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Sydney grammar 2024 adv
Mathematics Advanced
Barker College - Hornsby
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SYDNEY GRAMMAR SCHOOL NAME MATHS MASTER CANDIDATE NUMBER 2024 Trial Examination Form VI Mathematics Advanced Tuesday 13th August, 2024 8:40am General Instructions Reading time 10 minutes Working time 3 hours Attempt all questions. Write using black pen. Calculators approved NESA may be used. A loose reference sheet is provided separate to this paper. Carefully remove the central staple. Thirty Two Questions 100 Marks Section I (10 marks) Questions 1 10 This section is Each question is worth 1 mark. Record your answers on the provided answer sheet. Section II (90 marks) Questions Relevant mathematical reasoning and calculations are required. Answer the questions in this paper in the spaces provided. Collection Write your candidate number on this page and on the multiple choice sheet. Place everything inside this question booklet. Checklist Reference sheet answer sheet Candidature: 87 pupils Writer: NWS SYDNEY GRAMMAR SCHOOL NAME MATHS MASTER CANDIDATE NUMBER 2024 Trial Examination Form VI Mathematics Advanced Tuesday 13th August, 2024 8:40am General Instructions Reading time 10 minutes Working time 3 hours Attempt all questions. Write using black pen. Calculators approved NESA may be used. A loose reference sheet is provided separate to this paper. Carefully remove the central staple. Thirty Two Questions 100 Marks Section I (10 marks) Questions 1 10 This section is Each question is worth 1 mark. Record your answers on the provided answer sheet. Section II (90 marks) Questions Relevant mathematical reasoning and calculations are required. Answer the questions in this paper in the spaces provided. Collection Write your candidate number on this page and on the multiple choice sheet. Place everything inside this question booklet. Checklist Reference sheet answer sheet Candidature: 87 pupils Writer: NWS Form VI Mathematics Advanced Trial Examination August 2024 Section I Questions in this section are Record the single best answer for each question on the provided answer sheet. 1. What is the domain of the function (A) (B) (C) (D) (0,00) 2. Which equation best represents the graph drawn below? y x 0 (A) (B) (C) y (D) ) Form VI Mathematics Advanced Trial Examination August 2024 Section I Questions in this section are Record the single best answer for each question on the provided answer sheet. 1. What is the domain of the function (A) (B) (C) (D) (0,00) 2. Which equation best represents the graph drawn below? y x 0 (A) (B) (C) y (D) ) Form VI Mathematics Advanced Trial Examination August 2024 5. Consider a dataset with mean 10 and standard deviation 2. Two scores, 10 and 9 are added to the dataset. Which choice shows how the mean and standard deviation are affected the added scores? (A) Both mean and standard deviation will change. (B) Only standard deviation will decrease. (C) Only standard deviation will increase. (D) Both mean and standard deviation will stay the same. 6. Researchers are investigating population density, house size and the distance people live from the centre of a city. Data is plotted with distance on the horizontal axis. For the graph of population density VS distance, the correlation coefficient is r For house size VS distance, the correlation coefficient is r 0 Given this information, which one of the following statements must be true? (A) Population density tends to be higher when it is farther away from the city centre. (B) House size tends to be larger when it is closer to the city centre. (C) House size is positively correlated with population density. (D) Population density is more strongly correlated with distance from the city centre than house size. 7. Suppose the graph of y f(x) has an at 1, a at 2, a minimum turning point (5, and a single horizontal asymptote y What can you say is definitely true about the graph of 2)? (A) Its is (B) Its is (C) It has a maximum turning point at (7, 5). (D) Its horizontal asymptote cannot be determined. Form VI Mathematics Advanced Trial Examination August 2024 5. Consider a dataset with mean 10 and standard deviation 2. Two scores, 10 and 9 are added to the dataset. Which choice shows how the mean and standard deviation are affected the added scores? (A) Both mean and standard deviation will change. (B) Only standard deviation will decrease. (C) Only standard deviation will increase. (D) Both mean and standard deviation will stay the same. 6. Researchers are investigating population density, house size and the distance people live from the centre of a city. Data is plotted with distance on the horizontal axis. For the graph of population density VS distance, the correlation coefficient is r For house size VS distance, the correlation coefficient is r 0 Given this information, which one of the following