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Question Type
Question #1
Easy
[Maximum Mark: 2]
(a) Find the percentage error when approximating 125.50 to 126.
[1]
(b) Express the number \( 278 \times 10^6 \) in scientific notation.
[1]
Question #2
Easy
[Maximum Mark: 2]
(a) A car travels a distance of 150 km with an average speed of 60 km/h. Calculate the time taken for the journey.
[1]
(b) If the car had traveled the same distance with an average speed of 80 km/h, calculate the time taken for the journey.
[1]
Question #3
Easy
[Maximum Mark: 4]
(a) Calculate the future value (FV) of an investment of $520 after 5 years, with an annual interest rate of 1.2%, compounded quarterly.
[2]
(b) Calculate the future value (FV) of an investment of $800 after 3 years, with an annual interest rate of 1.8%, compounded monthly.
[2]
Question #4
Easy
[Maximum Mark: 5]
(a) Consider the arithmetic sequence where the first term \( a = 2 \) and the common difference \( d = 3 \). Find the 20th term of the sequence.
[2]
(b) The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). Using this formula, find the sum of the first 20 terms.
[3]
Question #5
Easy
[Maximum Mark: 6]
(a) An arithmetic sequence has the first term \( U_1 = 5 \) and a common difference \( d = 4 \). Calculate the sum of the first 15 terms.
[2]
(b) A geometric sequence has the first term \( U_1 = 3 \) and a common ratio \( r = 2 \). Calculate the sum of the first 10 terms.
[4]
Question #6
Easy
[Maximum Mark: 6]
A geometric sequence has the first term \( a = 3 \) and a common ratio \( r = 2 \).
(a) Find the 15th term of the sequence.
[2]
(b) The sum of the first 15 terms of the sequence is given by \( S_n = \frac{a(r^n – 1)}{r – 1} \). Using this formula, find the sum of the first 15 terms.
[4]
Question #7
Easy
[Maximum Mark: 6]
(a) For the data given below, find the value of the Pearson’s product-moment correlation coefficient, \( r \).
| Hours studied (\( x \)) | 10 | 12 | 15 | 18 | 20 | 22 | 25 | 28 |
|---|---|---|---|---|---|---|---|---|
| Exam score (\( y \)) | 60 | 62 | 70 | 74 | 80 | 85 | 88 | 90 |
[2]
(b) Lena found the following information about \( r \) in a statistics book:
| Value of \( |r| \) | Description of the correlation |
|---|---|
| 0 ≤ \( |r| \) < 0.4 | Weak |
| 0.4 ≤ \( |r| \) < 0.8 | Moderate |
| 0.8 ≤ \( |r| \) ≤ 1 | Strong |
Comment on your answer to part (a), using the information that Lena found.
[1]
(c) Write down the equation of the regression line of \( y \) on \( x \), in the form \( y = ax + b \).
[1]
(d) A student studied 30 hours and took the exam. Use the equation of the regression line to estimate his exam score.
[2]
Question #8
Easy
[Maximum Mark: 5]
The three cities Houston (H), Dallas (H), and San Antonio (S) form the Texas Triangle as seen in the diagram below.
Diagram not to scale.
(a) Calculate the value of ∠SDH in the Texas Triangle, where Houston (H), Dallas (D), and San Antonio (S) form the vertices. The distances between H, D, and S are given as 320 km, 360 km, and 300 km respectively.
[3]
(b) Determine the area of the triangle using the given distances.
[2]
Question #9
Easy
[Maximum Mark: 4]
(a) Sophia conducts an experiment on bacterial growth. She believes that the growth can be modeled by an exponential function
\( B(t) = C e^{kt} \)
where \( B \) is the number of bacteria, \( t \) is the time in hours since the start of the experiment, and \( C \) and \( k \) are constants. The number of bacteria is 150 at the start of the experiment and 600 after 4 hours. Write down the value of \( C \).
[1]
(b) Find the value of \( k \).
[3]
Question #10
Medium
[Maximum Mark: 13]
A craftsperson creates chimes for sale at a store. They plan to construct 10 chimes on the initial day and to increase production by 6 chimes each following day.
(a) Determine the quantity of chimes that would be produced on the 12th day.
[3]
The craftsperson intends to prepare chimes for a period of 15 days in anticipation of a sale.
(b) Calculate the number of chimes the craftsperson would have upon arrival at the sale.
[2]
To ensure a stock of 1000 chimes for a sale, the craftsperson opts to augment their daily production. The initial day will still see the creation of 10 chimes, with an increase by \(x\) chimes each subsequent day, exceeding the previous day’s output.
(c) Given the continuous work span of 15 days, compute the least integer value of \(x\) for the craftsperson to achieve this stockpile.
[3]
To examine the product, the craftsperson releases a chime to swing. The initial position is indicated on the left of the diagram. The craftsman begins recording once it has reached its full swing, on the right side of the diagram. In consecutive swings, the distance traversed by the chime diminishes to 92% of the previous swing’s extent. In the initial recorded swing, the chime covers a distance of 20.5 cm, hence \( U_1 \) = 20.5. During the second swing, the distance covered is 18.86 cm, hence \( U_2 \) = 18.86.
(d) Compute the span the chime would traverse during the fifth recorded swing.
[3]
(e) Evaluate the cumulative distance the chime would cover over the course of the first 16 recorded swings.
[2]
Question #11
Medium
[Maximum Mark: 6]
Nina is designing a quadrilateral kite ABCD on a set of coordinate axes where one unit represents 5 cm. The coordinates of A, B, and C are (2, 0), (0, 4), and (4, 5) respectively. Point D lies on the 𝑥 x-axis. [AC] is perpendicular to [BD]. This information is shown in the following diagram.
(a) Find the gradient of the line through points \( A(2, 0) \) and \( C(4, 5) \) on a coordinate plane.
[2]
(b) Write down the gradient of the line through points \( B(0, 4) \) and \( D \) on the same coordinate plane, given that \( [AC] \) is perpendicular to \( [BD] \).
[1]
(c) Find the equation of the line through points \( B \) and \( D \). Give your answer in the form \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are integers.
[2]
(d) Write down the \( x \)-coordinate of point \( D \) where the line intersects the \( x \)-axis.
[1]
Question #12
Medium
[Maximum Mark: 7]
(a) Let the function \( h(x) \) represent the height in centimeters of a cylindrical tank with diameter \( x \) cm.
The function is given by
\( h(x) = \frac{800}{x^2} + 0.8 \)
for \( 3 \leq x \leq 12 \). Find the range of \( h \).
[3]
(b) The function \( h^{-1} \) is the inverse function of \( h \).
(i) Find \( h^{-1}(15) \).
