Year 12 General Maths HSC Half Yearly Exam Practise Exam

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1 Year 12 General Maths HSC Half Yearly Exam Practise Exam Credit and Borrowing 1)! Minjy invests $2000 for 1 year and 5 months. The simple interest is calculated at a rate of 6% per annum. What is the total value of the investment at the end of this period? (A) $2170 (B) $2180 (C) $3003 (D) $3700 «1) C» 2)! Ali buys a television costing $1494 on interest-free terms over 2 years. If he pays a one-third deposit, how much will he be required to pay each month? (A) $20 75 (B) $41 50 (C) $43 58 (D) $83 00 «2) B» 3)! Frank has a credit card with an interest rate of 0 05% per day and no interest-free period. Frank used the credit card to pay for car repairs costing $480. He paid the credit card account 16 days later. What is the total amount (including interest) that he paid for the repairs? (A) $ (B) $ (C) $ (D) $ «3) B» 4)! Lynne invests $1000 for a term of 15 months. Interest is paid at a flat rate of 3 75% per annum. How much will Lynne s investment be worth at the end of the term? (A) $ (B) $ (C) $ (D) $ «4) A» 5)! The table shows monthly repayments for various amounts borrowed, and different annual interest rates, for a term of 20 years. Monthly repayment Amount borrowed 5% pa 6% pa 7% pa 8% pa $ $66 00 $71 64 $77 53 $83 64 $ $98 99 $ $ $ $ $ $ $ $ $ $ $ $ $ The total interest paid over 20 years on a loan of $ at 6% pa is (A) $ (B) $ (C) $ (D) $ «5) C» 6)! A motor car, advertised for $6900, is sold under the following terms: 40% deposit and the balance repaid over 5 years at $32 per week. Take 52 weeks in a year. Calculate: a. The deposit required. b. The amount paid as interest. c. The simple interest rate per annum charged on the balance. «6) a) $2796 b) $4126 c) 19 7% (to 1 d.p)»

2 7)! The table shows monthly payments for each $1000 borrowed. INTEREST RATE (% p.a.) PERIOD OF LOAN 5 years 10 years 15 years 20 years 25 years 5 $18 87 $10 61 $7 91 $6 60 $ $19 33 $11 10 $8 44 $7 10 $ $19 80 $11 61 $9 00 $7 75 $ $20 28 $12 13 $9 56 $8 36 $ $20 76 $12 67 $10 14 $9 00 $ $21 25 $13 22 $10 75 $9 65 $ $21 74 $13 78 $11 37 $10 32 $ $22 24 $14 35 $12 00 $11 01 $ $22 75 $14 93 $12 65 $11 72 $ $23 27 $15 53 $13 32 $12 44 $ $23 79 $16 13 $14 00 $13 17 $12 81 Christopher borrows $ to buy a house at 8% p.a. over twenty-five years. i. Use the information in the table to calculate Christopher s monthly payment on this loan. ii. How much does Christopher pay in total to repay this loan? iii. How much extra per month would Christopher pay if he were to repay the same loan over twenty years? Yang Yang wants to buy a house for $ She has saved some money for a deposit and will borrow the rest at 8% p.a. She will repay the loan over fifteen years, paying $1195 monthly. iv. How much will she borrow? v. How much has she saved for a deposit? «7) i) $1158 ii) $ iii) $96 iv) $ v) $25 000» 8)! Ted has borrowed $ at an interest rate of 6 24% per annum compounded monthly. The repayments have been set at $680 per month. The loan balance sheet shows the interest charged and the balance owing for the first month. Month Principal (at start of month) Monthly interest Monthly repayment Balance (at end of month) 1 $ $ $680 $ = $364 2 $ A $680 B i. Explain why is used to calculate the monthly interest. ii. Find the missing amounts at A and B. «8) i) The annual interest rate divided by 12 gives the monthly interest rate. ii) A $362 36, B = $ »

3 9)! John and Maria both take out, at the same time, bank housing loans of $ at an interest rate of 15% per annum. John repays the loan to the Bank by making monthly payments of $ over 25 years. Maria makes the same monthly payments as John, but also makes a single additional payment of $1400 at the end of the fifth year. The graph below shows how much each borrower still owes at any time in the repayment period. Use the graph to answer the following questions. Give all amounts to the nearest $1000 and all times to the nearest year. EFFECT OF ADDITIONAL PAYMENT AMOUNT OWING IN 40 THOUSANDS OF DOLLARS 30 Maria John TIME IN YEARS (FROM COMMENCEMENT OF MONTHLY PAYMENTS) i. How much does John still owe after 10 years of payments? ii. How long does it take for the amount that Maria owes to fall to $30 000? iii. How much longer does John take than Maria to repay the first $20 000? iv. How much does John still owe when Maria makes the last payment? v. What is the reduction in the amount owed by John over the last 5 years? vi. The amount of money owed by John changes over the twenty-five years. Comment on the rate of this change. «9) i) $ ii) 19 years iii) 2 years iv) $ v) $ vi) The amount owed decreases slowly in the first years of the loan. However, with each repayment made, the rate of reduction of the loan increases.»

