📖 Hover (or tap) any dotted word for its definition. Write your answers in the squared boxes with a pencil, finger or mouse — even draw straight onto the graph. Work saves on your device.
(b)The function g(x) is defined for 0 ≤ x ≤ 4, x ∈ ℝ. Its graph is shown on the right, and is made up of two line segments. Use the graph of g(x) to answer parts (b)(i), (b)(ii), and (b)(iii).
(i)State the range of values of x for which g′(x) > 2.
(b)(i)
(ii)Find the value of g(g(3)) (a composition). Give your answer in the form ab where a, b ∈ ℕ. Show your work on the graph.
(b)(ii)
(iii)The graph of y = g(x) is shown again below. Draw and label the graph of y = g−1(x) on the same diagram, for 0 ≤ x ≤ 4, where g−1 is the inverse of the function g. Hint: the graph of g−1 is the image of the graph of g under axial symmetry in the line y = x.
Draw y = g⁻¹(x) directly on the diagram (reflect g in the line y = x)
(b)Two functions, f(x) and g(x), are defined as follows, for x ∈ ℝ, x > 0: f(x) = e9x
g(x) = ln √x
g(x) = ln √x
(i)Find the value of f(1·2). Give your answer in the form a × 10n, where a ∈ ℝ, 1 ≤ a < 10, n ∈ ℕ, and a is correct to 1 decimal place.
(b)(i)
(ii)Find the value of x for which g(x) = 3·5. Give your answer in the form ep, where p ∈ ℝ.
(b)(ii)
(iii)Write the function g(f(x)) in terms of x, in its simplest form.
(b)(iii)
(a)A clothing company sells t-shirts. The daily sales of t-shirts over a 360-day period can be modelled using the following function, T(x), where 0 ≤ x ≤ 360, x ∈ ℝ: T(x) = (x − 24060)3 + 70
(i)Fill in the table below, showing the value of T(x) for each of the given values of x. Give each value correct to the nearest whole number. (Two values are already filled in: T(60) = 43 and T(360) = 78.)
| x | 0 | 60 | 120 | 180 | 240 | 300 | 360 |
|---|---|---|---|---|---|---|---|
| T(x) | 43 | 78 |
(a)(i)
(ii)Draw the graph of y = T(x) on the axes, for 0 ≤ x ≤ 360, x ∈ ℝ.
(a)(ii)
The clothing company also sells scarves. The daily sales of scarves can be modelled using the following function, S(t), which is a periodic function with a period of 365 days: S(t) = 21 + 19 cos(2πt365). Here, t ∈ ℝ is the time, in days, from 1 January 2024, t ≥ 0, and 2πt365 is in radians.
(b)Work out the maximum and minimum daily sales of scarves, according to S(t). You do not need to use differentiation.
(b)
A different function, C(t), can also be used to model the daily sales of scarves, where: C(t) = S(t) − 2·4 + 0·03t. Again, t ∈ ℝ is the time, in days, from 1 January 2024, t ≥ 0.
(c)Find the value of t for which S(t) and C(t) give the same number of daily sales.
(c)
(d)The graphs, J, K, and L, of three functions are shown below (not to scale). One of these graphs shows the function C(t) over a number of years, where C(t) = S(t) − 2·4 + 0·03t. Write down which graph this is. Justify your answer.
(d)
(e)The derivative of C(t) is given by C′(t) = 0·03 − 38π365 sin(2πt365). Find the value of t that gives the first local maximum of C, that is, the first time that C′(t) = 0. Give your answer correct to the nearest day.
(e)
(c)The function g(x) is defined for −2 ≤ x ≤ 2, x ∈ ℝ. Its graph is shown in each of the two diagrams below.
(i)Draw the graph of g(x) − 2 on the co-ordinate diagram below, for x ∈ ℝ, on as large a domain as possible.
(c)(i)
(ii)Draw the graph of g(x + 3) on the co-ordinate diagram below, for x ∈ ℝ, on as large a domain as possible.
(c)(ii)
(c)The diagram below shows three sets, A, B, and C, and two functions, f and g, where f : A → B and g : B → C. #A = #C = 4 and #B = 3.
(i)Find the value of g(f(3)).
(c)(i)
(ii)Explain why g : B → C is injective but not surjective (that is, one-to-one but not onto).
(c)(ii)
Ava looks at the relationship between pairs of factors of a number.
(b)She makes a table to show the pairs of factors of 12 (the pairs of natural numbers x and y with xy = 12).