(ii) In the context of the question, interpret your answer to part (b)(i).
(iii) Write down the range of \( h^{-1} \).
[4]
Question #13
Medium
[Maximum Mark: 5]
(a) Inspectors are investigating the water usage of a factory. Let \( W \) be the rate, in liters per hour, at which water is used and \( t \) be the time in hours since the inspection began. When \( W \) is plotted against \( t \), the total amount of water used is represented by the area between the graph and the horizontal \( t \)-axis.
The rate, \( W \), is measured over the course of two hours. The results are shown in the following table.
Use the trapezoidal rule with an interval width of 0.5 to estimate the total amount of water used during these two hours.
| \( t \) (hours) | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| \( W \) (L/hr) | 20 | 40 | 60 | 50 | 30 |
[3]
(b) The real amount of water used during these two hours was 90 liters. Find the percentage error of the estimate found in part (a).
[2]
Question #14
Medium
[Maximum Mark: 6]
A nature conservation park is monitored by four weather stations. On the following Voronoi diagram, the coordinates of the stations are W(6, 4), X(12, 10), Y(8, 10), and Z(16, 4), where distances are measured in kilometers. The dotted lines represent the boundaries of the regions monitored by each station. The boundaries meet at points M(10, 6) and N(11, 5). To improve the monitoring efficiency, a new station is to be constructed within the quadrilateral WYXZ. The new station will be located so that it is as far as possible from the nearest existing station.
(a) Show that the new station should be built at point \( N \).
[3]
(b) The Voronoi diagram is to be updated to include the region around the new station at N. The edges defined by the perpendicular bisectors of [WN] and [NZ] have been added to the following diagram.
(i) Write down the equation of the perpendicular bisector of \([NY]\).
[2]
(ii) Hence draw the missing boundaries of the region around \( N \) on the diagram.
[1]
Question #15
Medium
[Maximum Mark: 5]
To assemble the lamp, Oliver joins the edges \([CA]\) and \([CB]\) together, forming a cone with a base radius of 6.5 cm.
Oliver is crafting a decorative lamp in the shape of a cone. The lamp’s shade is made from a sector, \( ACB \), of a circular piece of plastic with a radius of 25 cm and an angle \( ACB = \theta \), as shown in the diagram.
(a) (i) Write down the circumference of the base of the lamp in terms of \( \pi \).
[1]
(ii) Calculate the value of \( \theta \).
[2]
(b) Oliver plans to paint the exterior surface of the lamp. Find the surface area of the exterior of the lamp.
[2]
Question #16
Medium
[Maximum Mark: 8]
(a) The cumulative frequency graph below shows the scores obtained by a group of 100 students who sat a mathematics exam. Using the graph, find the median of the scores obtained.
[1]
(b) The students were awarded a grade from 1 to 5, depending on the score obtained in the exam. The number of students receiving each grade is shown in the following table:
Find an expression for \( a \) in terms of \( b \).
| Grade | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Number of Students | 5 | 15 | 30 | \( a \) | \( b \) |
[2]
(c) The mean grade for these students is 3.8.
(i) Find the number of students who obtained a grade 5.
(ii) Find the minimum score needed to obtain a grade 5.
[5]
Question #17
Medium
[Maximum Mark: 9]
Tom and Jane want to buy a new car and they need a loan of 150,000 dollars from a bank. The loan is for 20 years and the annual interest rate for the loan is 4%, compounded monthly. They will pay the loan in fixed monthly installments at the end of each month.
(a) Find the amount they will pay the bank each month.
[3]
(b) (i) Find the amount Tom and Jane will still owe the bank at the end of the first 7 years.
[3]
(ii) Using your answers to parts (a) and (b)(i), calculate how much interest they will have paid in total during the first 7 years.
[3]
Question #18
Hard
[Maximum Mark: 6]
(a) The surface area of an open box is given by \( S(x) = x^2 + \frac{160}{x} \). Find \( S'(x) \).
[3]
(b) (i) Solve \( S'(x) = 0 \).
[2]
(ii) Interpret your answer to part (b)(i) in context.
[1]
Question #19
Hard
[Maximum Mark: 5]
(a) The concentration of a certain chemical in the blood, \( C(t) \), \( t \) hours after administration is given by \( C(t) = 30(0.75)^t \), \( t \geq 0 \). The concentration before administration is zero. Write down:
(i) the initial concentration of the chemical.
(ii) the percentage of the chemical that leaves the blood each hour.
[3]
(b) Calculate the concentration remaining in the blood 8 hours after administration.
[2]
Question #20
Hard
[Maximum Mark: 5]
An inclined chairlift travels along a straight line on a steep slope, as shown in the diagram. The locations of the stations on the chairlift can be described by coordinates in reference to the x, y, and z-axes, where the x and y axes are in the horizontal plane and the z-axis is vertical. The ground level station A has coordinates (120, 20, 0) and station B, located near the top of the hill, has coordinates (25, 10, 300). All coordinates are given in metres.
(a) Find the distance between stations A and B.
[2]
(b) Station M is to be built halfway between stations A and B. Find the coordinates of station M.
[2]
(c) Write down the height of station M, in metres, above the ground.
[1]
Question #21
Hard
[Maximum Mark: 7]
A rectangular container has a length of 120 cm, a width of 50 cm, and a height of 45 cm. The lid is a hemisphere with the same diameter as the width of the box. The entire surface of the container is to be coated.
(a) Calculate the total surface area to be coated.
[7]
Question #22
Hard
[Maximum Mark: 5]
A drone is flying 450 meters above a field. A person is walking in a straight line from point \( C \) to point \( B \). The person observes the drone at an angle of \( 30^\circ \) from point \( C \) and at \( 45^\circ \) from point \( B \), 20 minutes later.
(a) Determine the angle of depression from a drone flying 450 meters above a field to a point \( C \) on the ground, where the person observes the drone at an angle of \( 30^\circ \).
[2]
(b) Calculate the horizontal distance between points \( C \) and \( B \), given that the person observes the drone at an angle of \( 45^\circ \) from point \( B \) after 20 minutes of walking.
[3]
Question #23
Hard
[Maximum Mark: 5]
The number of hours employees in a tech company worked overtime during a month is displayed in a box-and-whisker plot.
(a) Write down the minimum number of overtime hours worked according to the box-and-whisker plot provided.
[1]
(b) Write down the lower quartile and median number of overtime hours worked according to the plot.
[2]
(c) An analyst claims that fewer than 20% of employees worked less than 5 overtime hours while more than 30% worked more than 15 hours. Comment on the validity of this claim, providing justification.
[2]
Question #24
Hard
[Maximum Mark: 5]
A function is defined by \( g(x) = 3 – \frac{5}{x + 4} \) for \( -9 \leq x \leq 9 \), \( x \neq -4 \).
