| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Challenging +1.8 This AEA question combines scalar triple product for tetrahedron volume with logarithm manipulation. Part (b) of the vectors requires understanding that volume = |a·(b×c)|/6, which is A-level Further Maths content but straightforward application. The logarithm section requires systematic use of change of base and the given identity across multiple parts, demanding careful algebraic manipulation but following standard patterns. The multi-part structure and AEA context elevate it above typical A-level, but the techniques are well-defined without requiring novel geometric or algebraic insight. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
\text { and } \mathbf { d } = \left( \begin{array} { c }
- 4 \\
2 \\
- 11
\end{array} \right)$$
(a)Find the position vector of $E$ .
The volume of a tetrahedron is given by the formula
$$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
(b)Find the volume of the tetrahedron $A B C D$ .
4.(a)Given that $x > 0 , y > 0 , x \neq 1$ and $n > 0$ ,show that
$$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
(b)Solve the following,leaving your answers in the form $2 ^ { p }$ ,where $p$ is a rational number.\\
(i) $\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5$\\
(ii) $\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5$\\
(iii) $\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5$
\hfill \mbox{\textit{Edexcel AEA 2016 Q4 [11]}}