Date | May 2021 | Marks available | 2 | Reference code | 21M.1.AHL.TZ2.5 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 2 |
Command term | Estimate | Question number | 5 | Adapted from | N/A |
Question
Roger buys a new laptop for himself at a cost of £495. At the same time, he buys his daughter Chloe a higher specification laptop at a cost of £2200.
It is anticipated that Roger’s laptop will depreciate at a rate of 10% per year, whereas Chloe’s laptop will depreciate at a rate of 15% per year.
Roger and Chloe’s laptops will have the same value k years after they were purchased.
Estimate the value of Roger’s laptop after 5 years.
Find the value of k.
Comment on the validity of your answer to part (b).
Markscheme
£495×0.95=£292 (£292.292…) (M1)A1
[2 marks]
£495×0.9k=2200×0.85k (M1)
k=26.1 (26.0968…) A1
Note: Award M1A0 for k-1 in place of k.
[2 marks]
depreciation rates unlikely to be constant (especially over a long time period) R1
Note: Accept reasonable answers based on the magnitude of k or the fact that “value” depends on factors other than time.
[1 mark]
Examiners report
Syllabus sections
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18M.1.AHL.TZ1.H_9c:
Sketch the graph of y=f(x) showing clearly the position of the points A and B.
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22M.2.SL.TZ2.5d:
Sketch the graph of C against x, labelling the maximum point and the x-intercepts with their coordinates.
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22M.2.AHL.TZ2.6f:
Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.
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18M.2.AHL.TZ2.H_10c:
Sketch the graph of y=g(t) for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.
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19M.2.AHL.TZ2.H_4a:
Sketch the graphs y=sin3x+lnx and y=1+cosx on the following axes for 0 < x ≤ 9.
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22M.2.AHL.TZ2.7e:
Use the model to calculate the total amount of time when fishing should be stopped.
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22M.1.SL.TZ2.9a:
Write down one feature of this graph which suggests a cubic function might be appropriate to model this scenario.
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17M.1.SL.TZ1.T_15a:
Write down the value of c.
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22M.2.SL.TZ1.1d:
For the first year of the model, find the length of time when there are more than 10 hours and 30 minutes of daylight per day.
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17M.1.AHL.TZ1.H_6a:
Sketch the graphs on the same set of axes.
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18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
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17M.1.SL.TZ1.T_15c:
Write down the second x-intercept of the function.
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17N.2.AHL.TZ0.H_10d:
This region is now rotated through 2π radians about the x-axis. Find the volume of revolution.
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18M.1.AHL.TZ2.H_10b.ii:
Sketch the graph of y=g(x). State the equations of any asymptotes and the coordinates of any intercepts with the axes.
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18M.2.AHL.TZ2.H_10d.ii:
Show that α + β < −2.
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17M.1.SL.TZ2.T_14b:
Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.
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22M.1.AHL.TZ2.10b:
Solve f(x)=f-1(x).
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22M.2.SL.TZ2.3c:
Find the coordinates of X.
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22M.2.SL.TZ2.4d:
By considering the graph of h(t), determine the length of time during one revolution of the Ferris wheel for which the chair is higher than the cowboy statue.
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22M.2.SL.TZ2.5e:
Determine the maximum amount of coffee the cafe can make that will not result in a loss of money for the morning.
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22M.2.AHL.TZ2.6e:
Determine the two positions where the path of the arrow intersects the path of the ball.
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18N.2.SL.TZ0.T_4b.iii:
Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).
Give your answer in the form y = mx + c.
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21N.2.SL.TZ0.3f.ii:
The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.
At any given instant, find the probability that Tim can see point C from his window. Justify your answer.
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21N.2.SL.TZ0.3e.ii:
Find the time, in seconds, that point C is above a height of 100 m, during each complete rotation.
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22M.1.SL.TZ2.12a:
Determine an equation for the axis of symmetry of the parabola that models the archway.
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22M.1.SL.TZ2.12b:
Determine whether the crate will fit through the archway. Justify your answer.
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21N.2.SL.TZ0.3c.i:
Write down the amplitude of the function.
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21N.2.SL.TZ0.3e.i:
Find the height of C above the ground when t=2.
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21N.2.AHL.TZ0.2d:
Sketch the function h(t) for 0≤t≤5, clearly labelling the coordinates of the maximum and minimum points.
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21N.1.AHL.TZ0.10a:
Find g(0).
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18M.1.AHL.TZ2.H_2a:
Sketch the graphs of y=x2+1 and y=|x−2| on the following axes.
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16N.2.SL.TZ0.T_6b:
Express this volume in cm3.
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17M.1.AHL.TZ1.H_6b:
Given that the graphs enclose a region of area 18 square units, find the value of b.
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18N.2.SL.TZ0.T_4b.i:
Use your graphic display calculator to find the zero of f (x).
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17M.1.SL.TZ2.T_14a:
Write down the amount of money Jashanti saves per month.
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17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation f(x)=5.