(a) Determine the range of \( g(x) \).
[3]
(b) Find the value of \( g^{-1}(1) \).
[2]
Question #25
Hard
[Maximum Mark: 6]
(a) In a research study, the weights (in kg) of two different breeds of dogs, namely Beagles and Poodles, are recorded as follows:
Formulate the null and alternative hypotheses for a t-test to determine if Beagles are generally heavier than Poodles.
| Beagles (kg) | 15.3 | 16.1 | 14.8 | 16.4 | 15.6 | 16.2 | 15.0 | 16.0 | 15.7 | 16.3 |
|---|
| Poodles (kg) | 14.5 | 14.7 | 14.3 | 14.9 | 15.1 | 14.8 | 14.6 | 14.7 | 14.4 | 14.9 |
|---|
[2]
(b) Calculate the p-value for the t-test.
[2]
(c) State the conclusion of the test and justify your answer.
[2]
Question #26
Hard
[Maximum Mark: 7]
A park includes a flower bed enclosed by a circular arc \( AB \) of a circle centered at \( O \) with a radius of 8 m, where \( \angle AOB = 120^\circ \).
(a) Calculate the length of the circular arc AB.
[3]
(b) The straight border of the flower bed is formed by chord \( [AB] \). Determine the area of the flower bed.
[4]
Question #27
Hard
[Maximum Mark: 6]
Tommaso and Pietro have each been given 1500 euro to save for college.
(a) Pietro invests his money in an account with a nominal annual interest rate of 3.5%, compounded quarterly. Calculate the value of Pietro’s investment after 4 years. Round your answer to two decimal places.
[3]
(b) Tommaso wants his investment to double in 6 years with a nominal annual interest rate \( r \), compounded monthly. Find the value of \( r \).
[3]
Question #28
Hard
[Maximum Mark: 6]
A retail manager wants to predict daily sales of a particular product based on past data. The sales data for 5 consecutive days is recorded as follows:
| Day | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| Number of units sold | 150 | 120 | 135 | 140 | 155 |
(a) Estimate the expected number of units sold per day based on the sales data provided.
[1]
(b)
(i) State the degrees of freedom for the goodness-of-fit test.
(ii) Determine whether the manager’s prediction model is suitable at a 5% significance level. Provide justification for your conclusion.
(ii) Determine whether the manager’s prediction model is suitable at a 5% significance level. Provide justification for your conclusion.
[5]
Question #29
Hard
[Maximum Mark: 7]
Consider the quadratic function \( f(x) = ax^2 + bx + c \). The graph of \( y = f(x) \) has its vertex at \( (4, -4) \) and intersects the x-axis at points \( (p, 0) \) and \( (6, 0) \).
(a) Find the value of \( p \).
[1]
(b) Determine the values of:
(i) \( a \)
(ii) \( b \)
(iii) \( c \).
[5]
(c) Write down the equation of the axis of symmetry.
[1]
Question #30
Hard
[Maximum Mark: 7]
A company’s profit \( P \) (in thousands of dollars) from producing \( x \) units of a product is given by \( \frac{dP}{dx} = -2x + 80 \), where \( x \geq 0 \). The profit when producing 20 units is $600.
(a) Find the expression for \( P(x) \).
[5]
(b) Discuss how the company’s profit changes as production increases beyond 40 units per month. Justify your answer.
[2]
Question #31
Hard
[Maximum Mark: 4]
Lucas approximates \( q \), correct to four decimal places, by using the following expression: \[ 4 + \frac{1}{5 + \frac{11}{18}} \]
(a) Calculate Lucas’s approximation of \( q \), correct to four decimal places.
[2]
(b) Calculate the percentage error in using Lucas’s four decimal place approximation of \( q \), compared to the exact value of \( q \) in your calculator.
[2]
Question #32
Hard
[Maximum Mark: 4]
The daily maximum temperature over ten consecutive days was recorded by Emma. The temperatures, in degrees Celsius (\(^\circ C\)), are shown in the table below.
| 13 | 17 | 13 | 10 | 12 | 14 | 13 | 16 | 15 | 11 |
(a) Determine the mode of this data set.
[1]
(b) Calculate the mean of the temperature data set.
[2]
(c) Compute the standard deviation of the temperatures.
[1]
Question #33
Hard
[Maximum Mark: 6]
A decoration is made in the shape of a solid hemisphere. The radius of the hemisphere is 5 mm.
(a) Calculate the total surface area of one decoration.
[4]
(b) The total surface of the decoration is coated in gold. It is known that 1 gram of gold covers an area of 200 mm\(^2\). Calculate the weight of gold required to coat one decoration.
[2]
Question #34
Hard
[Maximum Mark: 7]
The price of petrol at Maria’s petrol station is $1.40 per litre. If a customer buys a minimum of 12 litres, a discount of $4 is applied.
This can be modeled by the following function, \( P(x) = 1.40x – 4 \), where \( x \geq 12 \), which gives the total cost when buying a minimum of 12 litres at Maria’s petrol station, where \( x \) is the number of litres of petrol that a customer buys.
(a) Find the total cost of buying 35 litres of petrol at Maria’s petrol station.
[2]
(b) Find the inverse of the function \( P(x) \), and use it to determine the number of litres, \( x \), required for the total cost to be $60.
[2]
(c) The price of petrol at John’s petrol station is $1.25 per litre. A customer must buy a minimum of 12 litres. The total cost at John’s petrol station is cheaper than Maria’s petrol station when \( x > k \). Find the minimum value of \( k \).
[3]
Question #35
Hard
[Maximum Mark: 6]
The Voronoi diagram below shows three identical cellular phone towers, \( C_1 \), \( C_2 \), \( C_3 \), and \( C_4 \). The coordinates for \( C_2 \), the tower in the region between \( C_1 \) and \( C_3 \), are not defined. The lines in the diagram below represent the edges in the Voronoi diagram.
Horizontal scale: 1 unit represents 1 km. Vertical scale: 1 unit.
(a) Sarah stands in the shaded region. Explain why Sarah will receive the strongest signal from tower \( C_4 \).
[1]
(b) Tower \( C_4 \) has coordinates \((12, 6)\) and the edge connecting vertices \( A \) and \( B \) has the equation \( y = 7 \). Write down the coordinates of tower \( C_2 \).
[2]
(c) Tower \( C_1 \) has coordinates \((6, 9)\). Find the gradient of the edge of the Voronoi diagram between towers \( C_1 \) and \( C_2 \).
[3]
Question #36
Hard
[Maximum Mark: 5]
Jackson has hens on his farm. He claims the mean weight of eggs from his black hens is less than the mean weight of eggs from his white hens.
