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Date	May 2022	Marks available	5	Reference code	22M.2.AHL.TZ2.11
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	11	Adapted from	N/A

Question

Two airplanes, $A$ and $B$ , have position vectors with respect to an origin $O$ given respectively by

$r_{A} = (\begin{matrix} 19 \\ - 1 \\ 1 \end{matrix}) + t (\begin{matrix} - 6 \\ 2 \\ 4 \end{matrix})$

$r_{B} = (\begin{matrix} 1 \\ 0 \\ 12 \end{matrix}) + t (\begin{matrix} 4 \\ 2 \\ - 2 \end{matrix})$

where $t$ represents the time in minutes and $0 \leq t \leq 2.5$ .

Entries in each column vector give the displacement east of $O$ , the displacement north of $O$ and the distance above sea level, all measured in kilometres.

The two airplanes’ lines of flight cross at point $P$ .

Find the three-figure bearing on which airplane $B$ is travelling.

[2]

Show that airplane $A$ travels at a greater speed than airplane $B$ .

[2]

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

[4]

Find the coordinates of $P$ .

[5]

d.i.

Determine the length of time between the first airplane arriving at $P$ and the second airplane arriving at $P$ .

[2]

d.ii.

Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .

[5]

Markscheme

let $ϕ$ be the required angle (bearing)

EITHER

$ϕ = 90 ° - arctan \frac{1}{2} (= arctan 2)$ (M1)

Note: Award M1 for a labelled sketch.

$\cos ϕ = \frac{(\begin{matrix} 0 \\ 1 \end{matrix}) \cdot (\begin{matrix} 4 \\ 2 \end{matrix})}{\sqrt{1} \times \sqrt{20}} (= 0.4472 \dots, = \frac{1}{\sqrt{5}})$ (M1)

$ϕ = arccos (0.4472 \dots)$

THEN

$063 °$ A1

Note: Do not accept $063.6 °$ or $63.4 °$ or $1 . 10^{c}$ .

[2 marks]

METHOD 1

let $|b_{A}|$ be the speed of $A$ and let $|b_{B}|$ be the speed of $B$

attempts to find the speed of one of $A$ or $B$ (M1)

$|b_{A}| = \sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}}$ or $|b_{B}| = \sqrt{4^{2} + 2^{2} + {(- 2)}^{2}}$

Note: Award M0 for $|b_{A}| = \sqrt{19^{2} + {(- 1)}^{2} + 1^{2}}$ and $|b_{B}| = \sqrt{1^{2} + 0^{2} + 12^{2}}$ .

$|b_{A}| = 7.48 \dots (= \sqrt{56})$ (km min^-1) and $|b_{B}| = 4.89 \dots (= \sqrt{24})$ (km min^-1) A1

$|b_{A}| > |b_{B}|$ so $A$ travels at a greater speed than $B$ AG

METHOD 2

attempts to use $speed = \frac{distance}{time}$

${speed}_{A} = \frac{|r_{A} (t_{2}) - r_{A} (t_{1})|}{t_{2} - t_{1}}$ and ${speed}_{B} = \frac{|r_{B} (t_{2}) - r_{B} (t_{1})|}{t_{2} - t_{1}}$ (M1)

for example:

${speed}_{A} = \frac{|r_{A} (1) - r_{A} (0)|}{1}$ and ${speed}_{B} = \frac{|r_{B} (1) - r_{B} (0)|}{1}$

${speed}_{A} = \frac{\sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}}}{1}$ and ${speed}_{B} = \frac{\sqrt{4^{2} + 2^{2} + 2^{2}}}{1}$

${speed}_{A} = 7.48 \dots (2 \sqrt{14})$ and ${speed}_{B} = 4.89 \dots (\sqrt{24})$ A1

${speed}_{A} > {speed}_{B}$ so $A$ travels at a greater speed than $B$ AG

[2 marks]

attempts to use the angle between two direction vectors formula (M1)

$\cos θ = \frac{(- 6) (4) + (2) (2) + (4) (- 2)}{\sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}} \sqrt{4^{2} + 2^{2} + {(- 2)}^{2}}}$ (A1)

$\cos θ = - 0.7637 \dots (= - \frac{7}{\sqrt{84}})$ or $θ = arccos (- 0.7637 \dots) (= 2.4399 \dots)$

attempts to find the acute angle $180 ° - θ$ using their value of $θ$ (M1)

$= 40.2 °$ A1

[4 marks]

for example, sets $r_{A} (t_{1}) = r_{B} (t_{2})$ and forms at least two equations (M1)

$19 - 6 t_{1} = 1 + 4 t_{2}$

$- 1 + 2 t_{1} = 2 t_{2}$

$1 + 4 t_{1} = 12 - 2 t_{2}$

Note: Award M0 for equations involving $t$ only.

