| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Interception: verify/find meeting point (position vector method) |
| Difficulty | Moderate -0.3 This is a standard M1 mechanics question on vector interception requiring equating position vectors and solving simultaneous equations, then calculating speeds using Pythagoras. The method is routine and well-practiced, though it involves multiple steps (4-5 marks typical). Slightly easier than average A-level due to being a textbook application with no conceptual surprises. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
# Question 1
**(i)**
- M1: Forming an equation for $t$. Accept vector equation for this mark. May be implied by a statement that $t = 2$.
Either $-2 + 8t = 7t$ or $t = 10 - 4t$
- A1: $\Rightarrow t = 2$
oe, eg showing $t = 2$ satisfies both equations or a vector equation.
- B1: Substituting $t = 2$ in both expressions
- B1: They meet at $(14, 2)$
Accept $\begin{pmatrix} 14 \\ 2 \end{pmatrix}$
[4]
**(ii)**
- B1: Ashok's speed is $\sqrt{8^2 + 1^2} = \sqrt{65}$
CAO from correct speeds
- B1: Kumar's speed is $\sqrt{7^2 + (-4)^2} = \sqrt{65}$ km h$^{-1}$
- B1: They both walk at the same speed
SC1 for finding both velocities correctly but neither speed
[3]
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Follow through between parts should be allowed for the value of $h$ (when $t = 10$) found in part (iii) if it is used in part (iv) or in part (v)(A).1 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors $\binom { 1 } { 0 }$ and $\binom { 0 } { 1 }$ are in the directions east and north.
At time $t$ hours the position vectors of two hikers, Ashok and Kumar, are given by:
$$\begin{array} { l l }
\text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t , \\
\text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } .
\end{array}$$
(i) Prove that the two hikers meet and give the coordinates of the point where this happens.\\
(ii) Compare the speeds of the two hikers.
\hfill \mbox{\textit{OCR MEI M1 Q1 [7]}}