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Date	May 2018	Marks available	5	Reference code	18M.1.AHL.TZ1.H_10
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	H_10	Adapted from	N/A

Question

The following figure shows a square based pyramid with vertices at O(0, 0, 0), A(1, 0, 0), B(1, 1, 0), C(0, 1, 0) and D(0, 0, 1).

The Cartesian equation of the plane $Π_{2}$ ${\Pi _2}$ , passing through the points B , C and D , is $y + z = 1$ $y + z = 1$ .

The plane $Π_{3}$ ${\Pi _3}$ passes through O and is normal to the line BD.

$Π_{3}$ ${\Pi _3}$ cuts AD and BD at the points P and Q respectively.

Find the Cartesian equation of the plane $Π_{1}$ ${\Pi _1}$ , passing through the points A , B and D.

[3]

Find the angle between the faces ABD and BCD.

[4]

Find the Cartesian equation of $Π_{3}$ ${\Pi _3}$ .

[3]

Show that P is the midpoint of AD.

[4]

Find the area of the triangle OPQ.

[5]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

recognising normal to plane or attempting to find cross product of two vectors lying in the plane (M1)

for example, $\to AB \times \to AD = ⎛ ⎝ \begin{matrix} 010 \end{matrix} ⎞ ⎠ \times ⎛ ⎝ \begin{matrix} - 1 01 \end{matrix} ⎞ ⎠ = ⎛ ⎝ \begin{matrix} 101 \end{matrix} ⎞ ⎠$ $\mathop {{\text{AB}}}\limits^ \to \,\, \times \mathop {{\text{AD}}}\limits^ \to = \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) \times \left( \begin{gathered} - 1 \hfill \\ \,0 \hfill \\ \,1 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right)$ (A1)

$Π_{1} : x + z = 1$ ${\Pi _1}\,{\text{:}}\,\,x + z = 1$ A1

[3 marks]

EITHER

$⎛ ⎝ \begin{matrix} 101 \end{matrix} ⎞ ⎠ ∙ ⎛ ⎝ \begin{matrix} 011 \end{matrix} ⎞ ⎠ = 1 = \sqrt{2} \sqrt{2} cos θ$ $\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \bullet \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{gathered} \right) = 1 = \sqrt 2 \sqrt 2 \,{\text{cos}}\,\theta$ M1A1

$∣ ∣ ∣ ∣ ⎛ ⎝ \begin{matrix} 101 \end{matrix} ⎞ ⎠ \times ⎛ ⎝ \begin{matrix} 011 \end{matrix} ⎞ ⎠ ∣ ∣ ∣ ∣ = \sqrt{3} = \sqrt{2} \sqrt{2} sin θ$ $\left| {\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \times \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{gathered} \right)} \right| = \sqrt 3 = \sqrt 2 \sqrt 2 \,{\text{sin}}\,\theta$ M1A1

Note: M1 is for an attempt to find the scalar or vector product of the two normal vectors.

$\Rightarrow θ = 60^{\circ} (= \frac{π}{3})$ $\Rightarrow \theta = 60^\circ \left( { = \frac{\pi }{3}} \right)$ A1

angle between faces is $20^{\circ} (= \frac{2 π}{3})$ $20^\circ \left( { = \frac{{2\pi }}{3}} \right)$ A1

[4 marks]

$\to DB = ⎛ ⎝ \begin{matrix} 11 - 1 \end{matrix} ⎞ ⎠$ $\mathop {{\text{DB}}}\limits^ \to = \left( \begin{gathered} \,1 \hfill \\ \,1 \hfill \\ - 1 \hfill \\ \end{gathered} \right)$ or $\to BD = ⎛ ⎝ \begin{matrix} - 1 - 1 1 \end{matrix} ⎞ ⎠$ $\mathop {{\text{BD}}}\limits^ \to = \left( \begin{gathered} - 1 \hfill \\ - 1 \hfill \\ \,1 \hfill \\ \end{gathered} \right)$ (A1)

$Π_{3} : x + y - z = k$ ${\Pi _3}\,{\text{:}}\,\,x + y - z = k$ (M1)