EITHER

attempts to solve the system of equations for one of $t_{1}$ or $t_{2}$ (M1)

$t_{1} = 2$ or $t_{2} = \frac{3}{2}$ A1

attempts to solve the system of equations for $t_{1}$ and $t_{2}$ (M1)

$t_{1} = 2$ or $t_{2} = \frac{3}{2}$ A1

THEN

substitutes their $t_{1}$ or $t_{2}$ value into the corresponding $r_{A}$ or $r_{B}$ (M1)

$P (7, 3, 9)$ A1

Note: Accept $\vec{OP} = (\begin{matrix} 7 \\ 3 \\ 9 \end{matrix})$ . Accept $7$ km east of $O$ , $3$ km north of $O$ and $9$ km above sea level.

[5 marks]

d.i.

attempts to find the value of $t_{1} - t_{2}$ (M1)

$t_{1} - t_{2} = 2 - \frac{3}{2}$

$0.5$ minutes ( $30$ seconds) A1

[2 marks]

d.ii.

EITHER

attempts to find $r_{B} - r_{A}$ (M1)

$r_{B} - r_{A} = (\begin{matrix} - 18 \\ 1 \\ 11 \end{matrix}) + t (\begin{matrix} 10 \\ 0 \\ - 6 \end{matrix})$

attempts to find their $D (t)$ (M1)

$D (t) = \sqrt{{(10 t - 18)}^{2} + 1 + {(11 - 6 t)}^{2}}$ A1

attempts to find $r_{A} - r_{B}$ (M1)

$r_{A} - r_{B} = (\begin{matrix} 18 \\ - 1 \\ - 11 \end{matrix}) + t (\begin{matrix} - 10 \\ 0 \\ 6 \end{matrix})$

attempts to find their $D (t)$ (M1)

$D (t) = \sqrt{{(18 - 10 t)}^{2} + {(- 1)}^{2} + {(- 11 + 6 t)}^{2}}$ A1

Note: Award M0M0A0 for expressions using two different time parameters.

THEN

either attempts to find the local minimum point of $D (t)$ or attempts to find the value of $t$ such that $D' (t) = 0$ (or equivalent) (M1)

$t = 1.8088 \dots (= \frac{123}{68})$

$D (t) = 1.01459 \dots$

minimum value of $D (t)$ is $1.01 (= \frac{\sqrt{1190}}{34})$ (km) A1

Note: Award M0 for attempts at the shortest distance between two lines.

[5 marks]

Examiners report

General comment about this question: many candidates were not exposed to this setting of vectors question and were rather lost.

Part (a) Probably the least answered question on the whole paper. Many candidates left it blank, others tried using 3D vectors. Out of those who calculated the angle correctly, only a small percentage were able to provide the correct true bearing as a 3-digit figure.

Part (b) Well done by many candidates who used the direction vectors to calculate and compare the speeds. A number of candidates tried to use the average rate of change but mostly unsuccessfully.

Part (c) Most candidates used the correct vectors and the formula to obtain the obtuse angle. Then only some read the question properly to give the acute angle in degrees, as requested.

Part (d) Well done by many candidates who used two different parameters. They were able to solve and obtain two values for time, the difference in minutes and the correct point of intersection. A number of candidates only had one parameter, thus scoring no marks in part (d) (i). The frequent error in part (d)(ii) was providing incorrect units.

Part (e) Many correct answers were seen with an efficient way of setting the question and using their GDC to obtain the answer, graphically or numerically. Some gave time only instead of actually giving the minimal distance. A number of candidates tried to find the distance between two skew lines ignoring the fact that the lines intersect.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion

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22M.1.AHL.TZ2.6b:
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SPM.1.SL.TZ0.8c.ii:
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22M.2.AHL.TZ1.12c.iii:
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Write down an equation in $r$ and $h$ that shows this information.
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21M.3.AHL.TZ1.1c.i:
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21M.3.AHL.TZ1.1c.iii:
no points where the gradient is equal to zero.
21M.3.AHL.TZ1.1c.ii:
one local maximum point and one local minimum point.
21M.3.AHL.TZ1.1e.i:
exactly one $x$ -axis intercept.
20N.2.SL.TZ0.T_4f:
To prevent leaks, a sealant is applied to the interior surface of the bird bath.

Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.
19N.2.SL.TZ0.S_8c.i:
Find the coordinates of ${\text{B}}$ .
SPM.1.SL.TZ0.9d:
The region R is enclosed by the graph of $f$ , the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of $k$ .
16N.1.AHL.TZ0.H_11e:
Sketch the graph of $f$ , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.
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 $y = - x$ .
19M.2.SL.TZ2.S_5b:
Find the rate at which the population of fish is increasing at $t$ = 10.
SPM.1.SL.TZ0.9b:
Find the x-coordinate of P.
18N.1.AHL.TZ0.H_10d:
Find the area enclosed by the curve and the $x$ -axis between B and D, as shaded on the diagram.
18M.2.SL.TZ2.S_9c:
Write down the number of times that the acceleration of P is 0 m s⁻² .
EXN.1.SL.TZ0.7b:
Find the dimensions of rectangle $ORST$ that has maximum perimeter and determine the value of the maximum perimeter.
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
19N.1.SL.TZ0.S_8e:
Find the maximum volume.
19N.1.SL.TZ0.S_8d.ii:
Justify your answer.
19N.1.SL.TZ0.S_8c:
Find $\frac{{{\text{d}}V}}{{{\text{d}}x}}$ .
19M.1.SL.TZ2.T_15c:
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EXN.1.SL.TZ0.7a:
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21M.3.AHL.TZ2.1f:
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17N.1.SL.TZ0.S_6:
Let $f(x) = 15 - {x^2}$ , for $x \in \mathbb{R}$ . The following diagram shows part of the graph of $f$ and the rectangle OABC, where A is on the negative $x$ -axis, B is on the graph of $f$ , and C is on the $y$ -axis.

