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Date	November 2019	Marks available	2	Reference code	19N.1.AHL.TZ0.H_11
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	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 3.18—Intersections of lines & planes

Show 55 related questions

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
16N.1.AHL.TZ0.H_1:
Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3,{\text{ }}x - y + z = 5$ and $x + y + 2z = 6$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $\Pi$ .
18M.1.AHL.TZ1.H_10c:
Find the Cartesian equation of ${\Pi _3}$ .
19N.1.AHL.TZ0.H_11b.ii:
Comment on the positions of ${\text{V}}$ in relation to the plane ${\text{ABC}}$ .
19N.1.AHL.TZ0.H_8:
A straight line, ${L_\theta }$ , has vector equation r $= \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 0 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right){\text{, }}\lambda {\text{, }}\theta \in \mathbb{R}$ .

The plane ${\Pi _p}$ , has equation $x = p{\text{, }}p \in \mathbb{R}$ .

Show that the angle between ${L_\theta }$ and ${\Pi _p}$ is independent of both $\theta$ and $p$ .
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.
18M.1.AHL.TZ1.H_10b:
Find the angle between the faces ABD and BCD.
19M.2.AHL.TZ2.H_11d.ii:
Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.
16N.1.AHL.TZ0.H_8a:
find the value of $a$ ;
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 $Π$ .
19N.1.AHL.TZ0.H_3:
Three planes have equations:

$2x - y + z = 5$

$x + 3y - z = 4$ , where $a{\text{, }}b \in \mathbb{R}$ .

$3x - 5y + az = b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.
18N.1.AHL.TZ0.H_9b:
Find, in terms of $b$ , the equation of the line L which passes through M and is perpendicular to the plane П.
19M.2.AHL.TZ2.H_11a:
Find the Cartesian equation of the plane containing P, Q and R.
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
20N.2.AHL.TZ0.H_10c:
Find the equation of $Π_{2}$ , giving your answer in the form $r . n = d$ .
19M.2.AHL.TZ2.H_11d.i:
Show that $5a - 7c = 4$ .
19N.1.AHL.TZ0.H_11b.i:
Find the two possible coordinates of ${\text{V}}$ .
18M.1.AHL.TZ1.H_10a:
Find the Cartesian equation of the plane ${\Pi _1}$ , passing through the points A , B and D.
16N.1.AHL.TZ0.H_8b:
determine the coordinates of the point of intersection P.
20N.2.AHL.TZ0.H_10b:
Find the shortest distance from the point $O (0, 0, 0)$ to $Π_{1}$ .
18M.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
18M.1.AHL.TZ1.H_10d:
Show that P is the midpoint of AD.
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B \hat{A} C = 60 °$ .
19M.2.AHL.TZ2.H_11b:
Given that П₁ and П₂ meet in a line $L$ , verify that the vector equation of $L$ can be given by r $= \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)$ .
20N.2.AHL.TZ0.H_10a:
Given that $L$ meets $Π_{1}$ at the point $P$ , find the coordinates of $P$ .
17N.1.AHL.TZ0.H_2b:
Find the coordinates of the point of intersection of the line L with the plane Π.
16N.2.AHL.TZ0.H_2:
Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$ .
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 $P$ .
18M.1.AHL.TZ2.H_9f.ii:
Find YZ.
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.
18M.1.AHL.TZ1.H_10e:
Find the area of the triangle OPQ.
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 $L$ .
19N.1.AHL.TZ0.H_11c.i:
At ${\text{X}}$ , find the value of $p$ and the value of $\theta$ .
17N.1.AHL.TZ0.H_2a:
Find a vector equation of the line L passing through the points A and B.
21M.2.AHL.TZ1.6b:
Find the coordinates of the point where $Π_{1}$ , $Π_{2}$ and $Π_{3}$ intersect.
18N.1.AHL.TZ0.H_9c:
Show that L does not intersect the $y$ -axis for any negative value of $b$ .
19M.2.AHL.TZ2.H_11c:
Given that П₃ is parallel to the line $L$ , show that $a + 2b - 5c = 0$ .
18N.1.AHL.TZ0.H_9a:
Find, in terms of $b$ , a Cartesian equation of the plane Π containing this triangle.
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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