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Date	November Example questions	Marks available	3	Reference code	EXN.2.AHL.TZ0.11
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	11	Adapted from	N/A

Question

The points $A (5, - 2, 5)$ , $B (5, 4, - 1)$ , $C (- 1, - 2, - 1)$ and $D (7, - 4, - 3)$ are the vertices of a right-pyramid.

The line $L$ passes through the point $D$ and is perpendicular to $Π$ .

Find the vectors $\vec{AB}$ and $\vec{AC}$ .

[2]

a.

Use a vector method to show that $B \hat{A} C = 60 °$ .

[3]

b.

Show that the Cartesian equation of the plane $Π$ that contains the triangle $ABC$ is $- x + y + z = - 2$ .

[3]

c.

Find a vector equation of the line $L$ .

[1]

d.i.

Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .

[4]

d.ii.

Find the volume of right-pyramid $ABCD$ .

[4]

e.

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$\vec{AB} = (\begin{matrix} 0 \\ 6 \\ - 6 \end{matrix}) (= 6 (\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}))$ A1

$\vec{AC} = (\begin{matrix} - 6 \\ 0 \\ - 6 \end{matrix}) (= 6 (\begin{matrix} - 1 \\ 0 \\ - 1 \end{matrix}))$ A1

[2 marks]

a.

attempts to use $\cos B \hat{A} C = \frac{\vec{AB} \cdot \vec{AC}}{|\vec{AB}| |\vec{AC}|}$ (M1)

$= \frac{(\begin{matrix} 0 \\ 6 \\ - 6 \end{matrix}) \cdot (\begin{matrix} - 6 \\ 0 \\ - 6 \end{matrix})}{\sqrt{72} \times \sqrt{72}}$ A1

$= \frac{1}{2}$ A1

so $B \hat{A} C = 60 °$ AG

[3 marks]

b.

attempts to find a vector normal to $Π$ M1

for example, $\vec{AB} \times \vec{AC} = (\begin{matrix} - 36 \\ 36 \\ 36 \end{matrix}) (= 36 (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix}))$ leading to A1

a vector normal to $Π$ is $n = (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix})$

EITHER

substitutes $(5, - 2, - 5)$ (or $(5, 4, - 1)$ or $(- 1, - 2, - 1)$ ) into $- x + y + z = d$ and attempts to find the value of $d$

for example, $d = - 5 - 2 + 5 (= - 2)$ M1

OR

attempts to use $r \cdot n = a \cdot n$ M1

for example, $(\begin{matrix} x \\ y \\ z \end{matrix}) \cdot (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix}) = (\begin{matrix} 5 \\ - 2 \\ 5 \end{matrix}) \cdot (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix})$

THEN

leading to the Cartesian equation of $Π$ as $- x + y + z = - 2$ AG

[3 marks]

c.

$r = (\begin{matrix} 7 \\ - 4 \\ - 3 \end{matrix}) + λ (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix}) (λ \in ℝ)$ A1

[1 mark]

d.i.

substitutes $x = 7 - λ, y = - 4 + λ, z = - 3 + λ$ into $- x + y + z = - 2$ (M1)

$- (7 - λ) + (- 4 + λ) + (- 3 + λ) = - 2 (3 λ = 12)$

$λ = 4$ A1

shows a correct calculation for finding $d_{min}$ , for example, attempts to find

$|4 (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix})|$ M1

$d_{min} = 4 \sqrt{3} (= 6.93)$ A1

[4 marks]

d.ii.

let the area of triangle $ABC$ be $A$

EITHER

attempts to find $A = \frac{1}{2} |\vec{AB} \times \vec{AC}|$ , for example M1

$A = \frac{1}{2} |(\begin{matrix} - 36 \\ 36 \\ 36 \end{matrix})|$

OR

attempts to find $\frac{1}{2} |\vec{AB}| |\vec{AC}| \sin θ$ , for example M1

$A = \frac{1}{2} \times 6 \sqrt{2} \times 6 \sqrt{2} \times \frac{\sqrt{3}}{2}$ (where $\sin \frac{π}{3} = \frac{\sqrt{3}}{2}$ )

THEN

$A = 18 \sqrt{3} (= 31.2)$ A1

uses $V = \frac{1}{3} A h$ where $A$ is the area of triangle $ABC$ and $h = d_{min}$ M1