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19M.2.SL.TZ1.T_4b.ii:
State the domain of P.
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19M.1.AHL.TZ1.H_8d:
Sketch the curve y=f(x), 0 ≤ x ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.
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18M.2.AHL.TZ2.H_10a.ii:
With reference to your graph, explain why f is a function on the given domain.
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17M.2.AHL.TZ1.H_12e:
Find the inverse function g−1 and state its domain.
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17M.1.AHL.TZ1.H_11a.ii:
Factorize x2+3x+2.
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18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of y=f(x) for −5π8⩽x⩽π8.
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17M.2.SL.TZ2.T_6d.i:
Write down the x-coordinates of these two points;
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17M.2.SL.TZ2.T_6a:
Write down the y-intercept of the graph.
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17M.2.SL.TZ2.T_6c.ii:
Find f(2).
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18M.1.AHL.TZ2.H_10b.i:
Express g(x) in the form A+Bx−2 where A, B are constants.
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18M.2.AHL.TZ2.H_10a.iii:
Explain why f has no inverse on the given domain.
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17M.1.AHL.TZ1.H_11e:
Sketch the graph of y=f(|x|).
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17M.2.AHL.TZ1.H_12c:
Explain why f is an even function.
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16N.1.AHL.TZ0.H_3b:
find the value of b.
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21M.2.SL.TZ1.2i:
The endpoints of the minute hand and hour hand meet when θ=k.
Find the smallest possible value of k.
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18N.2.SL.TZ0.T_4b.ii:
Use your graphic display calculator to find the coordinates of the local minimum point.
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16N.2.AHL.TZ0.H_5b:
State the range of f.
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17M.2.SL.TZ2.T_6c.i:
Show that a=8.
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17M.1.AHL.TZ1.H_11d:
Hence find the value of p if ∫10f(x)dx=ln(p).
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17N.2.AHL.TZ0.H_10a.i:
Show that the x-coordinate of the minimum point on the curve y=f(x) satisfies the equation tanx=2x.
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19M.1.AHL.TZ1.H_8a:
Write down the x-coordinate of the point of inflexion on the graph of y=f(x).
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17M.1.SL.TZ2.T_14c:
Calculate how much extra money Jashanti needs.
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19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.
State whether the straight road will ever cross the river. Justify your answer.
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17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to (g−1)′(x)=0.
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18M.1.AHL.TZ2.H_10c:
The function h is defined by h(x)=√x, for x ≥ 0.
State the domain and range of h∘g.
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17N.1.AHL.TZ0.H_6a:
Sketch the graph of y=1−3xx−2, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.
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18M.1.AHL.TZ1.H_9b.i:
Show that there is exactly one point of inflexion, B, on the graph of y=f(x).
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18M.1.AHL.TZ2.H_2b:
Solve the equation x2+1=|x−2|.
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18M.1.AHL.TZ1.H_9a:
The graph of y=f(x) has a local maximum at A. Find the coordinates of A.
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17M.1.AHL.TZ1.H_11a.i:
Express x2+3x+2 in the form (x+h)2+k.
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17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of y=f(x) is positive.
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16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of r which minimizes A.
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16N.1.AHL.TZ0.H_3a:
state the value of a and the value of c;
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17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of f where the normal to the graph is parallel to the line y=−x.
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17N.2.AHL.TZ0.H_10b:
Sketch the graph of y=f(x) showing clearly the minimum point and any asymptotic behaviour.
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20N.2.SL.TZ0.S_1a:
Find f(1).
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20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of 34.50 represents.
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20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of 8.50 represents.
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20N.1.SL.TZ0.T_2c:
Kaelani has 450 PGK.
Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
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20N.1.SL.TZ0.T_4a.ii:
Write down the coordinates of the local minimum point.
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17N.2.AHL.TZ0.H_10a.ii:
Determine the values of x for which f(x) is a decreasing function.
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17M.1.AHL.TZ1.H_11c:
Show that 1x+1−1x+2=1x2+3x+2.
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20N.2.SL.TZ0.S_1b:
Solve f(x)=0.
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20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
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17M.2.SL.TZ2.T_6g:
The equation f(x)=m, where m∈R, has four solutions. Find the possible values of m.
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16N.2.AHL.TZ0.H_5c:
Solve the inequality |3xarccos(x)|>1.
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17M.2.AHL.TZ1.H_12f:
Find g′(x).
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16N.2.SL.TZ0.T_6e:
Find dAdr.
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17M.2.SL.TZ2.T_6b:
Find f′(x).
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17M.1.SL.TZ1.T_15b:
Find the value of a and of b.
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18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B(2a,b×2−3a) where a, b∈Q. Find the value of a and the value of b.
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17M.2.AHL.TZ1.H_12d:
Explain why the inverse function f−1 does not exist.
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17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to g′(x)=0;
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17N.1.AHL.TZ0.H_6b:
Hence or otherwise, solve the inequality |1−3xx−2|<2.
