| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the dot product for perpendicularity (setting u·v = 0 and solving a quadratic) followed by a standard angle calculation using the cosine formula. Both parts require only direct recall of standard vector formulas with minimal problem-solving, making it easier than average for A-level. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2p^2 - 2p + 2 + 12p + 6 \to 2p^2 + 10p + 8\) | M1 | Correct method for scalar product |
| \(\mathbf{u}.\mathbf{v} = 0\) | B1 | Scalar product \(= 0\) |
| \((p+1)(p+4) = 0 \to p = -1\) or \(p = -4\) | A1 [3] | cao. Both solutions required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{u}.\mathbf{v} = 2 + 0 + 18 = 20\) | M1 | Use of \(x_1x_2 + y_1y_2 + z_1z_2\) |
| \( | \mathbf{u} | = \sqrt{41}\) or \( |
| \(20 = \sqrt{41} \times \sqrt{13} \times \cos\theta\) oe | M1 | All connected correctly |
| \(\theta = 30.0°\) or \(0.523\) rads | A1 [4] | cao |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2p^2 - 2p + 2 + 12p + 6 \to 2p^2 + 10p + 8$ | M1 | Correct method for scalar product |
| $\mathbf{u}.\mathbf{v} = 0$ | B1 | Scalar product $= 0$ |
| $(p+1)(p+4) = 0 \to p = -1$ **or** $p = -4$ | A1 [3] | cao. Both solutions required |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{u}.\mathbf{v} = 2 + 0 + 18 = 20$ | M1 | Use of $x_1x_2 + y_1y_2 + z_1z_2$ |
| $|\mathbf{u}| = \sqrt{41}$ or $|\mathbf{v}| = \sqrt{13}$ | M1 | Correct method for moduli |
| $20 = \sqrt{41} \times \sqrt{13} \times \cos\theta$ oe | M1 | All connected correctly |
| $\theta = 30.0°$ or $0.523$ rads | A1 [4] | cao |
---6 Two vectors $\mathbf { u }$ and $\mathbf { v }$ are such that $\mathbf { u } = \left( \begin{array} { c } p ^ { 2 } \\ - 2 \\ 6 \end{array} \right)$ and $\mathbf { v } = \left( \begin{array} { c } 2 \\ p - 1 \\ 2 p + 1 \end{array} \right)$, where $p$ is a constant.\\
(i) Find the values of $p$ for which $\mathbf { u }$ is perpendicular to $\mathbf { v }$.\\
(ii) For the case where $p = 1$, find the angle between the directions of $\mathbf { u }$ and $\mathbf { v }$.
\hfill \mbox{\textit{CAIE P1 2012 Q6 [7]}}