4 Further Applications of Measurement 10)! A balcony is in the shape of a right triangle and a semicircle, as shown in the diagram. 6 m 8 m NOT TO SCALE Calculate the area of the balcony correct to the nearest square metre. (A) 49 m 2 (B) 73 m 2 (C) 125 m 2 (D) 149 m 2 «10) A» 11)! What radius of a sphere is increased by 10%. What is the percentage increase in its surface area? (A) 10% (B) 20% (C) 21% (D) 33% «11) C» 12)! This is a sketch of a sector of a circle. 13)! 9 cm cm Calculate the area of this sector (correct to one decimal place). (A) 9 4 m 2 (B) 18 8 m 2 (C) 36 8 m 2 (D) 84 8 m 2 A NOT TO SCALE «12) D» 33 metres 25 metres 29 metres 19 metres D 10 metres B 17 metres 17 metres 17 metres 17 metres C h Use Simpson's Rule [ Area ( d f d L 4d M )] twice to estimate the area of ABCD to the nearest 3 square metre. «13) 1570 m 2» 14)! A surveyor sketched this diagram of a pond in a rectangular field. S T POND V U NOT TO SCALE Measurements in metres. i. Calculate the area of the rectangle STUV. ii. Use Simpson's Rule to calculate the area of the unshaded region PVUQ. h [ Area ( d f d L 4d M )] 3 iii. The surveyor calculated the area of the shaded region STQP to be 430 m 2. Use this result and your calculations to find the area of the pond. 17 Q 18

5 «14) i) 1400 m 2 ii) 560 m 2 iii) 410 m 2» 15)! A clay brick is made in the shape of a rectangular prism with dimensions as shown. 9 cm NOT TO SCALE 8 cm 21 cm i. Calculate the volume of the clay brick. Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1 4 cm. ii. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.) iii. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.) «15) i) 1512 cm 3 ii) 1364 cm 3 iii) 9 8%» Further Algebra Skills 16)! Which of the following is the correct simplification of 8x 3 5x 3? (A) 3x 6 (B) 3x 3 (C) 3x (D) 3 17)! If d = 6t 2, what is a possible value of t when d = 2400? (A) 0 05 (B) 20 (C) 120 (D) 400 «16) B» «17) B» 18)! Simplify 2m 2 3mp 2. (A) 5m 2 p 2 (B) 5m 3 p 2 (C) 6m 2 p 2 (D) 6m 3 p 2 «18) D» 19)! Using the formula d = 5t 3 2, Marcia tried to find the value of t when d = 137. Here is her solution. She made one mistake. d = 5t = 5t = 5t 3 27 = t 3 t = 3 Line A Line B Line C Line D Which line does NOT follow correctly from the previous line? (A) Line A (B) Line B (C) Line C (D) Line D 20)! What is the formula for q as the subject of 4p = 5t + 2q 2? (A) q 5t 4 p 4 p 5t (B) q 2 2 (C) q 5t 4 p 4 p 5t (D) q 2 2 «19) B» «20) D»

6 2A 21)! The formula D is used to calculate the dosage of Hackalot cough medicine to be given to a child. 15 D is the dosage of Hackalot cough medicine in millilitres (ml). A is the age of the child in months. i. If George is nine months old, what dosage of Hackalot cough medicine should he be given? ii. The correct dosage of Hackalot cough medicine for Sam is 4 ml. What is the difference in the ages of Sam and George, in months? «21) i) 1 2 ml ii) 21 months» ab 2 4w 22)! Simplify. w 3b 23)! A factory makes boots and sandals. In any week the total number of pairs of boots and sandals that are made is 200 the maximum number of pairs of boots made is 120 the maximum number of pairs of sandals made is 150 y «22) 4ab 3» 200 A Number of pairs of sandals (y) B C The factory manager has drawn a graph to show the numbers of pairs of boots (x) and sandals (y) that can be made. i. Find the equation of the line AD. ii. Explain why this line is only relevant between B and C for this factory. iii. The profit per week, $P, can be found by using the equation P = 24x + 15y. Compare the profits at B and C. «23) i) x + y = 1 ii) If the max number of manufactured boots is attained, 120 boots only 80 sandals can be manufactured, thus the point C. If the max number of manufactured sandals is attained, 150 sandals only 50 boots can be manufactured, thus the point B. Any combinations of numbers of boots and sandals may be manufactured with the sum equal to 200, thus all points between B and C are available, thus the line segment between B and C. iii) The profit at B is $3450. The profit at C is $4080» D Number of pairs of boots (x) x