He recorded the weights of eggs, in grams, from a random selection of hens. The data is shown in the table below.
In order to test his claim, Jackson performs a \( t \)-test at a 5% level of significance. It is assumed that the weights of eggs are normally distributed and the samples have equal variances.
| Black Hens | 140 | 135 | 145 | 142 | 130 | 128 |
|---|---|---|---|---|---|---|
| White Hens | 138 | 140 | 144 | 142 | 137 | 135 |
(a) State, in words, the null hypothesis.
[1]
(b) Calculate the \( p \)-value for this test.
[2]
(c) State whether the result of the test supports Jackson’s claim. Justify your reasoning.
[2]
Question #37
Hard
[Maximum Mark: 6]
Professor Kim observed that students have difficulty remembering the information presented in her lectures. She modeled the percentage of information retained, \( R \), by the function \( R(t) = 100 e^{-qt} \), \( t \geq 0 \), where \( t \) is the number of days after the lecture. She found that 1 day after a lecture, students had forgotten 40% of the information presented.
(a) Find the value of \( q \).
[2]
(b) Use this model to find the percentage of information retained by her students 2 days after Professor Kim’s lecture.
[2]
(c) Based on her model, Professor Kim believes that her students will always retain some information from her lecture. State a mathematical reason why Professor Kim might believe this.
[1]
(d) Write down one possible limitation of the domain of the model.
[1]
Question #38
Hard
[Maximum Mark: 8]
Alex and Brian each began a fitness programme. On day one, they both ran 600 m. On each subsequent day, Alex ran 150 m more than the previous day whereas Brian increased his distance by 3% of the distance ran on the previous day.
(a)
(i) Calculate how far Alex ran on day 25 of his fitness programme.
[2]
(ii) Calculate how far Brian ran on day 25 of his fitness programme.
[3]
(b) On day \( n \) of the fitness programmes, Brian runs more than Alex for the first time. Find the value of \( n \).
[3]
Question #39
Hard
[Maximum Mark: 5]
A triangular plot of land \( XYZ \) is such that \( XY = 60 \, \text{m} \) and \( YZ = 90 \, \text{m} \), each measured correct to the nearest metre, and the angle at \( Y \) is equal to 100\(^\circ\), measured correct to the nearest 1\(^\circ\).
(a) Calculate the maximum possible area of the plot.
[5]
Question #40
Hard
[Maximum Mark: 7]
A game is played where two unbiased dice are rolled and the score in the game is the lesser of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on the dice. As each dice
(a) Complete the table to show the probability distribution of \( S \), where \( S \) is the random variable representing the score in a game.
| t | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| P(T = t) |
[2]
(b) Find the probability that:
- (i) a player scores at least 4 in a game.
- (ii) a player scores 6, given that they scored at least 4.
[3]
(c) Find the expected score of a game.
[2]
Question #41
Hard
[Maximum Mark: 6]
When an intruder comes close to a castle an alarm is sounded. The sound intensity, \( I \), of the alarm varies inversely with the square of the distance, \( d \), from the alarm, where \( d > 0 \). It is known that at a distance of 2 metres from the alarm, the sound intensity is 5 watts per square metre (W m\(^{-2}\)).
(a) Show that \( I = \frac{20}{d^2} \).
[2]
(b) Sketch the curve of \( I \) on the axes below showing clearly the point (2, 5).
[2]
(c) Whilst walking, Maya can hear the alarm only if the sound intensity at her location is greater than \( 2 \times 10^{-6} \, \text{Wm}^{-2} \). Find the values of where Maya cannot hear the alarm.
[2]
Question #42
Hard
[Maximum Mark: 8]
Grace kicks foot balls into the air. Each time she hits a ball, she records \( \theta \), the angle at which the ball is launched into the air, and \( l \), the horizontal distance, in metres, which the ball travels from the point of contact to the first time it lands.
Grace analyses her results and concludes:
\[ \frac{dl}{d\theta} = -0.25\theta + 7.5 \quad 30^\circ \leq \theta \leq 70^\circ \]
(a) Determine whether the graph of \( l \) against \( \theta \) is increasing or decreasing at \( \theta = 45^\circ \).
[3]
(b) Grace observes that when the angle is \( 35^\circ \), the ball will travel a horizontal distance of 195.2 m. Find an expression for the function \( l(\theta) \).
[5]
Question #43
Hard
[Maximum Mark: 5]
The front view of a garden shed is made up of a rectangle with a semicircle on top. The shed is 2.4 m high, 1.2 m wide, and 1.5 m long and sits on a rectangular base.
The top of the curved surfaces of the roof of the shed is to be painted. Find the area to be painted.
[5]
Question #44
Hard
[Maximum Mark: 4]
A telephone pole stands on flat ground. The base of the pole is taken as the origin, R, of a coordinate system in which the top, S, of the pole has coordinates (0, 0, 7.4). All units are in meters.
(a) Calculate the length of the cable connecting T to S if point T has coordinates (5.0, 2.5, 0).
(a) Calculate the length of the cable connecting T to S if point T has coordinates (5.0, 2.5, 0).
[2]
(b) Determine the angle θ that the cable makes with the ground.
[2]
Question #45
Hard
[Maximum Mark: 5]
The height of a soccer ball after it is kicked is modeled by the function
\(h(t) = -4.9t^2 + 15t + 1.0\)
where \(h(t)\) is the height in meters above the ground and \(t\) is the time in seconds after the ball was kicked.
(a) Write down the height of the ball above the ground at the moment it was kicked.
[1]
(b) Calculate the value of \(t\) when the ball hits the ground.
[2]
(c) Specify an appropriate domain for \(t\) in this model.
[2]
Question #46
Hard
[Maximum Mark: 6]
The relationship between temperature in Kelvin \(K\) and degrees Fahrenheit \(F\) is
\[ F = 1.8K – 459.67. \]
(a) (i) Derive a formula that converts a temperature measured in degrees Fahrenheit to Kelvin. [2]
(ii) Using your formula, find the temperature (in Kelvin) that corresponds to \(77^\circ\text{F}\). [1]
Over one year, the mean daily temperature in Mexico City was \(290\,\text{K}\) with a standard deviation of \(9\,\text{K}\).
(b) (i) Express, in degrees Fahrenheit, the mean daily temperature for that year. [2]
(ii) Express, in degrees Fahrenheit, the standard deviation of the daily temperature. [1]
Question #47
Hard
[Maximum Mark: 7]
(a) Determine the equation of the line that the road follows, given that the towns M and N are located at (5, -3) and (10, 9) respectively, and the road runs along the perpendicular bisector of [MN].
Three towns, M, N, and O are plotted as coordinates on a map, where the x and y axes represent distances east and north of a central point, respectively, measured in kilometers.
