| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line-plane intersection and related angle/perpendicularity |
| Difficulty | Moderate -0.3 This is a straightforward two-part question testing standard techniques: (i) applying the plane equation formula r·n = a·n directly, and (ii) substituting the parametric line into the plane equation to solve for λ, then finding coordinates. Both parts are routine applications of Core 4 methods with no problem-solving insight required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Filled circle (•) corresponds to \(\leq\) (included endpoint); unfilled circle (○) corresponds to \(<\) (excluded endpoint) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Line 102 states \(\frac{V_k}{N_k+1} < a \leq \frac{V_k}{N_k}\) | M1 | Starting from correct inequality |
| Multiply through: \(V_k < a(N_k+1)\) and \(aN_k \leq V_k\) giving \(aN_k \leq V_k < a(N_k+1)\) hence \(0 \leq V_k - N_k a < a\) | A1 | Correct rearrangement shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(V_k - N_k a \geq 0\) (residual is non-negative) uses \(\leq\) at upper end; strict inequality \(< a\) because residual must be less than \(a\) to not gain another seat, uses \(<\) at lower end | B1 |
# Question 5(i): Filled vs unfilled circles
| Answer | Mark | Guidance |
|--------|------|----------|
| Filled circle (•) corresponds to $\leq$ (included endpoint); unfilled circle (○) corresponds to $<$ (excluded endpoint) | B1 | |
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# Question 5(ii): Rearrange inequality
| Answer | Mark | Guidance |
|--------|------|----------|
| Line 102 states $\frac{V_k}{N_k+1} < a \leq \frac{V_k}{N_k}$ | M1 | Starting from correct inequality |
| Multiply through: $V_k < a(N_k+1)$ and $aN_k \leq V_k$ giving $aN_k \leq V_k < a(N_k+1)$ hence $0 \leq V_k - N_k a < a$ | A1 | Correct rearrangement shown |
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# Question 5(iii): Justify inequality signs
| Answer | Mark | Guidance |
|--------|------|----------|
| $V_k - N_k a \geq 0$ (residual is non-negative) uses $\leq$ at upper end; strict inequality $< a$ because residual must be less than $a$ to not gain another seat, uses $<$ at lower end | B1 | |5 (i) Find the cartesian equation of the plane through the point ( $2 , - 1,4$ ) with normal vector
$$\mathbf { n } = \left( \begin{array} { r }
1 \\
- 1 \\
2
\end{array} \right)$$
(ii) Find the coordinates of the point of intersection of this plane and the straight line with equation
$$\mathbf { r } = \left( \begin{array} { r }
7 \\
12 \\
9
\end{array} \right) + \lambda \left( \begin{array} { l }
1 \\
3 \\
2
\end{array} \right)$$
\hfill \mbox{\textit{OCR MEI C4 2006 Q5 [7]}}