7 Interpreting Sets of Data 24)! Thirty students sat for tests in four different subjects. Each test was marked out of four. A histogram of the results for each subject is shown below. Number of students Number of students Number Art of students Number Computing of students Which subject had marks with the highest standard deviation? (A) Art (B) Biology (C) Computing (D) Drama Biology Drama «24) B» 25)! In five spelling tests, Jenny made the following numbers of mistakes: The mean number of mistakes is 4 and the standard deviation is 2. If she makes no mistakes in either of the next two tests, then (A) the mean increases and the standard deviation increases. (B) the mean increases and the standard deviation decreases. (C) the mean decreases and the standard deviation increases. (D) the mean decreases and the standard deviation decreases. «25) C» 26)! The ages of the children who live in Olympic Street are: A new family, with one 7-year-old child, moves into the street. Which one of the following changes? (A) The mean age (B) The median age (C) The range of the ages (D) The standard deviation of the ages «26) D» 27)! The dot plots below are drawn on the same scale. They show the class scores in tests taken before and after a unit of work was completed. Before After Which statement about the change in scores is correct? (A) The mean increased and the standard deviation decreased. (B) The mean increased and the standard deviation increased. (C) The mean decreased and the standard deviation decreased. (D) The mean decreased and the standard deviation increased. «27) A»

8 28)! The graph below shows the numbers of the two major types of cameras, analog and digital, sold in Australia in the years Camera market size by type (in thousands) Analog Digital In 2001, what percentage of the cameras sold were digital cameras? (To the nearest percent.) (A) 16% (B) 19% (C) 23% (D) 84% 997 «28) A» 29)! The lifetime in hours of 10 batteries of Brand X and 10 batteries of Brand Y are recorded below. Brand X Brand Y a. Calculate the mean and standard deviation of the lifetimes of each brand of battery. b. Which brand has the more consistent lifetimes? Give a reason for your answer. «29) a) X : x 82, = Y : y 81, = b) Brand Y. It has a lower standard deviation.» 30)! Andy and her biology class went to two large city parks and measured the heights of the trees in metres. In Central Park there were 25 trees. In East Park there were 27 trees. The data sets were displayed in two box-and-whisker plots. Central Park East Park Tree height (metres) i. In which park is the tallest tree, and how high is it? ii. What is the median height of trees in Central Park? iii. Compare and contrast the two data sets by examining the shape and skewness of the distributions, and the measures of location and spread. «30) i) East Park, 18 m ii) 7 m iii) The distribution for East Park trees is positively skewed, whereas the distribution for Central Park Trees is symmetrical. The median height for trees in Central Park is higher than that for East Park. The range and interquartile range in Central Park are lower than for East Park»

9 31)! Armand recorded the weights of a random sample of male students in his Year. The cumulative frequency graph displays the results. Number of Students Weight (kg) i. How many of the students surveyed were in the kg class? ii. Estimate the median weight of the students surveyed. iii. Of the 300 male students in Armand s Year, how many would you expect to weigh less than 70 kg? iv. 1. In order to select a sample, Armand s friend suggested selecting the first 50 male students in his Year to arrive at school on Monday morning. Explain why this would NOT be a random sample. 2. Describe a method that could have been used to select a random sample of the male students. «31) i) 16 ii) 77 kg iii) 96 iv) 1) It is a systematic sample, biased towards students who arrive early 2) Assign each male student a number and use a random number generator or draw the names from a hat.» Further Applications of Trigonometry 32)! Two yachts sailed in a straight line from a buoy B. One sailed 12 km in the direction 038 T and the other sailed 16 km in the direction 118 T. Which diagram is consistent with this information? (A) (B) B 12 80º B 12 68º º (C) (D) B 118º B 156º «32) A»

