IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May 2021	Marks available	2	Reference code	21M.1.SL.TZ1.11
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	11	Adapted from	N/A

Question

If a shark is spotted near to Brighton beach, a lifeguard will activate a siren to warn swimmers.

The sound intensity, $I$ , of the siren varies inversely with the square of the distance, $d$ , from the siren, where $d > 0$ .

It is known that at a distance of $1.5$ metres from the siren, the sound intensity is $4$ watts per square metre ( ${W m}^{- 2}$ ).

Show that $I = \frac{9}{d^{2}}$ .

[2]

a.

Sketch the curve of $I$ on the axes below showing clearly the point $(1.5, 4)$ .

[2]

b.

Whilst swimming, Scarlett can hear the siren only if the sound intensity at her location is greater than $1.5 \times 10^{- 6}$ ${W m}^{- 2}$ .

Find the values of $d$ where Scarlett cannot hear the siren.

[2]

c.

Markscheme

$I = \frac{k}{d^{2}}$ (M1)

$4 = \frac{k}{1 . 5^{2}}$ M1

$I = \frac{9}{d^{2}}$ AG

Note: The AG line must be seen for the second M1 to be awarded.
Award no marks for substituting $1.5$ and $4$ into $I = \frac{9}{d^{2}}$ (i.e., working backwards).

[2 marks]

a.

A1A1

Note: Award A1 for correct general shape (concave up) with no $I$ -intercept, passing through the marked point $(1.5, 4)$ ; the point must be labelled with either the coordinates or the values $1.5$ and $4$ on the $x$ and $y$ axes. Award A1 for the curve showing asymptotic behavior (i.e. $I$ tends to $0$ , as $d$ tends to infinity), extending to at least $d = 6$ ; the curve must not cross nor veer away from the horizontal asymptote.

[2 marks]

b.

$1.5 \times 10^{- 6} \geq \frac{9}{d^{2}}$ (M1)

Note: Award (M1) for a correct inequality.

$d \geq 2450 (m) (2449.48 \dots)$ A1

Note: Award A0 for $d = 2450$ .

[2 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology

Show 135 related questions

18M.1.AHL.TZ1.H_9c:
Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.
22M.2.SL.TZ2.5d:
Sketch the graph of $C$ against $x$ , labelling the maximum point and the $x$ -intercepts with their coordinates.
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.
18M.2.AHL.TZ2.H_10c:
Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.
19M.2.AHL.TZ2.H_4a:
Sketch the graphs $y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x$ and $y = 1 + {\text{cos}}\,x$ on the following axes for 0 < $x$ ≤ 9.
22M.2.AHL.TZ2.7e:
Use the model to calculate the total amount of time when fishing should be stopped.
22M.1.SL.TZ2.9a:
Write down one feature of this graph which suggests a cubic function might be appropriate to model this scenario.
17M.1.SL.TZ1.T_15a:
Write down the value of $c$ .
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.
17M.1.AHL.TZ1.H_6a:
Sketch the graphs on the same set of axes.
18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
17M.1.SL.TZ1.T_15c:
Write down the second $x$ -intercept of the function.
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
18M.1.AHL.TZ2.H_10b.ii:
Sketch the graph of $y = g\left( x \right)$ . State the equations of any asymptotes and the coordinates of any intercepts with the axes.
18M.2.AHL.TZ2.H_10d.ii:
Show that $\alpha$ + β < −2.
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.
22M.1.AHL.TZ2.10b:
Solve $f (x) = f^{- 1} (x)$ .
22M.2.SL.TZ2.3c:
Find the coordinates of $X$ .
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.
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.
22M.2.AHL.TZ2.6e:
Determine the two positions where the path of the arrow intersects the path of the ball.
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.
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.
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.
22M.1.SL.TZ2.12a:
Determine an equation for the axis of symmetry of the parabola that models the archway.
22M.1.SL.TZ2.12b:
Determine whether the crate will fit through the archway. Justify your answer.
21N.2.SL.TZ0.3c.i:
Write down the amplitude of the function.
21N.2.SL.TZ0.3e.i:
Find the height of $C$ above the ground when $t = 2$ .
21N.2.AHL.TZ0.2d:
Sketch the function $h (t)$ for $0 \leq t \leq 5$ , clearly labelling the coordinates of the maximum and minimum points.
21N.1.AHL.TZ0.10a:
Find $g (0)$ .
18M.1.AHL.TZ2.H_2a:
Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following axes.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
17M.1.AHL.TZ1.H_6b:
Given that the graphs enclose a region of area 18 square units, find the value of b.
18N.2.SL.TZ0.T_4b.i:
Use your graphic display calculator to find the zero of f (x).
17M.1.SL.TZ2.T_14a:
Write down the amount of money Jashanti saves per month.
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
19M.1.AHL.TZ1.H_8d:
Sketch the curve $y = f\left( x \right)$ , 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.
18M.2.AHL.TZ2.H_10a.ii:
With reference to your graph, explain why $f$ is a function on the given domain.
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
17M.1.AHL.TZ1.H_11a.ii:
Factorize ${x^2} + 3x + 2$ .
18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
18M.1.AHL.TZ2.H_10b.i:
Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
18M.2.AHL.TZ2.H_10a.iii:
Explain why $f$ has no inverse on the given domain.
17M.1.AHL.TZ1.H_11e:
Sketch the graph of $y = f\left( {\left| x \right|} \right)$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
16N.1.AHL.TZ0.H_3b:
find the value of $b$ .
21M.2.SL.TZ1.2i:
The endpoints of the minute hand and hour hand meet when $θ = k$ .

