OCR Further Additional Pure 2018 March — Question 2 5 marks

Exam BoardOCR
ModuleFurther Additional Pure (Further Additional Pure)
Year2018
SessionMarch
Marks5
TopicVector Product and Surfaces
TypeVolume of tetrahedron using scalar triple product
DifficultyStandard +0.8 This is a Further Maths question requiring knowledge of the scalar triple product formula for tetrahedron volume (V = 1/6|a·(b×c)|), which is beyond standard A-level. However, it's a direct application of a standard formula with straightforward coordinate arithmetic—no conceptual insight or problem-solving required beyond knowing and executing the method.
Spec8.04e Scalar triple product: volumes of tetrahedra and parallelepipeds
2 Four points \(A , B , C\) and \(D\) have coordinates \(( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )\) and \(( 0,3,7 )\) respectively. Find the volume of tetrahedron \(A B C D\).
AnswerMarks Guidance
E.g. \(\mathbf{x} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} 2 \\ 2 \\ -9 \end{pmatrix}\), \(\mathbf{y} = \mathbf{c} - \mathbf{a} = \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix}\), \(\mathbf{z} = \mathbf{d} - \mathbf{a} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}\)M1*, A1 Subtraction to find any 3 sides of the tetrahedron
Volume = \(\frac{1}{6}\\mathbf{x} \cdot \mathbf{y} \times \mathbf{z}\ \)
\(\begin{vmatrix} 2 & 2 & -9 \\ 5 & 0 & -2 \\ -1 & 1 & 2 \end{vmatrix} = -57\)B1 A correct (non-zero) scalar triple product (possibly BC). Condone sign error
\(\Rightarrow\) Volume = 9.5A1 [5] FT det; must be positive final answer 
| E.g. $\mathbf{x} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} 2 \\ 2 \\ -9 \end{pmatrix}$, $\mathbf{y} = \mathbf{c} - \mathbf{a} = \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix}$, $\mathbf{z} = \mathbf{d} - \mathbf{a} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$ | M1*, A1 | Subtraction to find any 3 sides of the tetrahedron |
| Volume = $\frac{1}{6}\|\mathbf{x} \cdot \mathbf{y} \times \mathbf{z}\|$ | dep*M1 | Attempted use of formula |
| $\begin{vmatrix} 2 & 2 & -9 \\ 5 & 0 & -2 \\ -1 & 1 & 2 \end{vmatrix} = -57$ | B1 | A correct (non-zero) scalar triple product (possibly BC). Condone sign error |
| $\Rightarrow$ Volume = 9.5 | A1 [5] | FT det; must be positive final answer |

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2 Four points $A , B , C$ and $D$ have coordinates $( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )$ and $( 0,3,7 )$ respectively. Find the volume of tetrahedron $A B C D$.

\hfill \mbox{\textit{OCR Further Additional Pure 2018 Q2 [5]}}
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