give a geometric reason for its size. Hint: You may need to extend the length of one of the existing lines. Given that angle EBD is size x, give the sizes of the other angles in the triangle in terms of x. A = 85 Any valid chain of reasoning is worth 8 No need for geometric reasons here, and one for each correct angle. 1=x 2=180-2x 3=2x 4=2x 5=180-4x 6=3x 7=3x 8=180-6x Page 17

18 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 7 Statistics Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================== QUESTION ONE (c) Explain why the data for this graph is most likely based on samples. Suggest how it may have been collected. It would be impossible to know the speed of every car, hence this is probably a sample. It could have been collected using speed camera data from randomly selected locations (or equivalent) (a) Describe the long-term trend in percentage of cars exceeding the speed limit in urban areas. It appears there is a decrease in the percentage of cars exceeding the speed limit as time has passed. (b) Sam thinks the trends for the two speed limits show similar movements. What kind of graph could he use to look for a correlation between the two sets of data? (d) Explain why we can t use this graph to find how many cars exceeded the speed limit in Because there is no information on the total number of cars checked. (or similar) (e) Estimate the percentage of cars that will break the rural speed limit in Any number between 20 and 23% A scatter diagram shows correlation Page 18

19 QUESTION TWO QUESTION THREE Sam s school is loaned a speed radar which records car speeds to the nearest km/h. Sam uses it for 10 minutes at the school gate and records the following speeds: Sam managed to obtain some data about car speeds in urban areas (where the speed limit is 50 km/h). (a) Is speed discrete or continuous data? Continuous (b) How does the graph show that all speeds were rounded? What were they rounded to? It is a bar graph rather than a histogram. Speed was rounded to the nearest kmh -1 (c) Describe features of the distribution of speeds. A for each correct statement up to 3 marks For example Almost symmetrical Unimodal Appears to be a couple of outliers at the top and at the bottom etc 52, 48, 55, 58, 53, 50, 49, 52, 53, 55, 59, 56, 51, 53, 53, 57, 54, 52, 56, 59, 51, 50, 49, 50, 53. (a) Create a dot plot for the data given above, using the scale below (b) Complete the table of summary statistics for the data. Range 11 Median 53 Mean Mode 53 Lower quartile 50.5 Upper quartile 55.5 (c) Sketch a box plot of the data above the scale below (d) In this sample, did the majority of cars stay within the speed limit? Give evidence for your claim. No it does not look like half the data is below 50. Or equivalent. (d) Comment on whether Sam s sample of cars is random. For what reasons might you question whether it is representative of all cars that pass by the school entrance? Page 19

20 Any valid comment about randomness (either way) is Eg It is a random sample only if the time he was there was randomly chosen. Any valid comment about representative make up of the sample is Eg It is not representative as it does not account for the different times of the day and their effect on the speed of cars. QUESTION FOUR Sam decides to hold a survey to find out why a lot of people speed past the school entrance. He puts a survey in every letterbox he passes on his way home from school. The survey includes these questions: 1. What speed do you normally drive at when passing Prince Albert High School? 2. How often do you break the speed limit? 3. Why do you break the speed limit? Identify some problems with Sam s sampling and question design. Any valid comment is worth to a max of 8 marks. Eg Biased sample Not all may have cars Not all may drive past the school Q3 makes the assumption that they break the limit they may not. Q2 people may be unwilling to admit they break the limit Q1 people may not be able to answer accurately. Page 20

21 NAME: TEACHER: YEAR 10 MATHEMATICS, 2014 Section 8 Probability Answer ALL questions in the spaces provided in this booklet. Show ALL working. For Assessor s use only Curriculum Level ========================================================================== QUESTION ONE Put a dot on the scale to represent the likelihood of each event. (a) Your teacher has a cat that can tap-dance and speak Mandarin. QUESTION TWO One study found that the probability of a female being left handed is 0.09, but for a male it is (a) Complete the tree diagram for this situation (b) The next baby to be born in Auckland will be a boy. (c) It will rain in your town sometime in the next fortnight. (d) All the kittens in a litter of 5 turn out to be males. (b) Calculate the probability that a randomly chosen person is female and left-handed. P(F LH) = = (c) Calculate the probability that a randomly chosen person is right-handed. Page 21 P(RH) = P(RH) = 0.895

22 (d) If three people are selected at random from the general population, what is the probability that all of them are left-handed males? P(M LH) = = 0.06 Three LH males in a row selected = P(3LHM) = (e) In a co-ed school (both genders attend) with 700 students, how many left-handers would we expect to have? P(LH) = = E(LH) = = 73.5 It would be expected to find 73 or 74 lefthanders in the school. (f) Identify at least one assumption we would have to make in order to calculate the answer to the previous question. Answers will vary e.g. assume the school has equal numbers of males and females. (a) The toast lands butter side down 28 times. Use this to give an estimate (as a fraction in its simplest form) for the probability of toast landing butter side down. P(BSD) 28 = (b) A group of schools got together to carry out trials of this experiment. They found that the toast landed butter side down times in their experiment. (i) Give an estimate for the probability of toast landing butter side down based on this experiment. P(BSD) 6248 = (ii) Another group of schools decide to carry out trials of buttered toast drops. Will they find that the toast lands butter side down 6248 times? Explain. No because (e.g.) every time you run a probability experiment, the results can be different. Each set of trials just gives one estimate QUESTION THREE Jeremy has a theory that toast is more likely to land with the butter side down. He tests this theory by dropping a piece of toast 50 times. Page 22 (c) Which estimate (the one from Jeremy s experiment or the one from the group of schools) is likely to be more accurate? Why? The trials will give a better estimate, as long-run experiments give more reliable results than smaller numbers of trials.

23 QUESTION FOUR An English teacher made a game involving two spinners. Students have to spin both spinners and put the parts together to make a word. Some words are not proper English. Each spinner has even-sized sections. Probability of one of these two words is 1 6. From 100 trials, we would expect to get them 16 or 17 times. (Based on this, either say it is more than what we would expect or not too far off to be truly unusual). (a) If you play the game, what is the probability of getting a word that ends in ing? P(ing) = 1 3 (b) How many words are possible? 12 (c) If you play the game and get a word ending in ing, what is the probability that it is a real word? P(real\ing) = 1 2 (d) Sarah gets hooked on the game and plays it a lot. If she has 5 turns, what is the probability that every word she makes begins with the letter b? P(b) = in a row ( 1 2 )5 = 1 32 = (e) Sarah then plays 100 games and gets either coldest or colder 25 times. Comment on whether this result seems unusual. Page 23

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