Find the smallest possible value of $k$ .
18N.2.SL.TZ0.T_4b.ii:
Use your graphic display calculator to find the coordinates of the local minimum point.
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
17M.1.AHL.TZ1.H_11d:
Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$ .
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 $\tan x = 2x$ .
19M.1.AHL.TZ1.H_8a:
Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .
17M.1.SL.TZ2.T_14c:
Calculate how much extra money Jashanti needs.
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.
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
18M.1.AHL.TZ2.H_10c:
The function $h$ is defined by $h\left( x \right) = \sqrt x$ , for $x$ ≥ 0.

State the domain and range of $h \circ g$ .
17N.1.AHL.TZ0.H_6a:
Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$ , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.
18M.1.AHL.TZ1.H_9b.i:
Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .
18M.1.AHL.TZ2.H_2b:
Solve the equation $\frac{x}{2} + 1 = \left| {x - 2} \right|$ .
18M.1.AHL.TZ1.H_9a:
The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.
17M.1.AHL.TZ1.H_11a.i:
Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
16N.1.AHL.TZ0.H_3a:
state the value of $a$ and the value of $c$ ;
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 $y = - x$ .
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
20N.2.SL.TZ0.S_1a:
Find $f (1)$ .
20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of $34.50$ represents.
20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
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.
20N.1.SL.TZ0.T_4a.ii:
Write down the coordinates of the local minimum point.
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
17M.1.AHL.TZ1.H_11c:
Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$ .
20N.2.SL.TZ0.S_1b:
Solve $f (x) = 0$ .
20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.1.SL.TZ1.T_15b:
Find the value of $a$ and of $b$ .
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
17N.1.AHL.TZ0.H_6b:
Hence or otherwise, solve the inequality $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
20N.2.SL.TZ0.S_1c:
The graph of $f$ has a local minimum at point $A$ .

Find the coordinates of $A$ .
20N.1.SL.TZ0.T_4a.i:
Write down the zero of $f (x)$ .
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.1.AHL.TZ1.H_8b:
find the value of $f\left( 1 \right)$ .
20N.1.SL.TZ0.T_4b:
Consider the function $g (x) = 3 - x$ .

Solve $f (x) = g (x)$ .
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 $k$ .
19M.2.AHL.TZ2.H_4b:
Hence solve ${\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0$ in the range 0 < $x$ ≤ 9.
18M.1.AHL.TZ2.H_10a:
Find the inverse function ${f^{ - 1}}$ , stating its domain.
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
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 $n$ .
18M.2.AHL.TZ2.H_10a.iv:
Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .
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.
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
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.
21M.1.AHL.TZ2.5a:
Estimate the value of Roger’s laptop after $5$ years.
21M.1.AHL.TZ2.5c:
Comment on the validity of your answer to part (b).
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
18M.2.AHL.TZ2.H_10b:
Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .
17M.1.AHL.TZ1.H_11f:
Determine the area of the region enclosed between the graph of $y = f\left( {\left| x \right|} \right)$ , the $x$ -axis and the lines with equations $x = - 1$ and $x = 1$ .
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.
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.
21M.1.AHL.TZ2.5b:
Find the value of $k$ .
21N.2.AHL.TZ0.2a.ii:
minimum value of $h$ .
21N.1.AHL.TZ0.10b:
On the same set of axes draw the graph of $y = g (x)$ , showing any intercepts and asymptotes.
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.
21N.2.SL.TZ0.3a.i:
maximum value of $h$ .
21N.2.SL.TZ0.3a.ii:
minimum value of $h$ .
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.
21N.2.SL.TZ0.3b.ii:
Calculate the angle, in degrees, that the blade $[BC]$ turns through in one second.
21N.2.SL.TZ0.3c.ii:
Find the period of the function.
21N.2.SL.TZ0.3d:
Sketch the function $h (t)$ for $0 \leq t \leq 5$ , clearly labelling the coordinates of the maximum and minimum points.
21N.2.SL.TZ0.3f.i:
At any given instant, find the probability that point $C$ is visible from Tim’s window.
21N.2.AHL.TZ0.2a.i:
maximum value of $h$ .
21N.2.AHL.TZ0.2b.ii:
Calculate the angle, in degrees, that the blade $[BC]$ turns through in one second.
21N.2.AHL.TZ0.2c.i:
Write down the amplitude of the function.
21N.2.AHL.TZ0.2c.ii:
Find the period of the function.
21N.2.AHL.TZ0.2e.i:
Find the height of $C$ above the ground when $t = 2$ .
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.
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.

Hide related questions

Topic 2—Functions