Town M is located at (5, -3) and town N is located at (10, 9). A road runs along the perpendicular bisector of [MN]. This information is shown in the following diagram.
(a) Determine the equation of the line that the road follows, given that the towns M and N are located at (5, -3) and (10, 9) respectively, and the road runs along the perpendicular bisector of [MN].
[5]
(b) Find the y-coordinate of town O if town O is directly south of town M, and the road passes through town O.
[2]
Question #48
Hard
[Maximum Mark: 5]
The ticket prices for a music festival are shown in the following table:
– A total of 800 tickets were sold.
– The total revenue from ticket sales was €14,700.
– The number of child tickets sold was 100 more than the number of senior tickets sold.
Let the number of adult tickets sold be \(x\), the number of child tickets sold be \(y\), and the number of senior tickets sold be \(z\).
| Ticket Type | Price (in euros, €) |
|---|---|
| Adult | 25 |
| Child | 15 |
| Senior | 18 |
– A total of 800 tickets were sold.
– The total revenue from ticket sales was €14,700.
– The number of child tickets sold was 100 more than the number of senior tickets sold.
Let the number of adult tickets sold be \(x\), the number of child tickets sold be \(y\), and the number of senior tickets sold be \(z\).
(a) Write down three equations that express the information provided above.
[3]
(b) Determine the number of each type of ticket sold.
[2]
Question #49
Hard
[Maximum Mark: 7]
An art piece is contained in a circular frame. The art piece has a shaded region bounded by a curve and a horizontal line.
When the art piece is placed on coordinate axes such that the bottom left corner of the piece has coordinates (1.0, 2.0) and the top right corner has coordinates (1.5, 2.5), the curve can be modeled by \(y = h(x)\), and the horizontal line can be modeled by the x-axis. Distances are measured in meters.
(a) Use the trapezoidal rule, with the values given in the following table, to approximate the area of the shaded region.
| x | 1.0 | 1.25 | 1.5 |
|---|---|---|---|
| y | 2 | 2.25 | 2.4 |
[3]
(b) The artist used the equation \(y = x^2 – 2x + 3\) to draw the curve. Calculate the exact area of the shaded region in the art piece.
[2]
(c) Calculate the area of the unshaded region in the art piece.
[2]
Question #50
Hard
[Maximum Mark: 7]
Maria is investigating whether a six-sided die is biased. She rolls the die 90 times and records the observed frequencies in the following table:
Maria performs a \(\chi^2\) goodness of fit test at a 5% significance level.
| Number on die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Observed frequency | 15 | 12 | 13 | 18 | 16 | 16 |
Maria performs a \(\chi^2\) goodness of fit test at a 5% significance level.
(a) State the null and alternative hypotheses.
[1]
(b) State the degrees of freedom.
[1]
(c) State the expected frequency of rolling a 1.
[1]
(d) Calculate the p-value for the test.
[2]
(e) Determine the conclusion of the test. Justify your answer.
[2]
Question #51
Hard
[Maximum Mark: 6]
A factory produces bottles of olive oil with a labeled volume of 500 ml. The volumes of the bottles are normally distributed with a mean of 500 ml and a standard deviation of 4 ml.
(a) State the percentage of bottles that have a volume greater than 500 ml.
[1]
(b) A bottle that has a volume less than 490 ml is rejected by the factory for being underfilled. Calculate the probability that a randomly chosen bottle is rejected for being underfilled.
[2]
(c) A bottle that has a volume greater than \(k\) ml is rejected by the factory for being overfilled. The factory rejects 2% of bottles for being overfilled. Determine the value of \(k\).
[3]
Question #52
Hard
[Maximum Mark: 7]
The function \(p\) is defined by \(p(x) = \frac{5x^3 + 4x^2 + 9}{x}\), \(x \neq 0\).
(a) Determine \(p'(x)\).
[3]
(b) Find the equation of the tangent to the curve \(x = 3\) at \((3, 60)\) in the form \(ax + by + d = 0\), where \(a, b, d \in \mathbb{Z}\).
[4]
Question #53
Hard
[Maximum Mark: 6]
Emily has three red beads and five green beads in her jewelry box. She picks two beads at random from the box.
(a) Complete the tree diagram.
[1]
(b) Calculate the probability that Emily picks two beads of the same color.
[2]
(c) Given that Emily picks two beads of the same color, calculate the probability that both beads are red.
[3]
Question #54
Hard
[Maximum Mark: 8]
The brightness of stars is measured in magnitudes, with values typically ranging between -1 and 15 where -1 is the brightest.
The Star Magnitude equation gives the average number of stars per night, \(L\), which have a magnitude of at least \(M\). For a particular region, the equation is:
\[\log_{10}(L) = cM\]
This region records an average of 300 stars per night with a magnitude of at least 2.
The Star Magnitude equation gives the average number of stars per night, \(L\), which have a magnitude of at least \(M\). For a particular region, the equation is:
\[\log_{10}(L) = cM\]
This region records an average of 300 stars per night with a magnitude of at least 2.
(a) Find the value of \(c\).
[2]
(b) The equation for this region can also be written as:
\[L = d \times 10^{M}\]
Determine the value of \(d\).
\[L = d \times 10^{M}\]
Determine the value of \(d\).
[2]
(c) Given \(0 \leq M \leq 15\), find the range for \(L\).
[2]
(d) The expected interval of time between stars with a magnitude of at least \(M\) is \(\frac{1}{L}\). In this region, the brightest star recorded had a magnitude of -0.5. Find the expected interval of time between this star and the next star of at least this magnitude. Give your answer to the nearest hour.
[2]
Question #55
Hard
[Maximum Mark: 6]
A company’s annual revenue was found to be changing at a rate of
\[\frac{dR}{dt} = 4t^2 + 10t\]
where \(R\) is the company’s revenue in thousands of dollars and \(t\) is the time since the company was founded, measured in years.
(a) Determine whether the revenue is increasing or decreasing when \(t = 3\).
[2]
(b) Two years after the company was founded, the revenue was 6 thousand dollars. Find an expression for \(R(t)\), when \(t = 0\).
[4]
Question #56
Hard
[Maximum Mark: 7]
A basketball is dropped from a height of 2.2 meters and bounces on the ground. The maximum height reached by the ball, after each bounce, is 80% of the previous maximum height.
(a) Show that the maximum height reached by the ball after it has bounced for the seventh time is 45 cm, to the nearest cm.
[2]
(b) Find the number of times, after the first bounce, that the maximum height reached is greater than 15 cm.
[2]
(c) Calculate the total vertical distance traveled by the ball from the point at which it is dropped until the fifth bounce.