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18M.2.AHL.TZ2.H_10d.i:
Find α and β in terms of k.
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17M.2.SL.TZ2.T_6e:
Write down the range of f(x).
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20N.2.SL.TZ0.S_1c:
The graph of f has a local minimum at point A.
Find the coordinates of A.
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20N.1.SL.TZ0.T_4a.i:
Write down the zero of f(x).
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19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
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19M.1.AHL.TZ1.H_8b:
find the value of f(1).
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20N.1.SL.TZ0.T_4b:
Consider the function g(x)=3-x.
Solve f(x)=g(x).
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21M.1.SL.TZ1.4c:
The price of gas at Erica’s gas station is $1.30 per litre. A customer must buy a minimum of 10 litres of gas. The total cost at Erica’s gas station is cheaper than Leon’s gas station when x>k.
Find the minimum value of k.
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19M.2.AHL.TZ2.H_4b:
Hence solve sin3x+lnx−cosx−1<0 in the range 0 < x ≤ 9.
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18M.1.AHL.TZ2.H_10a:
Find the inverse function f−1, stating its domain.
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16N.2.SL.TZ0.T_6c:
Write down, in terms of r and h, an equation for the volume of this water container.
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16N.2.SL.TZ0.T_6a:
Write down a formula for A, the surface area to be coated.
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16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
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21M.1.SL.TZ1.8b:
On day n of the fitness programmes Daniella runs more than Charlie for the first time.
Find the value of n.
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18M.2.AHL.TZ2.H_10a.iv:
Explain why f is not a function for −3π4⩽x⩽π4.
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16N.2.AHL.TZ0.H_5a:
Sketch the graph of f indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
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16N.2.SL.TZ0.T_6d:
Show that A=πr2+1000000r.
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17M.2.AHL.TZ1.H_12b:
Sketch the graph of y=f(x) showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
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21M.1.AHL.TZ2.5c:
Comment on the validity of your answer to part (b).
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17M.2.AHL.TZ1.H_12a:
Find the largest possible domain D for f to be a function.
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16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
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19M.1.AHL.TZ1.H_8c:
find the value of f(4).
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18M.2.AHL.TZ2.H_10b:
Show that g(t)=(1+t1−t)2.
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17M.1.AHL.TZ1.H_11f:
Determine the area of the region enclosed between the graph of y=f(|x|), the x-axis and the lines with equations x=−1 and x=1.
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17M.1.AHL.TZ1.H_11b:
Sketch the graph of f(x), indicating on it the equations of the asymptotes, the coordinates of the y-intercept and the local maximum.
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21M.2.SL.TZ2.3c:
Determine the first year in which this model predicts the average number of visitors per concert will exceed the total seating capacity of the concert hall.
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21M.1.SL.TZ1.11c:
Whilst swimming, Scarlett can hear the siren only if the sound intensity at her location is greater than 1.5×10-6 W m-2.
Find the values of d where Scarlett cannot hear the siren.
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21M.1.AHL.TZ2.5b:
Find the value of k.
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21N.2.AHL.TZ0.2a.ii:
minimum value of h.
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21N.1.AHL.TZ0.10b:
On the same set of axes draw the graph of y=g(x), showing any intercepts and asymptotes.
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21N.2.AHL.TZ0.2b.i:
Find the time, in seconds, it takes for the blade [BC] to make one complete rotation under these conditions.
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21N.2.SL.TZ0.3a.i:
maximum value of h.
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21N.2.SL.TZ0.3a.ii:
minimum value of h.
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21N.2.SL.TZ0.3b.i:
Find the time, in seconds, it takes for the blade [BC] to make one complete rotation under these conditions.
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21N.2.SL.TZ0.3b.ii:
Calculate the angle, in degrees, that the blade [BC] turns through in one second.
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21N.2.SL.TZ0.3c.ii:
Find the period of the function.
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21N.2.SL.TZ0.3d:
Sketch the function h(t) for 0≤t≤5, clearly labelling the coordinates of the maximum and minimum points.
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21N.2.SL.TZ0.3f.i:
At any given instant, find the probability that point C is visible from Tim’s window.
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21N.2.AHL.TZ0.2a.i:
maximum value of h.
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21N.2.AHL.TZ0.2b.ii:
Calculate the angle, in degrees, that the blade [BC] turns through in one second.
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21N.2.AHL.TZ0.2c.i:
Write down the amplitude of the function.
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21N.2.AHL.TZ0.2c.ii:
Find the period of the function.
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21N.2.AHL.TZ0.2e.i:
Find the height of C above the ground when t=2.
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21N.2.AHL.TZ0.2e.ii:
Find the time, in seconds, that point C is above a height of 100 m, during each complete rotation.
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21N.2.AHL.TZ0.2f:
The wind speed increases and the blades rotate faster, but still at a constant rate.
Given that point C is now higher than 110 m for 1 second during each complete rotation, find the time for one complete rotation.