10 33)! Using the sine rule, find the size of angle to the nearest degree º NOT TO SCALE sin A sin B sin C ( Sine rule : ) a b c (A) 33 (B) 49 (C) 91 (D) )! Three towns, Euclid, Gauss and Newton, are situated as shown in the diagram. Gauss is due east of Euclid. Newton N 90º «33) D» Euclid 60º Gauss NOT TO SCALE The bearing of Gauss from Newton is (A) 030 (B) 120 (C) 150 (D) 300 «34) B» 35)! Anderville (A) is 30 km due east of Daytown (D). Haston (H) is on a bearing of 040 from Dayton and 325 from Anderville. Which of the following diagrams best represents this information? (A) (B) H H (C) D 50º 55º 30 km H A (D) D 50º 35º 30 km H A 55º 50º D 30 km A 2 2 a b c 36)! Use the cosine rule, cos C = 2ab ABC when a = 3, b = 4, c = º 35º D 30 km A «35) A», to find, to the nearest degree, the size of angle C in triangle «36) 117»

11 37)! The sine rule for a triangle ABC states that A a b c. sin A sin B sin C h b NOT TO SCALE D C 150 metres B a. Use the information given on the diagram and the sine rule in ABC to show that 150 sin 44 b. sin 24 b. Hence find the value of b, correct to three decimal places. c. Use the right-angled triangle ADC, shown in the diagram, to find the value of h, correct to 2 decimal places. «37) a) Proof b) c) » 38)! The diagram below shows the results of a plane table survey. Q 008 P O S 224 R 120 a. Find the size of the angle POQ. b. 1 Find the area of the triangle POQ. (Area ab sin C) 2 «38) a) 53 b) m 2»

12 The Normal Distribution 39)! The frequency graph for the heights of a large group of people is shown. Frequency x Height The mean ( x ) of the heights is 155 cm and the standard deviation is 11 2 cm. A person is chosen at random from this group. Between which two values will the height of this person almost certainly lie? (A) cm and cm (B) 155 cm and cm (C) cm and cm (D) 155 cm and cm «39) C» 40)! Results for an aptitude test are given as z-scores. In this test Di gained a z-score of 3. The test has a mean of 55 and a standard deviation of 6. What was Di's actual mark in this test? (A) 57 (B) 58 (C) 64 (D) 73 «40) D» 41)! In the town of Burrow the ages of the residents are normally distributed. The mean age is 40 years and the standard deviation is 12 years. Approximately what percentage of the residents are younger than 52? (A) 16% (B) 32% (C) 68% (D) 84% «41) D» 42)! A factory produces bags of flour. The weights of the bags are normally distributed, with a mean of 900 g and a standard deviation of 50 g. What is the bast approximation for the percentage of bags that weigh more than 1000 g? (A) 0% (B) 2 5% (C) 5% (D) 16% «42) B» 43)! Which of the following frequency histograms shows data that could be normally distributed? (A) (B) Frequency Frequency Score Score (C) (D) Frequency Frequency Score «43) A»

13 44)! The normal distribution shown has a mean of 170 and a standard deviation of 10. NOT TO SCALE i. Roberto has a raw score in the shaded region. What could his z-score be? ii. What percentage of the data lies in the shaded region? «44) i) A number between 1 and 2 ii) 13 5%» 45)! The results of two class tests are normally distributed. The means and standard deviations of the tests are displayed in the table. Test 1 Test 2 Mean Standard deviation i. Stuart scored 63 in Test 1 and 62 in Test 2. He thinks that he has performed better in Test 1. Do you agree? Justify your answer using appropriate calculations. ii. If 150 students sat for Test 2, how many students would you expect to have scored less than 64? «45) i) I do not agree. He achieved a higher z score in test 2. Hence he performed better in test 2 iii) 123 students» Multi Stage Events and Applications of Probability 46)! There are three birds in a cage. Two are green and one is blue. If two birds escape, find the probability that one of them is blue and the other is green. (A) 2 1 (B) 3 1 (C) 3 2 (D) (D) 3 «46) C» 47)! One boy and two girls are about to sit in a row. Calculate the probability that the two girls will sit together. 1 (A) 4 1 (B) 3 3 (C) 8 «47) D» 48)! The diagram shows a spinner. When you spin, you can win either a $10 or a $5 prize. The arrow points to the amount won. $5 120º $10 In two spins, what is the probability of winning a total of $15? (A) (B) (C) (D) «48) C» 49)! Amy buys a $1 ticket in a raffle. There are 200 tickets in the raffle and two prizes. First prize is $100 and second prize is $50. Find Amy s financial expectation. (A) $1 00 (B) $0 75 (C) $0 25 (D) +$0 25 «49) C» «50) i) 2 1 ii) 4 1 iii) 8 1»

