| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Area of triangle using vector product |
| Difficulty | Standard +0.3 This is a standard Further Maths question testing routine application of vector product formulas. Part (i) uses the formula area = ½|AB × AC|, part (ii) applies area = ½ base × height to find perpendicular distance, and part (iii) uses the cross product to find the normal vector for the plane equation. All parts follow textbook methods with straightforward arithmetic, making it slightly easier than average even for Further Maths. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector4.04h Shortest distances: between parallel lines and between skew lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{AB} = \mathbf{i}+5\mathbf{j}-2\mathbf{k},\quad \overrightarrow{BC} = -4\mathbf{i}-2\mathbf{j}+5\mathbf{k},\quad \overrightarrow{AC} = -3\mathbf{i}+\mathbf{j}+3\mathbf{k}\) | B1 | 2 correct required |
| \(\overrightarrow{AB}\times\overrightarrow{BC} = 21\mathbf{i}+3\mathbf{j}+18\mathbf{k}\) | M1A1 | OE |
| Area of triangle \(ABC = \dfrac{1}{2}\sqrt{21^2+3^2+18^2} = 13.9\left(\dfrac{3}{2}\sqrt{86}\right)\) | A1 | |
| Alt method: Use scalar product to find angle | (M1A1) | |
| Find area using Area \(= \frac{1}{2}ab\sin C\) or equivalent | (M1A1) | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(d = \dfrac{ | \overrightarrow{AB}\times\overrightarrow{BC} | }{ |
| \(= 4.15\left(\dfrac{1}{5}\sqrt{430}\right)\) | A1 | Area triangle \(= \sin C \times |
| Alt method: Use equation of BC to find D (foot of perpendicular) in terms of parameter and scalar product to find parameter, \(\lambda = 8/15\). Find length | (M1A1) | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| From (*) Cartesian equation is \(7x + y + 6z = \text{const.}\) | M1 | |
| Through \((2, -1, 1)\): Hence \(7x + y + 6z = 19\) | A1 | |
| Total | 2 |
## Question 6:
### Part 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{AB} = \mathbf{i}+5\mathbf{j}-2\mathbf{k},\quad \overrightarrow{BC} = -4\mathbf{i}-2\mathbf{j}+5\mathbf{k},\quad \overrightarrow{AC} = -3\mathbf{i}+\mathbf{j}+3\mathbf{k}$ | B1 | 2 correct required |
| $\overrightarrow{AB}\times\overrightarrow{BC} = 21\mathbf{i}+3\mathbf{j}+18\mathbf{k}$ | M1A1 | OE |
| Area of triangle $ABC = \dfrac{1}{2}\sqrt{21^2+3^2+18^2} = 13.9\left(\dfrac{3}{2}\sqrt{86}\right)$ | A1 | |
| Alt method: Use scalar product to find angle | (M1A1) | |
| Find area using Area $= \frac{1}{2}ab\sin C$ or equivalent | (M1A1) | |
| **Total** | **4** | |
### Part 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $d = \dfrac{|\overrightarrow{AB}\times\overrightarrow{BC}|}{|\overrightarrow{BC}|} = \dfrac{\sqrt{21^2+3^2+18^2}}{\sqrt{4^2+2^2+5^2}}$ | M1A1 | Alt method: Find angle at $C$ |
| $= 4.15\left(\dfrac{1}{5}\sqrt{430}\right)$ | A1 | Area triangle $= \sin C \times |AC|$ |
| Alt method: Use equation of BC to find D (foot of perpendicular) in terms of parameter and scalar product to find parameter, $\lambda = 8/15$. Find length | (M1A1) | |
| **Total** | **3** | |
### Part 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| From (*) Cartesian equation is $7x + y + 6z = \text{const.}$ | M1 | |
| Through $(2, -1, 1)$: Hence $7x + y + 6z = 19$ | A1 | |
| **Total** | **2** | |
---6 The points $A , B$ and $C$ have position vectors $2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$ and $- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ respectively.\\
(i) Find the area of the triangle $A B C$.\\
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(ii) Find the perpendicular distance of the point $A$ from the line $B C$.\\
(iii) Find the cartesian equation of the plane through $A , B$ and $C$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q6 [9]}}