1.10c - OCR Spec

Edexcel M1 2012 January Q7

9 marks Moderate -0.8

7. [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively. Position vectors are relative to a fixed origin $O$.] A boat $P$ is moving with constant velocity $( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.

Calculate the speed of $P$. When $t = 0$, the boat $P$ has position vector $( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vector of $P$ is $\mathbf { p ~ k m }$.
Write down $\mathbf { p }$ in terms of $t$. A second boat $Q$ is also moving with constant velocity. At time $t$ hours, the position vector of $Q$ is $\mathbf { q } \mathrm { km }$, where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
the value of $t$ when $P$ is due west of $Q$,
the distance between $P$ and $Q$ when $P$ is due west of $Q$.

Edexcel M1 2003 June Q5

10 marks Moderate -0.3

5. A particle $P$ moves with constant acceleration $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. At time $t$ seconds, its velocity is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. When $t = 0 , \mathbf { v } = - 2 \mathbf { i } + 7 \mathbf { j }$.

Find the value of $t$ when $P$ is moving parallel to the vector $\mathbf { i }$.
Find the speed of $P$ when $t = 3$.
Find the angle between the vector $\mathbf { j }$ and the direction of motion of $P$ when $t = 3$.

Edexcel M1 2006 June Q7

15 marks Moderate -0.3

\hspace{0pt} [In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and north respectively.]

A ship $S$ is moving with constant velocity $( - 2.5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. At time 1200, the position vector of $S$ relative to a fixed origin $O$ is $( 16 \mathbf { i } + 5 \mathbf { j } )$ km. Find

the speed of $S$,
the bearing on which $S$ is moving. The ship is heading directly towards a submerged rock $R$. A radar tracking station calculates that, if $S$ continues on the same course with the same speed, it will hit $R$ at the time 1500.
Find the position vector of $R$. The tracking station warns the ship's captain of the situation. The captain maintains $S$ on its course with the same speed until the time is 1400 . He then changes course so that $S$ moves due north at a constant speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Assuming that $S$ continues to move with this new constant velocity, find
an expression for the position vector of the ship $t$ hours after 1400,
the time when $S$ will be due east of $R$,
the distance of $S$ from $R$ at the time 1600.

Edexcel M1 2008 June Q3

8 marks Moderate -0.8

3. A particle $P$ of mass 0.4 kg moves under the action of a single constant force $\mathbf { F }$ newtons. The acceleration of $P$ is $( 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. Find

the angle between the acceleration and $\mathbf { i }$,
the magnitude of $\mathbf { F }$. At time $t$ seconds the velocity of $P$ is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that when $t = 0 , \mathbf { v } = 9 \mathbf { i } - 10 \mathbf { j }$, (c) find the velocity of $P$ when $t = 5$.

Edexcel M1 2008 June Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-07_357_968_274_484} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two forces $\mathbf { P }$ and $\mathbf { Q }$ act on a particle at a point $O$. The force $\mathbf { P }$ has magnitude 15 N and the force $\mathbf { Q }$ has magnitude $X$ newtons. The angle between $\mathbf { P }$ and $\mathbf { Q }$ is $150 ^ { \circ }$, as shown in Figure 1. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is $\mathbf { R }$. Given that the angle between $\mathbf { R }$ and $\mathbf { Q }$ is $50 ^ { \circ }$, find

the magnitude of $\mathbf { R }$,
the value of $X$.

Edexcel M1 2012 June Q6

13 marks Moderate -0.8

6. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship $S$ is moving with constant velocity $( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.

Find the direction in which $S$ is moving, giving your answer as a bearing. At time $t$ hours after noon, the position vector of $S$ is $\mathbf { s } \mathrm { km }$. When $t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }$.
Write down $\mathbf { s }$ in terms of $t$. A fixed beacon $B$ is at the point with position vector $( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }$.
Find the distance of $S$ from $B$ when $t = 3$
Find the distance of $S$ from $B$ when $S$ is due north of $B$.

Edexcel M1 2014 June Q5

12 marks Moderate -0.3

5. A particle $P$ of mass 0.5 kg is moving under the action of a single force $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }$.