14 50)! A wheel has the numbers 1 to 20 on it, as shown in the diagram. Each time the wheel is spun, it stops with the marker on one of the numbers? The wheel is spun 120 times. How many times would you expect a number less than 6 to be obtained? (A) 20 (B) 24 (C) 30 (D) 36 «51) C» 51)! A teacher is arranging 7 children for a group photo. The children will sit in a row. In how many different ways can the teacher arrange the seven children in the row? «52) 5040» 52)! A committee of three people is to be chosen at random from a group of eight people. How many different committees can be formed? «53) 56» 53)! A tennis player gets a second serve only if the first serve does not go in. Pat's first serve has a probability of 0 4 of going in, and his second serve has a probability of 0 9 of going in. i. Copy the tree diagram shown below. Complete the tree diagram, showing the probability on each branch. First serve Second serve 0 4 IN ii. iii. NOT IN Find the probability that Pat serves a double fault. (A double fault occurs when both the first and the second serve do NOT go in.) What is the probability that ONE of Pat's serves goes in? First serve Second serve 0 4 IN 0.9 IN 0 6 NOT IN 0 1 «54) i) NOT IN ii) 0 06 iii) 0 94»

15 54)! WIN LOSE CONTINUE WIN On a spinning wheel, two sections are labelled WIN, on section Lose and the other section CONTINUE. If the wheel stops on CONTINUE, you have another spin. Find the probability of: i. winning on the first spin; ii. losing on the first spin; iii. taking exactly two spins to win.

16 Annuities and Loan Repayments 55)! Two families borrow different amounts of money on the same day. The Wang family has a flat rate loan. The Salama family has a reducing balance loan and repays the loan earlier than the Wang family. Which graph best represents this situation? (A) (B) (C) Balance of loan Balance of loan Time Time (D) Balance of loan Balance of loan Time Time «55) C» 56)! The table shows monthly payments for each $1000 borrowed. INTEREST RATE (% p.a.) PERIOD OF LOAN 5 years 10 years 15 years 20 years 25 years 5 $18 87 $10 61 $7 91 $6 60 $ $19 33 $11 10 $8 44 $7 10 $ $19 80 $11 61 $9 00 $7 75 $ $20 28 $12 13 $9 56 $8 36 $ $20 76 $12 67 $10 14 $9 00 $ $21 25 $13 22 $10 75 $9 65 $ $21 74 $13 78 $11 37 $10 32 $ $22 24 $14 35 $12 00 $11 01 $ $22 75 $14 93 $12 65 $11 72 $ $23 27 $15 53 $13 32 $12 44 $ $23 79 $16 13 $14 00 $13 17 $12 81 Christopher borrows $ to buy a house at 8% p.a. over twenty-five years. i. Use the information in the table to calculate Christopher s monthly payment on this loan. ii. How much does Christopher pay in total to repay this loan? iii. How much extra per month would Christopher pay if he were to repay the same loan over twenty years? Yang Yang wants to buy a house for $ She has saved some money for a deposit and will borrow the rest at 8% p.a. She will repay the loan over fifteen years, paying $1195 monthly. iv. How much will she borrow? v. How much has she saved for a deposit? «56) i) $1158 ii) $ iii) $96 iv) $ v) $25 000»

17 57)! Aaron decided to borrow $ over a period of 20 years at a rate of 7 0% per annum. MONTHLY REPAYMENTS TABLE Principal and interest per $1000 borrowed Interest Term of loan - years rate (pa) % % % % i. Using the Monthly Repayment Table, calculate Aaron s monthly repayment. ii. How much interest does he pay over the 20 years? iii. Aaron calculates that if he repays the loan over 15 years, his total repayments would be $ How much interest would he save by repaying the loan over 15 years instead of 20 years? «57) i) $ ii) $ iii) $36 270» 58)! Rod is saving for a holiday. He deposits $3600 into an account at the end of every year for four years. The account pays 5% per annum interest, compounding annually. The table shows future values of an annuity of $1. End of year Future values of an annuity of $1 Interest Rate 1% 2% 3% 4% 5% i. Use the table to find the value of Rod s investment at the end of four years. ii. How much interest does Rod earn on his investment over the four years? «58) i) $ ii) $ » 59)! An amount of $5000 is invested at 10% per annum, compounded six-monthly. Compounded values of $1 Period Interest rate per period 1% 5% 10% 15% 20% [[End Of Qns]] Use the table to find the value of this investment at the end of three years. «59) $8860»

18 [Answers]