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Date	May 2022	Marks available	2	Reference code	22M.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

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$ , which have a magnitude of at least $M$ . For a particular region the equation is

$\log_{10} N = a - M$ , for some $a \in ℝ$ .

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$ .

The equation for this region can also be written as $N = \frac{b}{10^{M}}$ .

The expected length of time, in years, between earthquakes with a magnitude of at least $M$ is $\frac{1}{N}$ .

Within this region the most severe earthquake recorded had a magnitude of $7.2$ .

Find the value of $a$ .

[2]

a.

Find the value of $b$ .

[2]

b.

Given $0 < M < 8$ , find the range for $N$ .

[2]

c.

Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year.

[2]

d.

Markscheme

$\log_{10} 100 = a - 3$ (M1)

$a = 5$ A1

[2 marks]

a.

EITHER

$N = 10^{5 - M}$ (M1)

$= \frac{10^{5}}{10^{M}} (= \frac{100000}{10^{M}})$

OR

$100 = \frac{b}{10^{3}}$ (M1)

THEN

$b = 100000 (= 10^{5})$ A1

[2 marks]

b.

$0.001 < N < 100000 (10^{- 3} < N < 10^{5})$ A1A1

Note: Award A1 for correct endpoints and A1 for correct inequalities/interval notation.

[2 marks]

c.

$N = \frac{10^{5}}{10^{7.2}} (= 0.0063095 \dots)$ (M1)

length of time $= \frac{1}{0.0063095 \dots} = 10^{2.2}$

$= 158$ years A1

[2 marks]

d.

Examiners report

Many candidates did not attempt this question. Of those who did attempt the question, most of these candidates arrived at the correct answer to this part with the most common incorrect answer being 103.

a.

Those that were successful in part (a) answered this well.

b.

This was only answered correctly by the strongest candidates.

c.

This part of the question was a discriminator as correct responses were few and far between.

d.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection

Show 123 related questions

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22M.2.SL.TZ2.4c.ii:
an appropriate range for $h (t)$ .
22M.2.SL.TZ2.4c.i:
an appropriate domain for $h (t)$ .
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.
17M.1.AHL.TZ1.H_6a:
Sketch the graphs on the same set of axes.
18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
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 $f (x) = f^{- 1} (x)$ .
EXN.2.SL.TZ0.3b.ii:
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22M.1.SL.TZ1.3c:
State an appropriate domain for $t$ in this model.
21N.1.SL.TZ0.5a:
Find the range of $h$ .
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.
18N.1.AHL.TZ0.H_3b:
Write down the least value of $a$ such that $g$ has an inverse.
18N.1.AHL.TZ0.H_3a:
For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.
17M.1.AHL.TZ1.H_6b:
Given that the graphs enclose a region of area 18 square units, find the value of b.
19M.2.SL.TZ1.S_9b:
Find u.
19M.2.SL.TZ2.T_5f:
Find the equation of the tangent line to the graph of $y = f\left( x \right)$ at $x = 2$ . Give the equation in the form $ax + by + d = 0$ where, $a$ , $b$ , and $d \in \mathbb{Z}$ .
18M.2.SL.TZ1.T_4a:
Find the value of k.
17M.1.SL.TZ2.T_14a:
Write down the amount of money Jashanti saves per month.
18N.1.AHL.TZ0.H_3c.i:
For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
18M.2.SL.TZ2.T_6f:
Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
19M.2.SL.TZ1.S_9d.iii:
Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x = 8$ and the tangent line to $f$ at $x = 2$ .
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
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.TZ2.H_2c:
Write down the domain and range of ${f^{ - 1}}$ .
16N.2.SL.TZ0.S_1b:
Find $(g \circ f)(x)$ .
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 $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
18M.2.SL.TZ2.T_6b:
A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
18M.2.SL.TZ1.T_4b:
Using your value of k , find f ′(x).
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.
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
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.
17M.1.AHL.TZ2.H_2a:
Write down the range of $f$ .
16N.1.AHL.TZ0.H_3b:
find the value of $b$ .
18M.1.SL.TZ1.S_3c:
On the grid above, sketch the graph of f ⁻¹.
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
16N.1.SL.TZ0.T_9b:
Find the value of $b$ .
17M.2.SL.TZ2.S_6b:
On the following grid, sketch the graph of $(f \circ g)(x)$ , for $0 \leqslant x \leqslant 2.25$ .
17M.1.AHL.TZ2.H_2b:
Find an expression for ${f^{ - 1}}(x)$ .
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 ${(x + h)^2} + k$ .
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.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.
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.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
17M.1.SL.TZ1.T_12a:
Write down the domain of the function.
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.S_1a:
Find $f(8)$ .
17M.2.SL.TZ2.S_6c:
The equation $(f \circ g)(x) = k$ has exactly two solutions, for $0 \leqslant x \leqslant 2.25$ . Find the possible values of $k$ .
EXN.2.SL.TZ0.3c:
Find the range of $h$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
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 $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$ .
21M.1.SL.TZ1.4a:
Find the total cost of buying $40$ litres of gas at Leon’s gas station.
18M.2.SL.TZ1.T_4c:
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
21M.1.SL.TZ1.4b:
Find $L^{- 1} (70)$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.T_12b.i:
Draw the line $y = - 6$ on the axes.
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 $f\left( 1 \right)$ .
19N.1.SL.TZ0.T_10b:
While in Kota Kinabalu, Criselda spent $440 MYR$ . She returned to the Currency Exchange counter and changed the remainder of her $MYR$ into $USD$ .

Calculate the amount of $USD$ she received.
17M.2.SL.TZ2.S_6a:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
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.
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
16N.2.SL.TZ0.S_1c:
Solve $(g \circ f)(x) = 0$ .
18M.1.AHL.TZ2.H_10a:
Find the inverse function ${f^{ - 1}}$ , stating its domain.
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19N.1.SL.TZ0.T_10a:
Calculate the amount of $MYR$ that Criselda received.
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
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.
17M.1.SL.TZ1.T_12c:
Find the range of values of $k$ for which $f(x) = k$ has no solution.
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.
17M.1.SL.TZ1.T_12b.ii:
Write down the number of solutions to $f(x) = - 6$ .
21M.1.SL.TZ2.7b:
Find the value of $f^{- 1} (0)$ .
21M.1.SL.TZ2.7a:
Find the range of $f$ .
21M.1.AHL.TZ2.2c:
Write down the range of $f^{- 1} (x)$ .
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
21M.1.AHL.TZ2.2b:
Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.
21M.1.AHL.TZ2.2a:
Find the range of $f$ .
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.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
EXN.2.SL.TZ0.3b.i:
Find the value of $t$ at which the ball hits the ground.
EXN.2.SL.TZ0.3a:
Find an expression for the height $h$ of the ball at time $t$ .
21N.1.SL.TZ0.5b.i:
Find $h^{- 1} (10)$ .
21N.1.SL.TZ0.5b.ii:
In the context of the question, interpret your answer to part (b)(i).
21N.1.SL.TZ0.5b.iii:
Write down the range of $h^{- 1}$ .

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