IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May Specimen paper	Marks available	1	Reference code	SPM.2.AHL.TZ0.4
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	4	Adapted from	N/A

Question

An aircraft’s position is given by the coordinates ( $x$ , $y$ , $z$ ), where $x$ and $y$ are the aircraft’s displacement east and north of an airport, and $z$ is the height of the aircraft above the ground. All displacements are given in kilometres.

The velocity of the aircraft is given as $\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,{\text{km}}\,{{\text{h}}^{ - 1}}$ .

At 13:00 it is detected at a position 30 km east and 10 km north of the airport, and at a height of 5 km. Let $t$ be the length of time in hours from 13:00.

If the aircraft continued to fly with the velocity given

When the aircraft is 4 km above the ground it continues to fly on the same bearing but adjusts the angle of its descent so that it will land at the point (0, 0, 0).

Write down a vector equation for the displacement, r of the aircraft in terms of $t$ .

[2]

a.

verify that it would pass directly over the airport.

[2]

b.i.

state the height of the aircraft at this point.

[1]

b.ii.

find the time at which it would fly directly over the airport.

[1]

b.iii.

Find the time at which the aircraft is 4 km above the ground.

[2]

c.i.

Find the direct distance of the aircraft from the airport at this point.

[3]

c.ii.

Given that the velocity of the aircraft, after the adjustment of the angle of descent, is $\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right){\text{km}}\,{{\text{h}}^{ - 1}}$ , find the value of $a$ .

[3]

d.

Markscheme

r $= \left( {\begin{array}{*{20}{c}} {30} \\ {10} \\ 5 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,$ A1A1

[2 marks]

a.

when $x = 0$ , $t = \frac{{30}}{{150}} = 0.2$ M1

EITHER

when $y = 0$ , $t = \frac{{10}}{{150}} = 0.2$ A1

since the two values of $t$ are equal the aircraft passes directly over the airport

OR

$t = 0.2$ , $y = 0$ A1

[2 marks]

b.i.

height = 5 − 0.2 × 20 = 1 km A1

[1 mark]

b.ii.

time 13:12 A1

[1 mark]

b.iii.

$5 - 20t = 4 \Rightarrow t = \frac{1}{{20}}$ (3 minutes) (M1)

time 13:03 A1

[2 marks]

c.i.

displacement is $\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)$ A1

distance is $\sqrt {{{22.5}^2} + {{7.5}^2} + {4^2}}$ (M1)

= 24.1 km A1

[3 marks]

c.ii.

METHOD 1

time until landing is $12 - 3 = 9$ minutes M1

height to descend = $4\,{\text{km}}$

$a = \frac{{ - 4}}{{\frac{9}{{60}}}}$ M1

$= - 26.7$ A1

METHOD 2

$\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right) = s\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)$ M1

$- 150 = 22.5\,s \Rightarrow s = - \frac{{20}}{3}$ M1

$a = - \frac{{20}}{3} \times 4$

$= - 26.7$ A1

[3 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.11—Vector equation of a line in 2d and 3d

Show 51 related questions

22M.1.AHL.TZ2.15a:
Find the vectors $a$ and $b$ in terms of $m$ and/or $c$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
22M.1.AHL.TZ1.13b:
Find the time at which the bearing of the ship from the harbour is $045 ˚$ .
21N.1.AHL.TZ0.16a:
Find the position vector $\vec{OS}$ of the ship at time $t$ hours.
22M.1.AHL.TZ1.13a:
Write down an expression for the position vector $r$ of the ship, $t$ hours after $1 : 00 pm$ .
18M.2.AHL.TZ1.H_11c.iii:
Find the distance between the two submarines at this time.
19M.2.SL.TZ1.S_9b:
Find u.
18M.1.SL.TZ1.S_9d:
Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.
16N.1.SL.TZ0.S_4a:
Find a vector equation of the line that passes through P and Q.
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 $x = 2$ .
16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .
18M.2.AHL.TZ1.H_11c.ii:
Find the value of t when the two submarines are closest together.
16N.1.AHL.TZ0.H_8b:
determine the coordinates of the point of intersection P.
SPM.2.AHL.TZ0.4c.ii:
Find the direct distance of the aircraft from the airport at this point.
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
SPM.2.AHL.TZ0.4b.ii:
state the height of the aircraft at this point.
SPM.2.AHL.TZ0.4c.i:
Find the time at which the aircraft is 4 km above the ground.
17N.1.SL.TZ0.S_9a.i:
Show that $\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)$ .
18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
19M.1.SL.TZ1.S_2b:
A second line, ${L_2}$ , is parallel to ${L_1}$ and passes through (1, 2, 3).

Write down a vector equation for ${L_2}$ .
SPM.2.AHL.TZ0.4b.i:
verify that it would pass directly over the airport.
17N.1.SL.TZ0.S_9b:
Find the value of $p$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
20N.1.SL.TZ0.S_9b:
Find the value of $m$ .
20N.1.SL.TZ0.S_9c:
Consider a unit vector $u$ , such that $u = p i - \frac{2}{3} j + \frac{1}{3} k$ , where $p > 0$ .

Point $C$ is such that $\vec{BC} = 9 u$ .

Find the coordinates of $C$ .
18N.2.SL.TZ0.S_8b.i:
Find the value of $y$ .
SPM.2.AHL.TZ0.4d:
Given that the velocity of the aircraft, after the adjustment of the angle of descent, is $\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right){\text{km}}\,{{\text{h}}^{ - 1}}$ , find the value of $a$ .
18M.1.SL.TZ2.S_1b:
The vector $\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)$ is perpendicular to $\mathop {{\text{AB}}}\limits^ \to$ . Find the value of p.
18M.2.AHL.TZ1.H_11b.ii:
Find the value of t when submarine B passes through P.
17N.1.SL.TZ0.S_9c:
The point D has coordinates $({q^2},{\text{ }}0,{\text{ }}q)$ . Given that $\overrightarrow {{\text{DC}}}$ is perpendicular to $L$ , find the possible values of $q$ .
18M.2.AHL.TZ1.H_11a:
Show that the two submarines would collide at a point P and write down the coordinates of P.
17N.1.SL.TZ0.S_9a.ii:
Find a vector equation for $L$ .
20N.1.SL.TZ0.S_9a:
Express $\vec{AB}$ in terms of $m$ .
SPM.2.AHL.TZ0.4a:
Write down a vector equation for the displacement, r of the aircraft in terms of $t$ .
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
18M.1.SL.TZ1.S_9a:
Show that $\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)$
16N.1.AHL.TZ0.H_8a:
find the value of $a$ ;
19M.2.SL.TZ2.S_7:
The vector equation of line $L$ is given by r $= \left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 4 \\ 5 \\ { - 1} \end{array}} \right)$ .

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.
18M.1.SL.TZ2.S_1a:
Find a vector equation for L₁.
19M.1.SL.TZ1.S_2a:
Find $c$ .
21M.1.AHL.TZ2.8a:
Find the possible value(s) for $p$ .
21M.1.AHL.TZ2.8b:
In the case that $p < 0$ , determine whether the lines intersect.
21N.1.AHL.TZ0.16b:
Find the value of $t$ when the ship will be closest to the lighthouse.
21N.1.AHL.TZ0.16c:
An alarm will sound if the ship travels within $20$ kilometres of the lighthouse.

State whether the alarm will sound. Give a reason for your answer.

Hide related questions

Topic 3—Geometry and trigonometry