| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line of intersection of planes |
| Difficulty | Standard +0.8 This is a Further Maths question requiring students to find the line of intersection of two planes given in scalar product form. It involves finding the direction vector via cross product of normals, then finding a point on both planes by solving simultaneous equations. While the technique is standard for FP3, it requires multiple steps (cross product, solving system, constructing vector equation) and is more sophisticated than typical A-level pure maths content, placing it moderately above average difficulty. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: \([3,1,-2]\times[1,5,4]\) | M1 | For attempt to find vector product of both normals |
| \(\Rightarrow \mathbf{b}=k[1,-1,1]\) | A1 | For correct vector identified with b |
| e.g. put \(x\) OR \(y\) OR \(z=0\) | M1 | For giving a value to one variable |
| Solve 2 equations in 2 unknowns | M1 | For solving the equations in the other variables |
| Obtain \([0,2,-1]\) OR \([2,0,1]\) OR \([1,1,0]\) | A1 | For a correct vector identified with a |
| OR: Solve \(3x+y-2z=4\), \(x+5y+4z=6\) | ||
| e.g. \(y+z=1\) OR \(x-z=1\) OR \(x+y=2\) | M1 | For eliminating one variable between 2 equations |
| Put \(x\) OR \(y\) OR \(z=t\) | M1 | For solving in terms of a parameter |
| \([x,y,z]=[t,2-t,-1+t]\) OR \([2-t,t,1-t]\) OR \([1+t,1-t,t]\) | M1 | For obtaining a parametric solution for \(x,y,z\) |
| Obtain \([0,2,-1]\) OR \([2,0,1]\) OR \([1,1,0]\) | A1 | For a correct vector identified with a |
| Obtain \(k[1,-1,1]\) | A1 5 | For correct vector identified with b |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: $[3,1,-2]\times[1,5,4]$ | M1 | For attempt to find vector product of both normals |
| $\Rightarrow \mathbf{b}=k[1,-1,1]$ | A1 | For correct vector identified with **b** |
| e.g. put $x$ OR $y$ OR $z=0$ | M1 | For giving a value to one variable |
| Solve 2 equations in 2 unknowns | M1 | For solving the equations in the other variables |
| Obtain $[0,2,-1]$ OR $[2,0,1]$ OR $[1,1,0]$ | A1 | For a correct vector identified with **a** |
| OR: Solve $3x+y-2z=4$, $x+5y+4z=6$ | | |
| e.g. $y+z=1$ OR $x-z=1$ OR $x+y=2$ | M1 | For eliminating one variable between 2 equations |
| Put $x$ OR $y$ OR $z=t$ | M1 | For solving in terms of a parameter |
| $[x,y,z]=[t,2-t,-1+t]$ OR $[2-t,t,1-t]$ OR $[1+t,1-t,t]$ | M1 | For obtaining a parametric solution for $x,y,z$ |
| Obtain $[0,2,-1]$ OR $[2,0,1]$ OR $[1,1,0]$ | A1 | For a correct vector identified with **a** |
| Obtain $k[1,-1,1]$ | A1 **5** | For correct vector identified with **b** |
---2 Find the equation of the line of intersection of the planes with equations
$$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$
giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.
\hfill \mbox{\textit{OCR FP3 2007 Q2 [5]}}