(i)Complete the table below, showing the 6 pairs of factors (x and y) of 12.
| x | 1 | 2 | 3 | 6 | ||
|---|---|---|---|---|---|---|
| y | 4 | 1 |
(b)(i)
(ii)Plot the 6 points above on the co-ordinate diagram below. One of the points is shown.
(b)(ii)
(iii)Ava realises that the relationship between x and y is y = 12x. On the co-ordinate diagram above, draw the graph of y = 12x for x ∈ ℝ, in the domain 1 ≤ x ≤ 12.
(b)(iii)
A Ferris wheel has a diameter of 120 m and completes exactly 10 full rotations in one hour. The point A starts at the lowest point, 12 m above ground. The height h of A after t minutes is h(t) = 72 − 60 cos(π3 t), where h is in metres, t ∈ ℝ, and π3 t is in radians.
(a)Complete the table below (h(1) = 42 is given).
| t | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| h(t) | 42 |
(a)
(b)Draw the graph of y = h(t) for 0 ≤ t ≤ 8, t ∈ ℝ.
(b)
(c)Find the period and range of h(t).
(c)
(d)During a 50-minute period, what is the greatest number of minutes for which the point A could be higher than 42 m?
(d)
(e)By solving the equation 72 − 60 cos(π3 t) = 110, find the second time (value of t) that the point A is at a height of 110 m, after it starts turning. Give your answer in minutes, correct to 2 decimal places.
(e)
The function h(x) = 0·001x3 − 0·12x2 + px + 5, x ∈ ℝ, in the domain 0 ≤ x ≤ 75.
(a)(i) Use h(10) = 30 to show that p = 3·6.
(a)(i)
(ii)Complete the table below and hence draw the graph of h(x) in the domain 0 ≤ x ≤ 75 on the grid.
| x | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 75 |
|---|---|---|---|---|---|---|---|---|---|
| h(x) | 30 | 21 | 5 | 21·875 |
(a)(ii)
(a)A cup of coffee is freshly brewed to 95°C. The temperature T (°C) as it cools is T(t) = Ae−0·081t + 20, where A is constant and t is time in minutes from when the coffee was brewed.
(i)Show that A = 75.
(a)(i)
(ii)Explain what the value 20 in the formula represents in the context of the coffee cooling.
(a)(ii)
(iii)Find the decrease in the temperature of the coffee 10 minutes after brewing. Give your answer correct to the nearest whole number.
(a)(iii)
(b)T(t) = 75e−0·081t + 20 gives the temperature of the coffee at time t. If the ideal temperature to drink coffee is 82°C, find the time, to the nearest second, that it takes for the coffee to reach this temperature.
(b)
(a)f(x) = 6x − 5 and g(x) = x + 56. Investigate if f(g(x)) = g(f(x)) (the composition of the two functions).
(a)
The number of bacteria in the early stages of a growing colony of bacteria can be approximated using the function N(t) = 450e0·065t, where t is the time, measured in hours, since the colony started to grow, and N(t) is the number of bacteria in the colony at time t.
(a)
(i)Find the number of bacteria in the colony after 4·5 hours. Give your answer correct to the nearest whole number.
(a)(i)
(ii)Find the time, in hours, that it takes the colony to grow to 790 bacteria. Give your answer correct to 1 decimal place.
(a)(ii)
(b)Using the function N(t) = 450e0·065t, find the average value of the number of bacteria in the colony during the period from t = 3 to t = 12. Give your answer correct to the nearest whole number.
(b)
(c)Find the rate of change at which N(t) = 450e0·065t is changing when t = 12. Give your answer correct to one decimal place. Interpret this value in the context of the question.
(c)
(d)After k hours, the rate of increase of N(t) is greater than 90 bacteria per hour. Find the least value of k, where k ∈ ℕ.
(d)
(e)The number of bacteria in a different colony can be approximated using P(t) = 220e0·17t, where P(t) is the number of bacteria and t is measured in hours. Assume that both colonies start growing at the same time. Find the time, to the nearest hour, at which the number of bacteria in both colonies will be equal.
(e)
The graph of the function f(x) = 3x, where x ∈ ℝ, cuts the y-axis at (0, 1) as shown in the diagram.
(a)(i)Draw the graph of the function g(x) = 4x + 1 on the diagram.
(a)(i)
(ii)Use substitution to verify that f(x) < g(x), for x = 1·9.