$Π_{3} : x + y - z = 0$ ${\Pi _3}\,{\text{:}}\,\,x + y - z = 0$ A1

[3 marks]

METHOD 1

line AD : (r =) $⎛ ⎝ \begin{matrix} 001 \end{matrix} ⎞ ⎠ + λ ⎛ ⎝ \begin{matrix} 10 - 1 \end{matrix} ⎞ ⎠$ $\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) + \lambda \left( \begin{gathered} \,1 \hfill \\ \,0 \hfill \\ - 1 \hfill \\ \end{gathered} \right)$ M1A1

intersects $Π_{3}$ ${\Pi _3}$ when $λ - (1 - λ) = 0$ $\lambda - \left( {1 - \lambda } \right) = 0$ M1

so $λ = \frac{1}{2}$ $\lambda = \frac{1}{2}$ A1

hence P is the midpoint of AD AG

METHOD 2

midpoint of AD is (0.5, 0, 0.5) (M1)A1

substitute into $x + y - z = 0$ $x + y - z = 0$ M1

0.5 + 0.5 − 0.5 = 0 A1

hence P is the midpoint of AD AG

[4 marks]

METHOD 1

$OP = \frac{1}{\sqrt{2}}, O \land P Q = 90^{\circ}, O \land Q P = 60^{\circ}$ ${\text{OP}} = \frac{1}{{\sqrt 2 }},\,\,{\text{O}}\mathop {\text{P}}\limits^ \wedge {\text{Q}} = 90^\circ ,\,\,{\text{O}}\mathop {\text{Q}}\limits^ \wedge {\text{P}} = 60^\circ$ A1A1A1

$PQ = \frac{1}{\sqrt{6}}$ ${\text{PQ}} = \frac{1}{{\sqrt 6 }}$ A1

area $= \frac{1}{2 \sqrt{12}} = \frac{1}{4 \sqrt{3}} = \frac{\sqrt{3}}{12}$ $= \frac{1}{{2\sqrt {12} }} = \frac{1}{{4\sqrt 3 }} = \frac{{\sqrt 3 }}{{12}}$ A1

METHOD 2

line BD : (r =) $⎛ ⎝ \begin{matrix} 110 \end{matrix} ⎞ ⎠ + λ ⎛ ⎝ \begin{matrix} - 1 - 1 1 \end{matrix} ⎞ ⎠$ $\left( \begin{gathered} 1 \hfill \\ 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) + \lambda \left( \begin{gathered} - 1 \hfill \\ - 1 \hfill \\ \,1 \hfill \\ \end{gathered} \right)$

$\Rightarrow λ = \frac{2}{3}$ $\Rightarrow \lambda = \frac{2}{3}$ (A1)

$\to OQ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} \frac{1}{3} \frac{1}{3} \frac{2}{3} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$ $\mathop {{\text{OQ}}}\limits^ \to = \left( \begin{gathered} \frac{1}{3} \hfill \\ \frac{1}{3} \hfill \\ \frac{2}{3} \hfill \\ \end{gathered} \right)$ A1

area = $\frac{1}{2} ∣ ∣ \to OP \times \to OQ ∣ ∣$ $\frac{1}{2}\left| {\mathop {{\text{OP}}}\limits^ \to \, \times \mathop {{\text{OQ}}}\limits^ \to } \right|$ M1

$\to OP = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} \frac{1}{2} 0 \frac{1}{2} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$ $\mathop {{\text{OP}}}\limits^ \to = \left( \begin{gathered} \frac{1}{2} \hfill \\ 0 \hfill \\ \frac{1}{2} \hfill \\ \end{gathered} \right)$ A1

Note: This A1 is dependent on M1.

area = $\frac{\sqrt{3}}{12}$ $\frac{{\sqrt 3 }}{{12}}$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.17—Vector equations of a plane