[3]
Question #57
Hard
[Maximum Mark: 6]
The triathlon is a competition where athletes compete in three events: swimming, cycling, and running. In all events, a shorter time means a better ranking.
The table below shows the results for the swimming and cycling events at a national competition.
The table below shows the results for the swimming and cycling events at a national competition.
| Athlete’s Country | Swimming Time (min) | Cycling Time (min) | Swimming Rank | Cycling Rank |
|---|---|---|---|---|
| Brazil | 25.3 | 45.1 | 1 | |
| Argentina | 26.7 | 43.8 | 2 | |
| Chile | 27.0 | 46.2 | 3 | |
| Peru | 28.5 | 47.5 | 4 | |
| Uruguay | 29.1 | 48.3 | 5 | |
| Paraguay | 29.9 | 49.1 | 6 | |
| Bolivia | 30.3 | 50.0 | 7 | |
| Ecuador | 31.4 | 51.2 | 8 | |
| Colombia | 32.7 | 52.3 | 9 | |
| Venezuela | 34.0 | 53.5 | 10 |
The Spearman’s rank correlation coefficient is used to determine if there is a linear correlation between an athlete’s ranking in swimming and their ranking in cycling.
(a) Complete the table to show the athletes’ rankings in cycling.
[2]
(b) Calculate the Spearman’s rank correlation coefficient \( r_s \).
[2]
The following guide is used by the coach to determine the strength of the correlation between the ranks for swimming and cycling.
| \( |r_s| \) | Strength |
|---|---|
| 0.000 to 0.199 | Very weak |
| 0.200 to 0.399 | Weak |
| 0.400 to 0.599 | Moderate |
| 0.600 to 0.799 | Strong |
| 0.800 to 1.000 | Very strong |
(c) State the strength of the correlation between the rankings and interpret this in the context of the question.
[2]
Question #58
Hard
[Maximum Mark: 6]
A drone is flying above two observers. One observer is on the roof of the a building, which is 120 meters tall. The other observer is at the base of the same building.
The angle of elevation from point A (on the roof) to the drone is \( 35^\circ \), and the angle of elevation from point B (at the base) to the drone is \( 50^\circ \). Point X is directly below the drone on the ground. The diagram below illustrates this setup.
(a) Determine the size of angle \( \hat{A}BX \).
[2]
(b) Calculate the distance from point B to the drone.
[3]
As the drone flies further away, maintaining the same altitude:
(c) Describe how the angle of depression from the drone to point B changes as the horizontal distance increases.
[1]
Question #59
Hard
[Maximum Mark: 5]
On 1 January 2023, Amira deposited $1500 into a savings account with an annual interest rate of 5%, compounded quarterly. At the end of every quarter, she deposits an additional $200 into the account.
(a) Calculate the amount of money in her account at the start of 2025. Give your answer to two decimal places.
[3]
(b) Find the number of complete quarters, starting from 1 January 2023, it will take for Amira’s account balance to exceed $7000.
[2]
Question #60
Hard
[Maximum Mark: 6]
Jonathan believes that, in a memory recall test, the mean score of students who regularly meditate (\(m_m\)) will be higher than the mean score of those who do not (\(m_n\)). Jonathan administered a memory recall test to a random sample of students. The scores are shown in the tables below.
| Scores of Meditating Students | 85 | 87 | 92 | 90 | 88 | 94 | 91 | 89 | 93 | 86 |
|---|---|---|---|---|---|---|---|---|---|---|
| Scores of Non-Meditating Students | 82 | 85 | 80 | 88 | 79 | 84 | 86 | 83 |
Jonathan conducts a one-tailed t-test at a 5% significance level. Assume the scores are normally distributed and the samples have equal variances.
(a) State the null and alternative hypotheses.
[2]
(b) Calculate the p-value for this test.
[2]
(c) State the conclusion of the test in the context of the question. Justify your answer.
[2]
Question #61
Hard
[Maximum Mark: 5]
Line \( L_1 \) is tangent to the graph of the function \( g(x) \) at the point \( Q(4, -2) \). Line \( L_2 \) is given by the equation \( y = \frac{1}{3}x – 4 \) and is perpendicular to \( L_1 \).
(a) Write down the gradient of \( L_1 \).
[1]
(b) Find the equation of \( L_1 \) in the form \( y = mx + c \).
[2]
(c) Show that \( L_2 \) is not the line that is normal to \( g(x) \) at point Q.
[2]
Question #62
Hard
[Maximum Mark: 5]
When the brakes of a bicycle are fully applied, it will continue to travel some distance before it stops completely. This stopping distance, \( d \), in meters, is directly proportional to the square of the speed, \( v \), of the bicycle in kilometers per hour (km/h).
When a bicycle is traveling at 20 km/h, it will travel 4.8 meters after the brakes are fully applied before stopping completely.
When a bicycle is traveling at 20 km/h, it will travel 4.8 meters after the brakes are fully applied before stopping completely.
(a) Determine an equation for \( d \) in terms of \( v \).
[2]
The police use this equation to estimate whether cyclists are exceeding the speed limit.
A cyclist is found to have traveled 15 meters after fully applying the brakes before stopping.
(b) Use your equation from part (a) to estimate the speed at which this cyclist was traveling before the brakes were applied.
[2]
(c) Identify one other factor besides speed that could affect the stopping distance after the brakes are fully applied.
[1]
Question #63
Hard
[Maximum Mark: 6]
A rectangular tray with an open top is to be made from a piece of sheet metal
measuring 60 cm by 40 cm. Squares of equal size will be cut from the corners of the sheet
metal, as indicated in the diagram.
The sides will then be folded along the grey lines to form the tray.
The sides will then be folded along the grey lines to form the tray.
The volume of the tray, in cubic centimeters, can be modeled by the function
\( V(x) = (60 – 2x)(40 – 2x)x \), for \( 0 < x < k \), where \( x \) is the length of the sides of
the squares removed (in centimeters).
(a) State the maximum possible value of \( k \) in this context.
[1]
(b) Determine the value of \( x \) that maximizes the volume of the tray.
[2]
A second piece of 60 cm by 40 cm sheet metal is damaged, requiring a strip 3 cm wide
to be removed from all four sides. A tray will then be constructed in a similar manner from the remaining
sheet metal.
(c) Calculate the maximum possible volume of the tray made from the second piece of sheet metal.
[3]
Question #64
Hard
[Maximum Mark: 7]
Password strength is a measure of how secure a computer password is. The higher the strength, the more difficult it is to guess the password. The relationship between password strength, \( s \) (measured in bits), and the number of guesses, \( N \), required to crack the password is given by \( 0.301s = \log_{10} N \).
(a) Calculate the value of \( s \) for a password that requires 8000 guesses to crack.