Show that the magnitude of the acceleration of $P$ is $2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At time $t = 0$, the velocity of $P$ is $( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Find the velocity of $P$ at time $t = 2$ seconds. Another particle $Q$ moves with constant velocity $\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Find the distance moved by $Q$ in 2 seconds.
Show that at time $t = 3.5$ seconds both particles are moving in the same direction.

Edexcel M1 2014 June Q6

9 marks Moderate -0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-11_472_908_285_520} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two forces $\mathbf { P }$ and $\mathbf { Q }$ act on a particle at $O$. The angle between the lines of action of $\mathbf { P }$ and $\mathbf { Q }$ is $120 ^ { \circ }$ as shown in Figure 4. The force $\mathbf { P }$ has magnitude 20 N and the force $\mathbf { Q }$ has magnitude $X$ newtons. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is the force $\mathbf { R }$. Given that the magnitude of $\mathbf { R }$ is $3 X$ newtons, find, giving your answers to 3 significant figures

the value of $X$,
the magnitude of $( \mathbf { P } - \mathbf { Q } )$.

Edexcel M1 2015 June Q6

8 marks Easy -1.3

A particle $P$ is moving with constant velocity. The position vector of $P$ at time $t$ seconds $( t \geqslant 0 )$ is $\mathbf { r }$ metres, relative to a fixed origin $O$, and is given by

$$\mathbf { r } = ( 2 t - 3 ) \mathbf { i } + ( 4 - 5 t ) \mathbf { j }$$

Find the initial position vector of $P$. The particle $P$ passes through the point with position vector $( 3.4 \mathbf { i } - 12 \mathbf { j } )$ m at time $T$ seconds.
Find the value of $T$.
Find the speed of $P$.

Edexcel M1 2016 June Q7

11 marks Moderate -0.3

7. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle $P$. The force $\mathbf { F } _ { 1 }$ is given by $\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector $( \mathbf { i } + \mathbf { j } )$.
Given that the resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector ( $\mathbf { i } + 3 \mathbf { j }$ ),

find $\mathbf { F } _ { 2 }$ (7) The acceleration of $P$ is $( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. At time $t = 0$, the velocity of $P$ is $( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Find the speed of $P$ when $t = 3$ seconds.

Edexcel M1 2017 June Q7

14 marks Standard +0.3

7. [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 $O$.] Two ships, $P$ and $Q$, are moving with constant velocities.
The velocity of $P$ is $( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and the velocity of $Q$ is $( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$

Find the direction of motion of $P$, giving your answer as a bearing to the nearest degree. When $t = 0$, the position vector of $P$ is $( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }$ and the position vector of $Q$ is $( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vectors of $P$ and $Q$ are $\mathbf { p } \mathrm { km }$ and $\mathbf { q } \mathrm { km }$ respectively.
Find an expression for
1. $\mathbf { p }$ in terms of $t$,
2. $\mathbf { q }$ in terms of $t$.
Hence show that, at time $t$ hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
Find the values of $t$ when the ships are 10 km apart.

Edexcel M1 2018 June Q6

13 marks Moderate -0.3

6. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively] Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle $P$ of mass 0.5 kg . $\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$.
Given that the resultant force of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is in the same direction as $- 2 \mathbf { i } - \mathbf { j }$,

show that $p - 2 q = - 16$ Given that $q = 3$
find the magnitude of the acceleration of $P$,
find the direction of the acceleration of $P$, giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

Edexcel M1 2002 November Q2

7 marks Moderate -0.8

2. A particle $P$ of mass 1.5 kg is moving under the action of a constant force ( $3 \mathbf { i } - 7.5 \mathbf { j }$ ) N. Initially $P$ has velocity $( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find

the magnitude of the acceleration of $P$,
the velocity of $P$, in terms of $\mathbf { i }$ and $\mathbf { j }$, when $P$ has been moving for 4 seconds.