Show 51 related questions

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ $P (1, - 2, 0)$ lies on both $\prod_{1}$ $\prod_{1}$ and $\prod_{2}$ $\prod_{2}$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $Π$ $\Pi$ .
18M.1.AHL.TZ1.H_10c:
Find the Cartesian equation of $Π_{3}$ ${\Pi _3}$ .
17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π $_{3}$ $_3$ .
17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
18M.1.AHL.TZ2.H_9d:
Find the vector equation of the straight line passing through M and normal to the plane $Π$ $\Pi$ containing ABCD.
17M.2.AHL.TZ2.H_9d:
Show that the line (BC) lies in the plane Π $_{1}$ $_1$ .
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.
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ $L$ , the line of intersection of $\prod_{1}$ $\prod_{1}$ and $\prod_{2}$ $\prod_{2}$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ $L$ and $\prod_{3}$ $\prod_{3}$ .
21M.2.AHL.TZ1.6a:
Find a Cartesian equation of the plane $Π_{3}$ $Π_{3}$ which is perpendicular to $Π_{1}$ $Π_{1}$ and $Π_{2}$ $Π_{2}$ and passes through the origin $(0, 0, 0)$ $(0, 0, 0)$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ $d_{min}$ , from $D$ $D$ to $Π$ $Π$ .
18N.1.AHL.TZ0.H_9b:
Find, in terms of $b$ $b$ , 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.
17M.2.AHL.TZ2.H_9c:
Find the Cartesian equation of the plane Π $_{1}$ $_1$ , which passes through C and is perpendicular to $- - \to O A$ $\overrightarrow {{\rm{OA}}}$ .
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\to AD = \to BC$ $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
17M.2.AHL.TZ1.H_7:
Find the Cartesian equation of plane Π containing the points $A (6, 2, 1)$ ${\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)$ and $B (3, - 1, 1)$ ${\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)$ and perpendicular to the plane $x + 2 y - z - 6 = 0$ $x + 2y - z - 6 = 0$ .
19M.2.AHL.TZ2.H_11d.i:
Show that $5 a - 7 c = 4$ $5a - 7c = 4$ .
18M.1.AHL.TZ1.H_10a:
Find the Cartesian equation of the plane $Π_{1}$ ${\Pi _1}$ , passing through the points A , B and D.
17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
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 $(1, 0, - 1)$ $(1,{\text{ }}0,{\text{ }} - 1)$ .
18M.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
EXN.1.AHL.TZ0.8b.ii:
the condition on the value of $p$ $p$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $- - \to AB$ $\vec{AB}$ and $- - \to AC$ $\vec{AC}$ .
18M.1.AHL.TZ1.H_10d:
Show that P is the midpoint of AD.
EXN.1.AHL.TZ0.8b.i:
the value of $m$ $m$ .
16N.1.AHL.TZ0.H_4a:
Find a $\times$ $\times$ b.
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B ˆ A C = 60 °$ $B \hat{A} C = 60 °$ .
19M.2.AHL.TZ2.H_11b:
Given that П₁ and П₂ meet in a line $L$ $L$ , verify that the vector equation of $L$ $L$ can be given by r $= ⎛ ⎜ ⎜ ⎝ \begin{matrix} \frac{5}{4} 0 - \frac{7}{4} \end{matrix} ⎞ ⎟ ⎟ ⎠ + λ ⎛ ⎜ ⎜ ⎝ \begin{matrix} \frac{1}{2} 1 - \frac{5}{2} \end{matrix} ⎞ ⎟ ⎟ ⎠$ $= \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)$ .
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
21N.2.AHL.TZ0.11a.i:
Find the vector $- - \to AB$ $\vec{AB}$ and the vector $- - \to AC$ $\vec{AC}$ .
21N.2.AHL.TZ0.11a.ii:
Hence find the equation of $Π_{1}$ $Π_{1}$ , expressing your answer in the form $a x + b y + c z = d$ $a x + b y + c z = d$ , 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.
17M.2.AHL.TZ2.H_9g:
Find the acute angle between the planes Π $_2$ and Π $_3$ .
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.
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:
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 $b$ .
17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
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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Topic 3— Geometry and trigonometry » AHL 3.18—Intersections of lines & planes

Topic 3— Geometry and trigonometry

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