Find the $x$ -coordinate of A that gives the maximum area of OABC.
21M.3.AHL.TZ2.1g.ii:
a point of inflexion with zero gradient for odd values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .
21M.3.AHL.TZ2.1h:
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State the conditions on $n$ and $k$ such that the equation $x^{n} {(a - x)}^{n} = k$ has four solutions for $x$ .
SPM.1.SL.TZ0.9c:
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21M.3.AHL.TZ1.1d.i:
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20N.2.SL.TZ0.S_10c:
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EXN.1.SL.TZ0.7d:
Find the dimensions of rectangle $ORST$ that has maximum area.
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
21N.1.SL.TZ0.9a:
Find all the values of $x$ where the graph of $f$ is increasing. Justify your answer.
21N.1.SL.TZ0.9c.i:
Find the value of $x$ where the graph of $f$ has a local minimum. Justify your answer.
21N.1.SL.TZ0.9d:
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Given that $f (p) + f (t) = 4$ , find the value of $f (0)$ .
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.3.AHL.TZ2.1b.i:
Write down the coordinates of the two points of inflexion on the curve $y^{2} = x^{3} + 1$ .
22M.3.AHL.TZ2.1e:
Find the value of this $x$ -coordinate, giving your answer in the form $x = \sqrt{\frac{p \sqrt{3} + q}{r}}$ , where $p, q, r \in ℤ$ .
21N.1.SL.TZ0.9b:
Find the value of $x$ where the graph of $f$ has a local maximum.
19M.1.SL.TZ1.T_15a:
Write down an equation for the area, $A$ , of the curved surface in terms of $r$ .
17N.1.SL.TZ0.T_11a:
Find the equation of the axis of symmetry of the graph of $y = f(x)$ .
19M.1.SL.TZ2.T_15a:
Find the value of $P$ if no vases are sold.
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
17M.2.SL.TZ2.T_5c:
Find the value of $x$ that maximizes the area of the plot.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
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_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
18M.2.SL.TZ1.T_6a:
Write down the height of the cylinder.
18M.1.SL.TZ2.T_13b:
Find the value of s.
16N.1.AHL.TZ0.H_11g:
Find the value of the curvature of the graph of $f$ at the local maximum point.
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
17N.2.AHL.TZ0.H_11a.ii:
Sketch a graph of $y = f’(x)$ for $0 \leqslant x < \frac{\pi }{2}$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
SPM.1.SL.TZ0.8d.ii:
Hence, use your answer to part (d)(i) to show that the graph of $f$ has a local minimum point at $x = b$ .
17N.1.SL.TZ0.T_15d:
Find the price, $p$ , that will give Maria the highest weekly profit.
EXN.1.AHL.TZ0.11e:
Show that the maximum value of $m$ is $\frac{27}{32} \sqrt{\frac{2}{3}}$ .
20N.2.SL.TZ0.T_4d:
Using your answer to part (c), find the value of $h$ for which $V$ is a maximum.
17N.2.AHL.TZ0.H_11c:
Solve the equation $f(x) = 0$ , giving your answers in the form $\arctan k$ where $k \in \mathbb{Z}$ .
SPM.1.SL.TZ0.9a:
Show that $f'\left( x \right) = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}$ .
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 $\tan x = 2x$ .
17N.1.SL.TZ0.T_15b:
Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.
21M.3.AHL.TZ1.1e.iii:
exactly three $x$ -axis intercepts.
17N.2.AHL.TZ0.H_11a.iii:
Find the $x$ -coordinate(s) of the point(s) of inflexion of the graph of $y = f(x)$ , labelling these clearly on the graph of $y = f’(x)$ .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
18N.1.AHL.TZ0.H_10c:
Find the $x$ -coordinates of A and of C , giving your answers in the form $a + {\text{arctan}}\,b$ , where $a$ , $b \in \mathbb{R}$ .
17N.2.AHL.TZ0.H_11b.i:
Express $\sin x$ in terms of $\mu$ .
16N.1.AHL.TZ0.H_11d:
Find the $x$ -coordinate of the point of inflexion of the graph of $f$ .
20N.2.SL.TZ0.S_10d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ . The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
17N.2.AHL.TZ0.H_11b.ii:
Express $\sin 2x$ in terms of $u$ .
EXN.2.AHL.TZ0.12d:
Use the differential equation $\frac{d y}{d x} = \frac{y^{2} + 3 x y + 2 x^{2}}{x^{2}}$ to show that the points of zero gradient on the curve lie on two straight lines of the form $y = m x$ where the values of $m$ are to be determined.
19M.2.SL.TZ1.S_4b:
Find the $x$ -coordinates of the points of inflexion of the graph of $f$ .
21M.2.AHL.TZ1.12e:
Find the value of $t$ when the rate of change of the population is at its maximum.
21M.3.AHL.TZ1.1d.ii:
the $y$ -coordinate of the local minimum point is $- 2 c^{\frac{3}{2}} + 2$ .
21M.1.SL.TZ1.8c:
Given that $f'' (x) = \frac{20 \ln x - 9}{x^{6}}$ , show that $P$ is a local maximum point.
21M.3.AHL.TZ1.1e.ii:
exactly two $x$ -axis intercepts.
21M.3.AHL.TZ2.1g.i:
a local minimum point for even values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
17M.2.SL.TZ1.S_10c:
Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y = x$ . There is a point ${\text{P}}(a,{\text{ }}b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where $0.111 < a < 3.31$ .
17M.2.SL.TZ1.S_10a.ii:
Write down the value of $h$ ;
17M.2.SL.TZ1.S_10b.ii:
Hence, find the area of the region enclosed by the graphs of $h$ and ${h^{ - 1}}$ .
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
17N.1.SL.TZ0.T_11c:
Find the value of $a$ and of $b$ .
19M.1.SL.TZ1.T_15c:
Find the value of $r$ when the area of the curved surface is maximized.
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
19M.1.SL.TZ2.T_15b:
Differentiate $P\left( x \right)$ .
17N.1.SL.TZ0.T_15c:
Write down an expression for $W$ in terms of $p$ .
20N.2.SL.TZ0.T_4b:
Show that $V = \frac{2500 π h}{3} - \frac{π h^{3}}{3}$ .
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
20N.2.SL.TZ0.T_4e:
Find the maximum volume of the bird bath.
17M.2.SL.TZ2.T_5b:
Write down the area of the plot in terms of $x$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
17M.2.SL.TZ2.T_5a:
Show that the width of the plot, in metres, is given by $100 - x$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
20N.2.SL.TZ0.T_4c:
Find $\frac{d V}{d h}$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
19M.1.SL.TZ1.T_15b:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that

$A = 3{x^2} + \frac{{10368}}{x}$ .
18M.2.SL.TZ1.T_6d:
Show that the volume, V cm³ , of the new trash can is given by

$V = 110\pi {r^3}$ .
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).
18M.2.SL.TZ1.T_6f:
The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.

State whether the designer’s claim is correct. Justify your answer.
17M.2.SL.TZ2.T_5d:
Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.
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.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm³, of the bag.
18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
18N.2.SL.TZ0.T_6f:
Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .

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