Edexcel M1 2002 November Q7

11 marks Moderate -0.8

7. Two helicopters $P$ and $Q$ are moving in the same horizontal plane. They are modelled as particles moving in straight lines with constant speeds. At noon $P$ is at the point with position vector $( 20 \mathbf { i } + 35 \mathbf { j } ) \mathrm { km }$ with respect to a fixed origin $O$. At time $t$ hours after noon the position vector of $P$ is $\mathbf { p } \mathrm { km }$. When $t = \frac { 1 } { 2 }$ the position vector of $P$ is $( 50 \mathbf { i } - 25 \mathbf { j } ) \mathrm { km }$. Find

the velocity of $P$ in the form $( a \mathbf { i } + b \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$,
an expression for $\mathbf { p }$ in terms of $t$. At noon $Q$ is at $O$ and at time $t$ hours after noon the position vector of $Q$ is $\mathbf { q } \mathrm { km }$. The velocity of $Q$ has magnitude $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ in the direction of $4 \mathbf { i } - 3 \mathbf { j }$. Find
(d) an expression for $\mathbf { q }$ in terms of $t$,
(e) the distance, to the nearest km , between $P$ and $Q$ when $t = 2$. \section*{8.} \section*{Figure 4}
\includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-6_695_1153_322_562}
Two particles $A$ and $B$, of mass $m \mathrm {~kg}$ and 3 kg respectively, are connected by a light inextensible string. The particle $A$ is held resting on a smooth fixed plane inclined at $30 ^ { \circ }$ to the horizontal. The string passes over a smooth pulley $P$ fixed at the top of the plane. The portion $A P$ of the string lies along a line of greatest slope of the plane and $B$ hangs freely from the pulley, as shown in Fig. 4. The system is released from rest with $B$ at a height of 0.25 m above horizontal ground. Immediately after release, $B$ descends with an acceleration of $\frac { 2 } { 5 } g$. Given that $A$ does not reach $P$, calculate
(a) the tension in the string while $B$ is descending,
(b) the value of $m$. The particle $B$ strikes the ground and does not rebound. Find
the magnitude of the impulse exerted by $B$ on the ground,
the time between the instant when $B$ strikes the ground and the instant when $A$ reaches its highest point.

Edexcel M1 2014 January Q2

6 marks Moderate -0.8

2. A particle $P$ is moving with constant velocity ( $2 \mathbf { i } - 3 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the speed of $P$. The particle $P$ passes through the point $A$ and 4 seconds later passes through the point with position vector ( $\mathbf { i } - 4 \mathbf { j }$ ) m.
Find the position vector of $A$.

Edexcel M1 2014 January Q7

12 marks Moderate -0.3

7. A force $\mathbf { F }$ is given by $\mathbf { F } = ( 9 \mathbf { i } + 13 \mathbf { j } )$ N.

Find the size of the angle between the direction of $\mathbf { F }$ and the vector $\mathbf { j }$. The force $\mathbf { F }$ is the resultant of two forces $\mathbf { P }$ and $\mathbf { Q }$. The line of action of $\mathbf { P }$ is parallel to the vector ( $2 \mathbf { i } - \mathbf { j }$ ). The line of action of $\mathbf { Q }$ is parallel to the vector ( $\mathbf { i } + 3 \mathbf { j }$ ).
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$,
1. the force $\mathbf { P }$,
2. the force $\mathbf { Q }$.

Edexcel M1 2017 January Q3

8 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-05_520_730_264_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two forces $\mathbf { P }$ and $\mathbf { Q }$ act on a particle at a point $O$. Force $\mathbf { P }$ has magnitude 6 N and force $\mathbf { Q }$ has magnitude 7 N . The angle between the line of action of $\mathbf { P }$ and the line of action of $\mathbf { Q }$ is $120 ^ { \circ }$, as shown in Figure 1. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is $\mathbf { R }$. Find

the magnitude of $\mathbf { R }$,
the angle between the line of action of $\mathbf { R }$ and the line of action of $\mathbf { P }$.

Edexcel M1 2018 January Q6

9 marks Moderate -0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors.]

A particle $P$ of mass 2 kg moves under the action of two forces, $( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }$ and $( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }$.

Find the magnitude of the acceleration of $P$. At time $t = 0 , P$ has velocity ( $- u \mathbf { i } + u \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$, where $u$ is a positive constant. At time $t = T$ seconds, $P$ has velocity $( 10 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Find
1. the value of $T$,
2. the value of $u$.

