AQA Further Paper 2 Specimen — Question 16 9 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
SessionSpecimen
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors: Lines & Planes
Type3D geometry applications
DifficultyChallenging +1.8 This is a substantial 3D coordinate geometry problem requiring multiple steps: finding point A from plane and line intersection, determining the triangular base area using vectors, calculating the prism length, and computing volume. While each individual step uses standard Further Maths techniques (vector equations, cross products, plane-line intersection), the multi-stage nature, spatial visualization required, and integration of several topics makes this significantly harder than average but not exceptionally difficult for Further Maths students who have practiced similar problems.
Spec4.04b Plane equations: cartesian and vector forms4.04f Line-plane intersection: find point
A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism \(ABCDEF\) with parallel triangular ends \(ABC\) and \(DEF\), and a rectangular base \(ACFD\). He uses the metre as the unit of length. \includegraphics{figure_16} The coordinates of \(B\), \(C\) and \(D\) are \((3, 1, 11)\), \((9, 3, 4)\) and \((-4, 12, 4)\) respectively. He uses the equation \(x - 3y = 0\) for the plane \(ABC\). He uses \(\mathbf{r} - \begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix} \times \begin{pmatrix} 4 \\ -12 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) for the equation of the line \(AD\). Find the volume of the space enclosed inside this section of the roof. [9 marks]
Question 16:
AnswerMarks
16Uses the mathematical model to
find the volume by first finding the
coordinate of A.
To award this mark must see an
attempt to find coords of A, and an
AnswerMarks Guidance
attempt at volume of prismAO3.4 M1
y=1212t
z=4
4t43(1212t)0
40t-40 = 0
t = 1
0 0 4
OR
x 4  4  0
       
 y    12   12    0 
    z     4       0     0  
3y4  4  0
     
y12  12  0
     
     
z4 0 0
     
12(z4)0z4
12(3y+4)4(y12)0
 y=0, x=0
A has coordinates (0,0,4)
3

 
AB 1
 
 
7
 
9

 
AC 3
 
 
0
 
21
 
 
AB×AC= 63
 
 
0
 
21 10
Area ABC 
2
d 4 10
Volume =
21 10
V  4 10 420 m3
2
Selects method involving both
equation of plane and equation of
line to find coords of A
Either using parametric form or
using cross product
AnswerMarks Guidance
Ignore sign errorsAO3.1a M1
Either collects terms together and
solves to find value of parameter
for ‘their’ equation
Or correctly calculates cross
AnswerMarks Guidance
product for ‘their’ vectorsAO1.1b A1F
Deduces the correct coordinates of
AnswerMarks Guidance
AAO2.2a A1
Selects a correct approach to
AnswerMarks Guidance
calculate the volume of the prism.AO3.1a M1
Finds two sides of the triangle ABC
in vector form
AnswerMarks Guidance
FT ‘their’ AAO1.2 A1F
Finds area of ABC FT ‘their’ AAO1.1b A1F
Finds length of prism
AnswerMarks Guidance
FT ‘their’ AAO1.1b A1F
Gives their answer in context by
correctly finding the volume of the
roof with correct units.
AnswerMarks Guidance
FT ‘their’ prismAO1.1b A1F
Total9
Total100 
Question 16:
16 | Uses the mathematical model to
find the volume by first finding the
coordinate of A.
To award this mark must see an
attempt to find coords of A, and an
attempt at volume of prism | AO3.4 | M1 | x4t4
y=1212t
z=4
4t43(1212t)0
40t-40 = 0
t = 1
0 0 4
OR
x 4  4  0
       
 y    12   12    0 
    z     4       0     0  
3y4  4  0
     
y12  12  0
     
     
z4 0 0
     
12(z4)0z4
12(3y+4)4(y12)0
 y=0, x=0
A has coordinates (0,0,4)
3

 
AB 1
 
 
7
 
9

 
AC 3
 
 
0
 
21
 
 
AB×AC= 63
 
 
0
 
21 10
Area ABC 
2
d 4 10
Volume =
21 10
V  4 10 420 m3
2
Selects method involving both
equation of plane and equation of
line to find coords of A
Either using parametric form or
using cross product
Ignore sign errors | AO3.1a | M1
Either collects terms together and
solves to find value of parameter
for ‘their’ equation
Or correctly calculates cross
product for ‘their’ vectors | AO1.1b | A1F
Deduces the correct coordinates of
A | AO2.2a | A1
Selects a correct approach to
calculate the volume of the prism. | AO3.1a | M1
Finds two sides of the triangle ABC
in vector form
FT ‘their’ A | AO1.2 | A1F
Finds area of ABC FT ‘their’ A | AO1.1b | A1F
Finds length of prism
FT ‘their’ A | AO1.1b | A1F
Gives their answer in context by
correctly finding the volume of the
roof with correct units.
FT ‘their’ prism | AO1.1b | A1F
Total | 9
Total | 100
A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism $ABCDEF$ with parallel triangular ends $ABC$ and $DEF$, and a rectangular base $ACFD$. He uses the metre as the unit of length.

\includegraphics{figure_16}

The coordinates of $B$, $C$ and $D$ are $(3, 1, 11)$, $(9, 3, 4)$ and $(-4, 12, 4)$ respectively.

He uses the equation $x - 3y = 0$ for the plane $ABC$.

He uses $\mathbf{r} - \begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix} \times \begin{pmatrix} 4 \\ -12 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ for the equation of the line $AD$.

Find the volume of the space enclosed inside this section of the roof.
[9 marks]

\hfill \mbox{\textit{AQA Further Paper 2  Q16 [9]}}
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