statements must be true? (A) Population density tends to be higher when it is farther away from the city centre. (B) House size tends to be larger when it is closer to the city centre. (C) House size is positively correlated with population density. (D) Population density is more strongly correlated with distance from the city centre than house size. 7. Suppose the graph of y f(x) has an at 1, a at 2, a minimum turning point (5, and a single horizontal asymptote y What can you say is definitely true about the graph of 2)? (A) Its is (B) Its is (C) It has a maximum turning point at (7, 5). (D) Its horizontal asymptote cannot be determined. Form VI Mathematics Advanced Trial Examination August 2024 8. A bag contains six white socks and eight black socks. Two socks are selected at random without replacement. Given the two selected socks are the same colour, which of the following is the closest to the probability that they are both black? (A) (B) (C) (D) 9. The graph of y is drawn below. Y y a C X Which of the following statements is true? (A) f(x) has a minimum turning point at I a. (B) f(x) has a point of inflection at (C) f(x) has a maximum turning point at I C. (D) f(x) has a point of inflection at I c. Form VI Mathematics Advanced Trial Examination August 2024 8. A bag contains six white socks and eight black socks. Two socks are selected at random without replacement. Given the two selected socks are the same colour, which of the following is the closest to the probability that they are both black? (A) (B) (C) (D) 9. The graph of y is drawn below. Y y a C X Which of the following statements is true? (A) f(x) has a minimum turning point at I a. (B) f(x) has a point of inflection at (C) f(x) has a maximum turning point at I C. (D) f(x) has a point of inflection at I c. Form VI Mathematics Advanced Trial Examination August 2024 CANDIDATE NUMBER Section II Part A QUESTION ELEVEN (2 marks) Find the derivative of: (a) 3x4 1 (b) y XVX 1 Form VI Mathematics Advanced Trial Examination August 2024 CANDIDATE NUMBER Section II Part A QUESTION ELEVEN (2 marks) Find the derivative of: (a) 3x4 1 (b) y XVX 1 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWELVE (4 marks) Find: (a) I 2 (b) sin x d r 2 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWELVE (4 marks) Find: (a) I 2 (b) sin x d r 2 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION FOURTEEN (3 marks) The table shows the probability distribution of a discrete random variable. X 2 4 10 20 P(X x) k 0 0 3k (a) Show that k 0. 1 (b) State the mode. 1 (c) Find the expected value. 1 12 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION FOURTEEN (3 marks) The table shows the probability distribution of a discrete random variable. X 2 4 10 20 P(X x) k 0 0 3k (a) Show that k 0. 1 (b) State the mode. 1 (c) Find the expected value. 1 12 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION FIFTEEN (4 marks) (a) Find the equation of the tangent to the curve y 2) at the point (2,0). 3 (b) Find the angle of inclination of the tangent, correct to the nearest degree. 1 13 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION FIFTEEN (4 marks) (a) Find the equation of the tangent to the curve y 2) at the point (2,0). 3 (b) Find the angle of inclination of the tangent, correct to the nearest degree. 1 13 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION SEVENTEEN (4 marks) Two athletic teams, each with 12 members, participated in a 100 m sprint competition. The times for each person in Team A are different, and the times for each person in Team B are different. The results for Team A are displayed in the plot below. Team A Team B 11 12 13 14 time (seconds) (a) The summary for Team results is 11, 12, 12, 13, 13. 1 Draw the plot to display Team results in the space provided above. (b) State the type of skewness of the distribution of Team A. 1 (c) One member from each team is selected at random. Find the probability that at least 2 one of them finished in under 12 seconds in the 100 m sprint. Form VI Mathematics Advanced Trial Examination August 2024 QUESTION SEVENTEEN (4 marks) Two athletic teams, each with 12 members, participated in a 100 m sprint competition. The times for each person in Team A are different, and the times for each person in Team B are different. The results for Team A are displayed in the plot below. Team A Team B 11 12 13 14 time (seconds) (a) The summary for Team results is 11, 12, 12, 13, 13. 1 Draw the plot to display Team results in the space provided above. (b) State the type of skewness of the distribution of Team A. 1 (c) One member from each team is selected at random. Find the probability that at least 2 one of them finished in under 12 seconds in the 100 m sprint. Form VI Mathematics Advanced Trial Examination August 2024 QUESTION EIGHTEEN (2 marks) Let and g(x) e. (a) Find g (f(x)). 1 (b) Show that g (f(x)) is an even function. 