[2]
(b) Express \( N \) as a function of \( s \).
[1]
(c) Calculate the number of guesses required to crack a password with a strength of 32 bits. Write your answer in the form \( a \times 10^k \), where \( 1 \leq a < 10 \), \( k \in \mathbb{Z} \).
[3]
The graph of the function \( N(s) \) passes through the point \( (0, 1) \).
(d) Explain the significance of the coordinate values in the context of password security.
[1]
Question #65
Hard
[Maximum Mark: 8]
The cross-section of a model of a hill is described by the following graph.
The heights of the model are measured at horizontal intervals and are provided in the table.
| Horizontal distance, \( x \) (cm) | 0 | 12 | 24 | 36 | 48 |
|---|---|---|---|---|---|
| Vertical distance, \( y \) (cm) | 0 | 4 | 9 | 11 | 0 |
(a) Use the trapezoidal rule with \( h = 12 \) to estimate the cross-sectional area of the model.
[2]
It is given that the equation of the curve is \( y = 0.05x^2 – 0.0012x^3 \), for \( 0 \leq x \leq 48 \).
(b) (i) Write down an integral to find the exact cross-sectional area.
[2]
(ii) Calculate the exact value of the cross-sectional area, giving your answer to two decimal places.
[2]
(c) Determine the percentage error in the area found using the trapezoidal rule.
[2]
Question #66
Hard
[Maximum Mark: 7]
In a cricket match, Ria is batting near the wicket. The ball is delivered towards the wicket on a path described by the function .
In this model, is the horizontal distance of the ball from where it was delivered, and is the vertical height of the ball above the ground, both measured in meters.
The ball is considered a valid delivery if its height is between 0.4 m and 1.2 m as it passes over the wicket. The length of the wicket is 0.6 m.
When the ball reaches the start of the wicket, its height above the ground is 1.4 m. The height of the ball changes by meters as it travels over the length of the wicket.
(a) (i) Determine the value of .
[2]
(ii) Explain whether this delivery is valid.
[2]
On the next delivery, Ria hits the ball towards a boundary wall that is 4 meters high. The boundary wall is 90 meters from where the ball was hit. The path of the ball after it is hit can be modeled by the function .
(b) Determine whether the ball will clear the boundary wall. Justify your answer.
[3]
Question #67
Hard
[Maximum Mark: 7]
Wooden posts are to be installed around a community garden. Each post is made up of a cuboid base with a right square pyramid on top.
The cuboid part of the post is machine-made, with a width, and hence the width of the pyramid, of exactly 25 cm. The length from the apex of the pyramid, \( A \), to any corner of the base of the pyramid is 18.3 cm, accurate to the nearest tenth of a centimeter. The post is shown in the diagram.
(a) State the upper and lower bounds for the possible lengths of edge \( AC \).
[2]
Point \( H \) is the midpoint of \( BC \).
(b) Determine the upper and lower bounds for \( AH \), the slant height of the pyramid.
[3]
To ensure the posts are safe, the angle between the slant height and the base of the pyramid must be less than \( 24^\circ \).
(c) Prove that this post is safe. Justify your answer.
[2]
Question #68
Hard
[Maximum Mark: 7]
On a certain day, the speed of trains passing through a station can be modeled by a normal distribution with a mean of 72.5 km/h. A speed of 80.3 km/h is two standard deviations above the mean.
(a) Calculate the standard deviation for the speed of the trains.
[2]
Speeding fines are issued to all train operators whose trains travel at a speed greater than 77 km/h.
(b) Calculate the probability that a randomly selected train will incur a speeding fine.
[2]
It is observed that 84% of trains on this route travel at speeds between \( p \) km/h and \( q \) km/h, where \( p < q \). This range includes trains traveling at a speed of 78 km/h.
(c) Demonstrate that the range between \( p \) and \( q \) is not symmetrical about the mean.
[3]
Question #69
Hard
[Maximum Mark: 5]
A yacht travels 12 km on a bearing of \( 295^\circ \) and then a further 8 km on a bearing of \( 065^\circ \).
Determine the bearing on which the yacht should travel to return directly to the starting point.
[5]
Question #70
Hard
[Maximum Mark: 5]
Aisha is constructing a large sculpture and estimates the total weight of the materials to be approximately 28,500 kg. The actual weight of the materials needed for the sculpture is 30,150 kg.
(a) Calculate the percentage error in Aisha’s approximation.
[2]
Aisha decides to build three identical sculptures.
(b) (i) Determine the weight of the materials needed for these three sculptures, rounded to three significant figures.
[2]
(ii) Express your answer to part (b)(i) in the form \( a \times 10^k \), where \( 1 \leq a < 10 \), \( k \in \mathbb{Z} \).
[1]
Question #71
Hard
[Maximum Mark: 6]
Benita has $2,500 in her savings account. She considers investing the money for 3 years with a financial institution offering an annual interest rate of 2.5% compounded monthly.
(a) Calculate the amount of money Benita would have at the end of 3 years. Give your answer correct to two decimal places.
[3]
Instead of investing the money, Benita decides to purchase a laptop that costs $2,500. After 3 years, the laptop will have a value of $400. Assume that the depreciation rate per year is constant.
(b) Calculate the annual depreciation rate of the laptop.
[3]
Question #72
Hard
[Maximum Mark: 7]
In a university, 150 students participated in a coding competition. Their times to complete the competition were recorded, and the following cumulative frequency graph was generated.
(a) Use the graph to find:
(i) the median time;
[1]
(ii) the lower quartile;
[1]
(iii) the upper quartile;
[1]
(iv) the interquartile range.
[1]
Michael completed the competition in 25 minutes.
(b) Determine whether Michael’s time is an outlier.
[3]
Question #73
Hard
[Maximum Mark: 6]
At a local gym, Laura conducts a study to see if there is any relationship between a person’s age and the time it takes them to complete a fitness challenge. Eight participants were chosen at random, and their details are shown below.
| Participant | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Age (years) | 22 | 30 | 18 | 25 | 35 | 40 | 20 | 50 |
| Time (minutes) | 45.2 | 40.1 | 50.3 | 43.8 | 39.0 | 37.6 | 47.5 | 42.9 |
Laura decides to calculate the Spearman’s rank correlation coefficient for this data.
(a) Complete the table of ranks.
[2]
| Participant | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Age rank | ||||||||
| Time rank |
(b) Calculate the Spearman’s rank correlation coefficient, \( r_s \).
[2]
(c) Interpret this value of \( r_s \) in the context of the question.
[1]
(d) Provide a mathematical reason why Laura may have decided not to use Pearson’s product-moment correlation coefficient with the original data.