Edexcel M1 2020 January Q6

11 marks Moderate -0.8

6. A force $\mathbf { F }$ is given by $\mathbf { F } = ( 10 \mathbf { i } + \mathbf { j } ) \mathrm { N }$.

Find the exact value of the magnitude of $\mathbf { F }$.
Find, in degrees, the size of the angle between the direction of $\mathbf { F }$ and the direction of the vector $( \mathbf { i } + \mathbf { j } )$. The resultant of the force $\mathbf { F }$ and the force $( - 15 \mathbf { i } + a \mathbf { j } ) \mathrm { N }$, where $a$ is a constant, is parallel to, but in the opposite direction to, the vector $( 2 \mathbf { i } - 3 \mathbf { j } )$.
Find the value of $a$. \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-19_104_59_2613_1886}

Edexcel M1 2021 January Q5

7 marks Moderate -0.3

5. A particle is acted upon by two forces $\mathbf { F }$ and $\mathbf { G }$. The force $\mathbf { F }$ has magnitude 8 N and acts in a direction with a bearing of $240 ^ { \circ }$. The force $\mathbf { G }$ has magnitude 10 N and acts due South. Given that $\mathbf { R } = \mathbf { F } + \mathbf { G }$, find

the magnitude of $\mathbf { R }$,
the direction of $\mathbf { R }$, giving your answer as a bearing to the nearest degree. in a direction with a bearing of $240 ^ { \circ }$. The force $\mathbf { G }$ has magnitude 10 N and acts due South. Given that $\mathbf { R } = \mathbf { F } + \mathbf { G }$, find

Edexcel M1 2022 January Q8

14 marks Standard +0.3

8. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] A ship $A$ moves with constant velocity $( 3 \mathbf { i } - 10 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$ At time $t$ hours, the position vector of $A$ is $\mathbf { r } \mathrm { km }$.
At time $t = 0 , A$ is at the point with position vector $( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$.

Find $\mathbf { r }$ in terms of $t$. Another ship $B$ moves with constant velocity $( 15 \mathbf { i } + 14 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ At time $t = 0 , B$ is at the point with position vector $( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }$.
Show that, at time $t$ hours, $$\overrightarrow { A B } = [ ( 12 t - 10 ) \mathbf { i } + ( 24 t - 10 ) \mathbf { j } ] \mathrm { km }$$ Given that the two ships do not change course,
find the shortest distance between the two ships,
find the bearing of ship $B$ from ship $A$ when the ships are closest.

\includegraphics[max width=\textwidth, alt={}]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-28_2820_1967_102_100}

Edexcel M1 2023 January Q3

10 marks Moderate -0.8

A particle $P$ is moving with constant acceleration ( $- 4 \mathbf { i } + \mathbf { j }$ ) $\mathrm { ms } ^ { - 2 }$

At time $t = 0 , P$ has velocity $( 14 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$

Find the speed of $P$ at time $t = 2$ seconds.
Find the size of the angle between the direction of $\mathbf { i }$ and the direction of motion of $P$ at time $t = 2$ seconds. At time $t = T$ seconds, $P$ is moving in the direction of vector ( $2 \mathbf { i } - 3 \mathbf { j }$ )
Find the value of $T$

Edexcel M1 2024 January Q7

11 marks Standard +0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$.]

At midnight, a ship $S$ is at the point with position vector ( $19 \mathbf { i } + 22 \mathbf { j }$ )km
The ship travels with constant velocity $( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$

Find the speed of $S$. At time $t$ hours after midnight, the position vector of $S$ is $\mathbf { s } \mathrm { km }$.
Find an expression for $\mathbf { s }$ in terms of $\mathbf { i } , \mathbf { j }$ and $t$. A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector $( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }$. It is not safe for ships to be within 1.3 km of the lighthouse.
1. Find the value of $t$ when $S$ is closest to the lighthouse.
2. Hence determine whether it is safe for $S$ to continue its course.

Edexcel M1 2015 June Q7

5 marks Moderate -0.3

A particle $P$ moves from point $A$ to point $B$ with constant acceleration $( c \mathbf { i } + d \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$, where $c$ and $d$ are positive constants. The velocity of $P$ at $A$ is $( - 3 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $P$ at $B$ is $( 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The magnitude of the acceleration of $P$ is $2.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the value of $c$ and the value of $d$.

Edexcel M1 2024 June Q7