1 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION EIGHTEEN (2 marks) Let and g(x) e. (a) Find g (f(x)). 1 (b) Show that g (f(x)) is an even function. 1 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY (4 marks) The graph below is adapted from a diagram in the New York Times series It shows how the local sales change as the country prospers. The two variables in this scatterplot are the percentage change of gross domestic product (GDP) per capita (horizontal axis, x) and the percentage change in fast food sales from 2010 to 2015 (vertical axis, y). 120 100 80 60 40 20 B 0 0 20 40 60 80 100 Change in GDP per capita Using the data provided, the regression line was found to pass through points A (12, 28) and B (0, 20). The correlation coefficient was calculated to be T 0 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY (4 marks) The graph below is adapted from a diagram in the New York Times series It shows how the local sales change as the country prospers. The two variables in this scatterplot are the percentage change of gross domestic product (GDP) per capita (horizontal axis, x) and the percentage change in fast food sales from 2010 to 2015 (vertical axis, y). 120 100 80 60 40 20 B 0 0 20 40 60 80 100 Change in GDP per capita Using the data provided, the regression line was found to pass through points A (12, 28) and B (0, 20). The correlation coefficient was calculated to be T 0 Form VI Mathematics Advanced Trial Examination August 2024 (a) Describe the strength and direction of the correlation between the change in GDP per 1 capita and the change in fast food sales. (b) Find the equation of the regression line. 2 (c) Interpret the meaning of the gradient of the regression line, with reference to the context. 1 19 Form VI Mathematics Advanced Trial Examination August 2024 (a) Describe the strength and direction of the correlation between the change in GDP per 1 capita and the change in fast food sales. (b) Find the equation of the regression line. 2 (c) Interpret the meaning of the gradient of the regression line, with reference to the context. 1 19 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY TWO (5 marks) Recycled plastic materials can contain bacteria from previous usage. To ensure the plastic is clean and safe for recycling, it is heated to for a certain period to kill the bacteria present. The number of bacteria N in the plastic after t minutes at is given : N(t) 4800e , (a) Find the initial number of bacteria present in the plastic. 1 (b) Find the time in minutes to kill of the bacteria. Give your answer correct to the 2 nearest second. (c) Find the rate of decrease of the bacteria after 5 minutes. Give your answer to the 2 nearest whole number. 21 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY TWO (5 marks) Recycled plastic materials can contain bacteria from previous usage. To ensure the plastic is clean and safe for recycling, it is heated to for a certain period to kill the bacteria present. The number of bacteria N in the plastic after t minutes at is given : N(t) 4800e , (a) Find the initial number of bacteria present in the plastic. 1 (b) Find the time in minutes to kill of the bacteria. Give your answer correct to the 2 nearest second. (c) Find the rate of decrease of the bacteria after 5 minutes. Give your answer to the 2 nearest whole number. 21 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY THREE (5 marks) The Carpathia was a passenger steamship renowned for rescuing all the survivors of the Titanic in 1912. While the Carpathia was not the closest ship, once it received the distress signal from the Titanic it swiftly altered its course and reached the scene first. At the time the Titanic sent the distress signal, another ship, Mount Temple, was 50 nautical miles away, on a bearing of from the Titanic. The Carpathia was 97 nautical miles from Mount Temple, on a bearing of The diagram below illustrates the positions of the Titanic (T), the Carpathia (C), and Mount Temple (M) when the distress signal was sent. N1 N2 T 247 M C (a) Show that LTMC 1 22 Form VI Mathematics Advanced Trial Examination August 2024 QUESTION TWENTY THREE (5 marks) The Carpathia was a passenger steamship renowned for rescuing all the survivors of the Titanic in 1912. While the Carpathia was not the closest ship, once it received the distress signal from the Titanic it swiftly altered its course and reached the scene first. At the time the Titanic sent the distress signal, another ship, Mount Temple, was 50 nautical miles away, on a bearing of from the Titanic. The Carpathia was 97 nautical miles from Mount Temple, on a bearing of The diagram below illustrates the positions of the Titanic (T), the Carpathia (C), and Mount Temple (M) when the distress signal was sent. N1 N2 T 247 M C (a) Show that LTMC 1 22
Sydney grammar 2024 adv
Subject: Mathematics Advanced
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