[1]
Question #74
Hard
[Maximum Mark: 4]
The following frequency distribution table shows the number of books read by students in a reading challenge.
| Books Read | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Frequency | 3 | 5 | 6 | 10 | p | 7 | 4 |
For this distribution, the mean number of books read is 4.
(a) Write down the total number of students in terms of \( p \).
[1]
(b) Calculate the value of \( p \).
[3]
Question #75
Hard
[Maximum Mark: 6]
A bakery wants to determine if there are differences in the quality of its products during different shifts: morning, afternoon, and evening.
Their quality assurance team randomly selects 600 items to inspect. The quality of these items is categorized as excellent, good, or poor. The data is summarized in the following table.
| Shift | Excellent | Good | Poor | Total |
|---|---|---|---|---|
| Morning | 140 | 180 | 20 | 340 |
| Afternoon | 90 | 110 | 10 | 210 |
| Evening | 30 | 60 | 10 | 100 |
| Total | 260 | 350 | 40 | 650 |
An item is chosen at random from these 650.
(a) Find the probability that the item is not excellent, given that it is from the morning shift.
[2]
A \( \chi^2 \) test at the 5% significance level is conducted to determine if there is significant evidence of a difference in the quality of the products across the three shifts. The critical value for this test is 9.488.
The hypotheses for this test are:
\( H_0 \): The quality of the products and the shift are independent.
\( H_1 \): The quality of the products and the shift are not independent.
(b) Find the \( \chi^2 \) statistic.
[2]
(c) State, with justification, the conclusion for this test.
[2]
Question #76
Hard
[Maximum Mark: 5]
Tom owns three restaurants represented by points P, Q, and R on a map. Tom wants to divide the area into delivery zones. This process has begun in the following incomplete Voronoi diagram, where 1 unit represents 1 mile.
The midpoint of QR is \((4, 2)\).
(a) Show that the equation of the perpendicular bisector of [QR] is \(y = 2x – 6\).
[3]
Tom opens a new restaurant equidistant from the three points P, Q, and R.
(c) Show that the coordinates of the new restaurant is (3,0).
[2]
Question #77
Hard
[Maximum Mark: 5]
Kiran buys a scoop of sorbet in the shape of a sphere with a radius of 4.2 cm. The sorbet is served in a cup, and it may be assumed that \(\frac{1}{6}\) of the volume of the sorbet is inside the cup. This is shown in the following diagram.
(a) Calculate the volume of sorbet that is not inside the cup.
[3]
The cup has a height of 12 cm and a diameter of 5 cm. The outside of the cup is wrapped with a decorative paper.
(b) Calculate the surface area of the cup that is wrapped. Give your answer correct to the nearest cm².
[2]
Question #78
Hard
[Maximum Mark: 6]
The weights of oranges from a particular farm are normally distributed with a mean of 150 g and a standard deviation of 10 g. An orange from this farm is chosen at random.
(a) Calculate the probability that the weight of the orange is less than 140 g.
[2]
It is known that 20% of the oranges have a weight greater than \(k\) grams.
(b) Find the value of \(k\).
[2]
For an orange of weight \(w\) grams, chosen at random, \(P(150 – m < w < 150 + m) = 0.7\).
(c) Find the value of \(m\).
[2]
Question #79
Hard
[Maximum Mark: 8]
A football is kicked into the air. The height of the football is modeled by
\( h(t) = -5t^2 + 10t + 2, \quad t \geq 0, \)
where \( h \) is the height of the football above the ground, in meters, and \( t \) is the time, in seconds, after it was kicked.
(a) Find how long it takes for the football to reach its maximum height.
[2]
(b) Assuming that no one catches the football, find how long it would take for the football to hit the ground.
[2]
A player catches the football when it is at a height of 1.5 meters.
(c) Find the value of \( t \) when this player catches the football.
[2]
(d) State two limitations of using \( h(t) \) to model the height of the football.
[2]
Question #80
Hard
[Maximum Mark: 6]
Consider the function \(g(x) = \frac{x^2 – 4x + 3}{x + 1}\). The graph of \(g\) for \(-1 < x \leq 6\) is shown on the following axes.
(a) (i) Sketch the graph of \( g \) for \( -4 < x < -1 \) on the same axes.
[3]
(b) Use your graphic display calculator to find the solutions to the equation \( g(x) = 15 \).
[2]
(c) Write down the equation of the vertical asymptote for the graph of \( g \).
[1]
Question #81
Hard
[Maximum Mark: 5]
In a carnival game, players toss rings onto bottles. The random variable \( Y \) is the number of successful ring tosses out of five attempts. The probability distribution for \( Y \) is shown in the following table.
| y | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \(P(Y = y)\) | 0.1 | 0.2 | q | 0.15 | 2q | 0.3 |
(a) Find the value of \( q \).
[2]
The prize money awarded to the player, in dollars, is shown in the following table, where a negative value indicates a loss.
| y | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Prize (\$) | -5 | -3 | -2 | 1 | 3 | 6 |
(b) Determine whether this game is fair. Justify your answer.
[3]
Question #82
Hard
[Maximum Mark: 9]
A civil engineer wants to calculate the cross-sectional area of a water reservoir. The cross-section of the reservoir can be modeled by a curve and two straight lines as shown in the following diagram, where distances are measured in meters.
The curve is modeled by a function \( g(x) \). The following table gives values of \( g(x) \) for different values of \( x \) in the interval \( 0 \leq x \leq 4 \).
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y = g(x) | 0 | 2 | 2 | 1 | 0 |
(a) Calculate an estimate for the area in the interval \( 0 \leq x \leq 4 \) by using the trapezoidal rule with four equal intervals.
[2]
It is known that \( g'(x) = 4x^2 + 5 \) in the domain \( 0 < x < 1 \).
(b) Find an expression for \( g(x) \) in the domain \( 0 < x < 4 \).
[4]
(c) Hence find the actual area of the entire cross-section.
[3]
Question #83
Hard
[Maximum Mark: 7]
Consider the function \( g(x) = c \cos(dx) \) with \( c, d \in \mathbb{Z}^+ \). The following diagram shows part of the graph of \( g \).
(a) Write down the value of \( c \).
[1]
(b) (i) Write down the period of \( g \).
[1]
(ii) Hence, find the value of \( d \).
[2]
(c) Find the value of \( g\left(\frac{\pi}{10}\right) \).
[3]
Question #84
Hard
[Maximum Mark: 5]
Let the functions \( f(x) = 2x – 3 \) and \( h(x) = x^2 + 2m \), where \( m \) is a real number.
(a) Write an expression for \( (h \circ f)(x) \).
[2]
(b) If \( (h \circ f)(5) = 27 \), determine the possible values of \( m \).
[3]
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