(a)(ii)
(b)Prove, using induction, that f(n) ≥ g(n), where n ≥ 2 and n ∈ ℕ.
(b)
The weekly revenue produced by a company manufacturing air conditioning units is seasonal. The revenue (in euro) can be approximated by the function r(t) = 22 500 cos(π26 t) + 37 500, t ≥ 0, where t is the number of weeks measured from the beginning of July and (π26 t) is in radians.
(a)Find the approximate revenue produced 20 weeks after the beginning of July. Give your answer correct to the nearest euro.
(a)
(b)Find the two values of the time t, within the first 52 weeks, when the revenue is approximately €26 250.
(b)
(c)Find r′(t), the derivative of r(t) = 22 500 cos(π26 t) + 37 500.
(c)
(d)Use calculus to show that the revenue is increasing 30 weeks after the beginning of July.
(d)
(e)Find a value for the time t, within the first 52 weeks, when the revenue is at a minimum. Use r″(t) to verify your answer.
(e)
The time, in days of practice, it takes Jack to learn to type x words per minute (wpm) can be modelled by t(x) = k[ln(1 − x80)], where 0 ≤ x ≤ 70, x ∈ ℝ, and k is a constant.
(a)Based on the function t(x), Jack can learn to type 35 wpm in 35·96 days. Write the function in terms of k and hence show that k = −62·5, correct to 1 decimal place.
(a)
(b)Find the number of wpm that Jack can learn to type with 100 days of practice. Give your answer correct to the nearest whole number.
(b)
(c)Complete the table of t(x) values (correct to the nearest whole number) for x = 0, 10, 20, …, 70 and hence draw the graph of t(x) for 0 ≤ x ≤ 70.
(c)
(d)A simpler function that could also model the number of days needed to attain x wpm is p(x) = 1·5x. Draw, on the diagram above, the graph of p(x) for 0 ≤ x ≤ 70.
(d)
(e)Let h(x) = p(x) − t(x).
(i)Use your graphs above to estimate the solution to h(x) = 0 for x > 0.
(e)(i)
(ii)Use calculus to find the maximum value of h(x) for 0 ≤ x ≤ 70. Give your answer correct to the nearest whole number.
(e)(ii)
The population in Sapphire City over time is predicted by p(t) = Se0·1t × 106. The population in Avalon is predicted by q(t) = 3·9ekt × 106. Here t is time in years; t = 0 is the beginning of 2010; S and k are constants.
(a)The population in Sapphire City at the beginning of 2010 is 1 100 000 people. Find the value of S.
(a)
(b)Find the predicted population in Sapphire City at the beginning of 2015.
(b)
(c)Find the predicted change in the population in Sapphire City during 2015.
(c)
(d)The predicted population in Avalon at the beginning of 2011 is 3 709 795 people. Write down and solve an equation in k to show that k = −0·05, correct to 2 decimal places.
(d)
(e)Find the year during which the populations in both cities will be equal.
(e)
(f)Find the predicted average value of the population in Avalon from the beginning of 2010 to the beginning of 2025.
(f)
(g)Use the function q(t) = 3·9e−0·05t × 106 to find the predicted rate of change of the population in Avalon at the beginning of 2018.
(g)
The depth of water, in metres, at a point in a harbour varies with the tide and can be modelled by f(t) = a + b cos ct, where t is the time in hours from the first high tide on a particular Saturday and a, b, c are constants (ct in radians). On that Saturday: depth at high tide was 5·5 m; at low tide was 1·7 m; high tide occurred at 02:00 and again at 14:34.
(a)Add labelled and scaled axes to the diagram to show the graph of f over a portion of that Saturday. The point P should represent the depth at high tide on Saturday morning.
(a)
(b)(i)Find the value of a and the value of b.
(b)(i)
(ii)Show that c = 0·5, correct to 1 decimal place.
(b)(ii)
(c)Use the equation f(t) = a + b cos ct to find the times on that Saturday afternoon when the depth of the water was exactly 5·2 m. Give each answer correct to the nearest minute.
(c)
(a)(i)f(x) = 2ex and g(x) = ex − 1, where x ∈ ℝ. Complete the table of values (at x = 0, 0·5, 1, ln 4), correct to two decimal places where necessary.
(a)(i)
(ii)Use the table to draw the graphs of f(x) and g(x) in the domain 0 ≤ x ≤ ln 4. Label each graph clearly.
