| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Angle between line and plane |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard C4 vector techniques: finding direction vectors, vector equations of lines, angles between lines and planes using dot products, and line-plane intersection. All parts follow routine procedures with clear signposting, though it requires careful calculation across multiple steps. The Snell's Law context is cosmetic—the mathematics is straightforward application of formulas. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane |
| Answer | Marks |
|---|---|
| (i) They would agree to co-operate by spending less on advertising or by sharing equally. | B1 |
| (ii) Increased market share (or more money or more customers). | DB1 |
**(i)** They would agree to co-operate by spending less on advertising or by sharing equally. | B1 |
---|---|
**(ii)** Increased market share (or more money or more customers). | DB1 |When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point A $(1, 2, 2)$, and enters a glass object at point B $(0, 0, 2)$. The surface of the glass object is a plane with normal vector $\mathbf{n}$. Fig. 7 shows a cross-section of the glass object in the plane of the light ray and $\mathbf{n}$.
\includegraphics{figure_7}
\begin{enumerate}[label=(\roman*)]
\item Find the vector $\overrightarrow{AB}$ and a vector equation of the line AB. [2]
\end{enumerate}
The surface of the glass object is a plane with equation $x + z = 2$. AB makes an acute angle $\theta$ with the normal to this plane.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Write down the normal vector $\mathbf{n}$, and hence calculate $\theta$, giving your answer in degrees. [5]
\end{enumerate}
The line BC has vector equation $\mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ -2 \\ -1 \end{pmatrix}$. This line makes an acute angle $\phi$ with the normal to the plane.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that $\phi = 45°$. [3]
\item Snell's Law states that $\sin\theta = k\sin\phi$, where $k$ is a constant called the refractive index. Find $k$. [2]
\end{enumerate}
The light ray leaves the glass object through a plane with equation $x + z = -1$. Units are centimetres.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{4}
\item Find the point of intersection of the line BC with the plane $x + z = -1$. Hence find the distance the light ray travels through the glass object. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2009 Q7 [17]}}