Edexcel M1 2016 January — Question 6 13 marks

Exam BoardEdexcel
ModuleM1 (Mechanics 1)
Year2016
SessionJanuary
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors Introduction & 2D
TypeInterception: verify/find meeting point (position vector method)
DifficultyModerate -0.8 This is a standard M1 vectors question testing routine application of position vectors, distance formula, and constant velocity motion. All parts follow predictable patterns: (a) uses Pythagoras, (b-c) apply r = r₀ + vt, (d) equates position vectors and solves simultaneous equations, (e) substitutes back. No problem-solving insight required—purely procedural application of well-drilled techniques with straightforward arithmetic.
Spec1.10e Position vectors: and displacement1.10f Distance between points: using position vectors3.02a Kinematics language: position, displacement, velocity, acceleration
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm, the position vector of ship \(P\) is \((5\mathbf{i} - 3\mathbf{j})\) km and the position vector of ship \(Q\) is \((7\mathbf{i} + 5\mathbf{j})\) km.
  1. Find the distance between \(P\) and \(Q\) at 2 pm. [3]
Ship \(P\) is moving with constant velocity \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) is moving with constant velocity \((-3\mathbf{i} - 15\mathbf{j})\) km h\(^{-1}\).
  1. Find the position vector of \(P\) at time \(t\) hours after 2 pm. [2]
  2. Find the position vector of \(Q\) at time \(t\) hours after 2 pm. [1]
  3. Show that \(Q\) will meet \(P\) and find the time at which they meet. [5]
  4. Find the position vector of the point at which they meet. [2]
Question 6:
AnswerMarks
6(a)PQ(7i5j)(5i 3j)(2i8j)
PQ 22 82  68 8.2 or better
AnswerMarks
M1
M1 A1 (3)
AnswerMarks
(b)r (5i3j)t(2i5j)(2t5)i(5t3)j
PM1 A1
(2)
AnswerMarks
(c)r (7i5j)t(3i15j)(73t)i(515t)j
QA1 (1)
(d)(2t5)(73t)t  2
5
(5t3)(515t)t  2
5
time is 2.24 pm
AnswerMarks
Allow just t = 0.4M1 A1
M1 A1
A1 (5)
AnswerMarks
(e)r (5.8i j)
PM1 A1 (2)
13
Notes
Allow column vectors throughout.
AnswerMarks
6(a)First M1 for clear attempt to subtract in either order. Condone missing brackets.
Second M1 for attempt to find magnitude of their PQ or QP
A1 √68, 2√17 or 8.2 or better
AnswerMarks
(b)M1 for (either r or r ) a clear attempt at: (M0 if they use (t + 2))
P Q
r (5i3j)t(2i5j)(2t5)i(5t3)j
P
A1 if correct (i’s and j’s do not need to be collected.)
AnswerMarks
(c)A1 for r (7i5j)t(3i15j)(73t)i(515t)j
Q
AnswerMarks
(d)First M1 for equating coefficients of i (coeffs. of form a + bt)
First A1 for t = 2/5
Second M1 for equating coefficients of j (coeffs. of form a + bt)
Second A1 for t = 2/5
Third A1 for 2.24 (pm) , dependent on both previous M marks
AnswerMarks
(e)This answer must appear in part (e).
M1 for substituting their t value (allow even if they have only equated coefficients
once to obtain it) into their r or r expression
P Q
A1 for r (5.8i j)
P
Question 6:
--- 6(a) ---
6(a) | PQ(7i5j)(5i 3j)(2i8j)
PQ 22 82  68 8.2 or better
 | M1
M1 A1 (3)
(b) | r (5i3j)t(2i5j)(2t5)i(5t3)j
P | M1 A1
(2)
(c) | r (7i5j)t(3i15j)(73t)i(515t)j
Q | A1 (1)
(d) | (2t5)(73t)t  2
5
(5t3)(515t)t  2
5
time is 2.24 pm
Allow just t = 0.4 | M1 A1
M1 A1
A1 (5)
(e) | r (5.8i j)
P | M1 A1 (2)
13
Notes
Allow column vectors throughout.
6(a) | First M1 for clear attempt to subtract in either order. Condone missing brackets.
Second M1 for attempt to find magnitude of their PQ or QP
A1 √68, 2√17 or 8.2 or better
(b) | M1 for (either r or r ) a clear attempt at: (M0 if they use (t + 2))
P Q
r (5i3j)t(2i5j)(2t5)i(5t3)j
P
A1 if correct (i’s and j’s do not need to be collected.)
(c) | A1 for r (7i5j)t(3i15j)(73t)i(515t)j
Q
(d) | First M1 for equating coefficients of i (coeffs. of form a + bt)
First A1 for t = 2/5
Second M1 for equating coefficients of j (coeffs. of form a + bt)
Second A1 for t = 2/5
Third A1 for 2.24 (pm) , dependent on both previous M marks
(e) | This answer must appear in part (e).
M1 for substituting their t value (allow even if they have only equated coefficients
once to obtain it) into their r or r expression
P Q
A1 for r (5.8i j)
P
[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.]

At 2 pm, the position vector of ship $P$ is $(5\mathbf{i} - 3\mathbf{j})$ km and the position vector of ship $Q$ is $(7\mathbf{i} + 5\mathbf{j})$ km.

\begin{enumerate}[label=(\alph*)]
\item Find the distance between $P$ and $Q$ at 2 pm. [3]
\end{enumerate}

Ship $P$ is moving with constant velocity $(2\mathbf{i} + 5\mathbf{j})$ km h$^{-1}$ and ship $Q$ is moving with constant velocity $(-3\mathbf{i} - 15\mathbf{j})$ km h$^{-1}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the position vector of $P$ at time $t$ hours after 2 pm. [2]

\item Find the position vector of $Q$ at time $t$ hours after 2 pm. [1]

\item Show that $Q$ will meet $P$ and find the time at which they meet. [5]

\item Find the position vector of the point at which they meet. [2]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2016 Q6 [13]}}
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