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Date	November 2019	Marks available	5	Reference code	19N.1.AHL.TZ0.H_11
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Hence and Show that	Question number	H_11	Adapted from	N/A

Question

Points ${\text{A}}$ (0 , 0 , 10) , ${\text{B}}$ (0 , 10 , 0) , ${\text{C}}$ (10 , 0 , 0) , ${\text{V}}$ ( $p$ , $p$ , $p$ ) form the vertices of a tetrahedron.

Consider the case where the faces ${\text{ABV}}$ and ${\text{ACV}}$ are perpendicular.

The following diagram shows the graph of $\theta$ against $p$ . The maximum point is shown by ${\text{X}}$ .

Show that $\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)$ and find a similar expression for $\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}}$ .

[3]

a.i.

Hence, show that, if the angle between the faces ${\text{ABV}}$ and ${\text{ACV}}$ is $\theta$ , then ${\text{cos}}\,\theta = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}$ .

[5]

a.ii.

Find the two possible coordinates of ${\text{V}}$ .

[3]

b.i.

Comment on the positions of ${\text{V}}$ in relation to the plane ${\text{ABC}}$ .

[1]

b.ii.

At ${\text{X}}$ , find the value of $p$ and the value of $\theta$ .

[3]

c.i.

Find the equation of the horizontal asymptote of the graph.

[2]

c.ii.

Markscheme

$\overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right)$ A1

$\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} 0 \\ {10} \\ { - 10} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {10\left( {p - 10} \right) + 10p} \\ { - 10p} \\ { - 10p} \end{array}} \right)$ A1

$= \left( {\begin{array}{*{20}{c}} {20p - 100} \\ { - 10p} \\ { - 10p} \end{array}} \right) = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)$ AG

$\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} {10} \\ 0 \\ { - 10} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {10p} \\ {100 - 20p} \\ {10p} \end{array}} \right)\left( { = 10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right)} \right)$ A1

[3 marks]

a.i.

attempt to find a scalar product M1

$- 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right) \bullet 10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right) = 100\left( {3{p^2} - 20p} \right)$

OR $- \left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right) = 3{p^2} - 20p$ A1

attempt to find magnitude of either $\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}}$ or $\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}}$ M1

$\left| { - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)} \right| = \left| {10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right)} \right| = 10\sqrt {{{\left( {10 - 2p} \right)}^2} + 2{p^2}}$ A1

$100\left( {3{p^2} - 20p} \right) = 100{\left( {\sqrt {{{\left( {10 - 2p} \right)}^2} + 2{p^2}} } \right)^2}{\text{cos}}\,\theta$

${\text{cos}}\,\theta = \frac{{3{p^2} - 20p}}{{{{\left( {10 - 2p} \right)}^2} + 2{p^2}}}$ A1

Note: Award A1 for any intermediate step leading to the correct answer.

$= \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}$ AG

Note: Do not allow FT marks from part (a)(i).

[8 marks]

a.ii.

$p\left( {3p - 20} \right) = 0 \Rightarrow p = 0$ or $p = \frac{{20}}{3}$ M1A1

coordinates are (0, 0, 0) and $\left( {\frac{{20}}{3}{\text{, }}\frac{{20}}{3}{\text{, }}\frac{{20}}{3}} \right)$ A1

Note: Do not allow column vectors for the final A mark.

[3 marks]

b.i.

two points are mirror images in the plane
or opposite sides of the plane
or equidistant from the plane
or the line connecting the two Vs is perpendicular to the plane R1

[1 mark]

b.ii.

geometrical consideration or attempt to solve $- 1 = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}$ (M1)

$p = \frac{{10}}{3}{\text{, }}\theta = \pi$ or $\theta = 180^\circ$ A1A1

[3 marks]

c.i.

$p \to \infty \Rightarrow {\text{cos}}\,\theta \to \frac{1}{2}$ M1

hence the asymptote has equation $\theta = \frac{\pi }{3}$ A1

[2 marks]

c.ii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.16—Vector product

Show 45 related questions

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
19M.1.AHL.TZ1.H_11d:
Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .
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.
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.
17M.1.AHL.TZ1.H_5a:
Find the area of the parallelogram ABCD.
19M.1.AHL.TZ2.H_2a.i:
Find the vector $\overrightarrow {{\text{AB}}}$ .
19M.1.AHL.TZ2.H_2a.ii:
Find the vector $\overrightarrow {{\text{AC}}}$ .
17M.2.AHL.TZ2.H_7:
Given that a $\times$ b $=$ b $\times$ c $\ne$ 0 prove that a $+$ c $=$ sb where s is a scalar.
19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ and $\prod_{3}$ .
21M.2.AHL.TZ1.6a:
Find a Cartesian equation of the plane $Π_{3}$ which is perpendicular to $Π_{1}$ and $Π_{2}$ and passes through the origin $(0, 0, 0)$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
20N.2.AHL.TZ0.H_10c:
Find the equation of $Π_{2}$ , giving your answer in the form $r . n = d$ .
19M.1.AHL.TZ2.H_2b:
Hence or otherwise, find the area of the triangle ABC.
19N.1.AHL.TZ0.H_11a.i:
Show that $\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)$ and find a similar expression for $\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}}$ .
20N.2.AHL.TZ0.H_10b:
Find the shortest distance from the point $O (0, 0, 0)$ to $Π_{1}$ .
16N.1.AHL.TZ0.H_4b:
Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point $(1,{\text{ }}0,{\text{ }} - 1)$ .
19N.1.SL.TZ0.S_9b:
Given that ${\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}$ , show that $k = 7$ .
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 $\vec{AC}$ .
21M.1.AHL.TZ2.5:
Given any two non-zero vectors, $a$ and $b$ , show that ${|a \times b|}^{2} = {|a|}^{2} {|b|}^{2} - {(a \cdot b)}^{2}$ .
16N.1.AHL.TZ0.H_4a:
Find a $\times$ b.
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B \hat{A} C = 60 °$ .
20N.2.AHL.TZ0.H_10a:
Given that $L$ meets $Π_{1}$ at the point $P$ , find the coordinates of $P$ .
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
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 $P$ .
19M.1.AHL.TZ1.H_11b:
Verify that ${\text{P}}$ is the point of intersection of the two lines.
20N.2.AHL.TZ0.H_10d:
Determine the acute angle between $Π_{1}$ and $Π_{2}$ .
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 $L$ .
21M.2.AHL.TZ1.6b:
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19N.1.SL.TZ0.S_9c:
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17M.1.AHL.TZ1.H_5b:
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EXN.2.AHL.TZ0.11e:
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21M.1.AHL.TZ2.8a:
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21M.1.AHL.TZ2.8b:
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