(a)(ii)
(iii)Use your graphs to estimate the value of x for which f(x) = g(x).
(a)(iii)
(b)Solve f(x) = g(x) using algebra.
(b)
(b)(i)Show that f(x) = 3x − 2, where x ∈ ℝ, is an injective function.
(b)(i)
(ii)Given that f(x) = 3x − 2, where x ∈ ℝ, find a formula for f−1, the inverse function of f. Show your work.
(b)(ii)
The heptathlon scoring uses formulas where x is time (s) or distance (m) and y is points. 200 m race: y = a(b − x)c with a = 4·99087, b = 42·5, c = 1·81. Javelin: y = a(x − b)c with a = 15·9803, b = 3·8, c = 1·04.
(b)(i)Jessica ran 200 m in 23·8 s and threw the javelin 58·2 m. Use the formulas to find the number of points she scored in each event, correct to the nearest point.
(b)(i)
(ii)The world record javelin distance would merit 1295 points. Find the world record distance for the javelin, correct to two decimal places.
(b)(ii)
(iii)The 800 m race uses the same formula as the 200 m race but different constants. Jessica ran 800 m in 2 minutes 1·84 seconds, meriting 1087 points. If a = 0·11193 and b = 254 for the 800 m race, find the value of c, correct to two decimal places (using logs).
(b)(iii)
The number of hours of daylight in a particular location can be modelled by a function of the form f(t) = a + b cos(ct), where t is the time in months and a, b and c are constants. In this location the longest day, on 21 June, has 16·5 hours of daylight and the shortest day, on 21 December, has 7·5 hours of daylight.
(a)Taking 21 June as t = 0, and one month as 30 days, show that a = 12 and b = 4·5.
(a)
(b)The period of the function is 12 months. Show that c = 30, if ct is expressed in degrees.
(b)
(c)Find the number of hours of daylight on 21 March (t = 3), correct to one decimal place.
(c)
(d)Find the dates in the year on which there will be 14 hours of daylight. Give each answer correct to the nearest day.
(d)
(e)Sketch the graph of the function f(t) = 12 + 4·5 cos(30t) for one full year, 0 ≤ t ≤ 12.
(e)
In 2011, a new footbridge was opened at Mizen Head. The arch of the bridge is in the shape of a parabola. The length of the span of the arch, [AB], is 48 metres.
(a)Using the co-ordinate plane, with A(0, 0) and B(48, 0), the equation of the parabola is y = −0·013x2 + 0·624x. Find the co-ordinates of C, the highest point of the arch.
(a)
(b)The perpendicular distance between the walking deck, [DE], and [AB] is 5 metres. Find the co-ordinates of D and of E. Give your answers correct to the nearest whole number.
(b)
(c)Using integration, find the area of the shaded region, ABED. Give your answer correct to the nearest whole number.
(c)
(d)Write the equation of the parabola in part (a) in the form y − k = p(x − h)2, where k, p and h are constants.
(d)
(e)Using what you learned in part (d), or otherwise, write down the equation of a parabola for which the coefficient of x2 is −2 and the co-ordinates of the maximum point are (3, 4).
(e)
Ciarán is preparing food for his baby and must use cooled boiled water. The equation y = Aekt describes how the boiled water cools, where t is time in minutes from when the water boiled, y is the difference between the water temperature and room temperature at time t (in °C), and A, k are constants. The water boils at 100°C and room temperature is a constant 23°C.
(a)Write down the value of the temperature difference y when the water boils, and find the value of A.
(a)
(b)After five minutes, the temperature of the water is 88°C. Find the value of k, correct to three significant figures.
(b)
(c)Ciarán prepares the food when the water has cooled to 50°C. How long does it take, correct to the nearest minute, for the water to cool to this temperature?
(c)
(d)Using your values for A and k, sketch the curve f(t) = Aekt for 0 ≤ t ≤ 100, t ∈ ℝ.
(d)
(e)(i)On the same diagram, sketch a curve g(t) = Aemt, showing the water cooling at a faster rate, where A is the value from part (a) and m is a constant. Label each graph clearly.
(e)(i)
(ii)Suggest one possible value for m for the sketch you have drawn and give a reason for your choice.
(e)(ii)
(f)(i)Find the rates of change of the function f(t) after 1 minute and after 10 minutes. Give your answers correct to two decimal places.
(f)(i)
(ii)Show that the rate of change of f(t) will always increase over time.
(f)(ii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.