$V = \frac{1}{3} \times 18 \sqrt{3} \times 4 \sqrt{3}$

$= 72$ A1

[4 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.12—Vector definitions

Show 77 related questions

22M.2.AHL.TZ1.7b:
Given that $|a + b|$ is a minimum, find $p$ .
18M.2.AHL.TZ1.H_11c.iii:
Find the distance between the two submarines at this time.
18M.2.AHL.TZ1.H_11b.ii:
Find the value of t when submarine B passes through P.
18M.1.SL.TZ1.S_9d:
Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.
17M.1.SL.TZ1.S_8d.ii:
Hence or otherwise, find one point on ${L_2}$ which is $\sqrt 5$ units from C.
17N.1.AHL.TZ0.H_9a.i:
Find, in terms of a and b $\overrightarrow {{\text{OF}}}$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $\Pi$ .
19M.1.AHL.TZ1.H_11d:
Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .
18M.2.AHL.TZ1.H_11c.ii:
Find the value of t when the two submarines are closest together.
19N.1.SL.TZ0.S_9a.i:
$\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}}$ .
19M.1.AHL.TZ1.H_11a.i:
Find how many sets of four points can be selected which can form the vertices of a quadrilateral.
19N.2.SL.TZ0.S_2b:
Write down a direction vector for ${L_3}$ .
19M.1.AHL.TZ1.H_11c:
Write down the value of $\lambda$ corresponding to the point ${\text{A}}$ .
19M.1.AHL.TZ1.H_11a.ii:
Find how many sets of three points can be selected which can form the vertices of a triangle.
18M.2.AHL.TZ1.H_11c.i:
Find an expression for the distance between the two submarines in terms of t.
18M.1.AHL.TZ2.H_9d:
Find the vector equation of the straight line passing through M and normal to the plane $\Pi$ containing ABCD.
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
18M.2.AHL.TZ1.H_11a:
Show that the two submarines would collide at a point P and write down the coordinates of P.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
19M.1.AHL.TZ2.H_2a.i:
Find the vector $\overrightarrow {{\text{AB}}}$ .
19M.1.AHL.TZ2.H_2a.ii:
Find the vector $\overrightarrow {{\text{AC}}}$ .
19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.AHL.TZ1.7a:
Find the possible range of values for $|a + b|$ .
18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
22M.2.AHL.TZ2.11b:
Show that airplane $A$ travels at a greater speed than airplane $B$ .
18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
17M.1.SL.TZ1.S_8a.ii:
Hence, write down a vector equation for ${L_1}$ .
17N.1.AHL.TZ0.H_9b.i:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\lambda$ ;
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.1.SL.TZ2.S_2b:
perpendicular.
18N.2.SL.TZ0.S_8a.i:
Find $\overrightarrow {{\text{AB}}}$ .
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
20N.1.SL.TZ0.S_9b:
Find the value of $m$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
19N.2.SL.TZ0.S_2c:
${L_3}$ passes through the intersection of ${L_1}$ and ${L_2}$ .

Write down a vector equation for ${L_3}$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
17N.1.AHL.TZ0.H_9b.ii:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\mu$ .
19M.1.AHL.TZ2.H_2b:
Hence or otherwise, find the area of the triangle ABC.
19N.1.SL.TZ0.S_9b:
Given that ${\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}$ , show that $k = 7$ .
18M.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
19M.1.AHL.TZ1.H_11e:
Let ${\text{C}}$ be the point on ${l_1}$ with coordinates (1, 0, 1) and ${\text{D}}$ be the point on ${l_2}$ with parameter $\mu = - 2$ .

Find the area of the quadrilateral ${\text{CDBA}}$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
17N.2.SL.TZ0.S_3a:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
17M.1.SL.TZ1.S_8a.i:
Find $\overrightarrow {AB}$ .
18M.1.SL.TZ1.S_9a:
Show that $\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)$
18N.2.SL.TZ0.S_8b.i:
Find the value of $y$ .
19N.2.SL.TZ0.S_2a:
Find the point of intersection of ${L_1}$ and ${L_2}$ .
21N.2.AHL.TZ0.11a.i:
Find the vector $\vec{AB}$ and the vector $\vec{AC}$ .
21N.2.AHL.TZ0.11a.ii:
Hence find the equation of $Π_{1}$ , expressing your answer in the form $a x + b y + c z = d$ , where $a, b, c, d \in ℤ$ .
21N.2.AHL.TZ0.11d.i:
Find the reflection of the point $B$ in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11d.ii:
Hence find the vector equation of the line formed when $L$ is reflected in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11c.i:
Show that at the point $P, λ = \frac{3}{4}$ .
21N.2.AHL.TZ0.11c.ii:
Hence find the coordinates of $P$ .
18M.1.AHL.TZ2.H_9f.ii:
Find YZ.
19M.1.AHL.TZ1.H_11b:
Verify that ${\text{P}}$ is the point of intersection of the two lines.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
17N.2.SL.TZ0.S_3b:
Let $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)$ . Find ${\rm{B\hat AC}}$ .
18M.1.AHL.TZ2.H_9f.i:
Find the coordinates of X, Y and Z.
18M.1.AHL.TZ2.H_9c:
Find the area of the parallelogram ABCD.
17N.1.AHL.TZ0.H_9d:
Deduce an expression for $\overrightarrow {{\text{CD}}}$ in terms of a and b only.
20N.1.SL.TZ0.S_9a:
Express $\vec{AB}$ in terms of $m$ .
EXN.2.AHL.TZ0.11c:
Show that the Cartesian equation of the plane $Π$ that contains the triangle $ABC$ is $- x + y + z = - 2$ .
EXN.2.AHL.TZ0.11d.i:
Find a vector equation of the line $L$ .
17N.1.AHL.TZ0.H_9e:
Given that area $\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})$ , find the value of $k$ .
19M.1.SL.TZ2.S_2a:
parallel.
17N.1.AHL.TZ0.H_9c:
Show that $\mu = \frac{1}{{13}}$ , and find the value of $\lambda$ .
19N.1.SL.TZ0.S_9c:
Calculate the area of triangle ${\text{AOC}}$ .
17M.1.SL.TZ1.S_8c:
The lines ${L_1}$ and ${L_1}$ intersect at $C(9,{\text{ }}13,{\text{ }}z)$ . Find $z$ .
18M.2.AHL.TZ1.H_11b.i:
Show that submarine B travels in the same direction as originally planned.
17N.1.AHL.TZ0.H_9a.ii:
Find, in terms of a and b $\overrightarrow {{\text{AF}}}$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
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 $C$ .
17M.1.SL.TZ1.S_8b:
A second line ${L_2}$ , has equation r = $\left( {\begin{array}{*{20}{c}} 1 \\ {13} \\ { - 14} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right)$ .

Given that ${L_1}$ and ${L_2}$ are perpendicular, show that $p = 2$ .
EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid $ABCD$ .
21N.2.AHL.TZ0.11b:
The line $L$ is the intersection of $Π_{1}$ and $Π_{2}$ . Verify that the vector equation of $L$ can be written as $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

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