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PSLE

PSLE Foundation Mathematics

<p>At $1800$ hours, a boat sails north-west with a speed of $8\sqrt{2}$ km/h from point $P$ and travels to point $Q$ with position vector $\begin{pmatrix} 10 \\ -12 \end{pmatrix}$ km. Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">the velocity vector of the boat.</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">the initial position vector at point $P$ if the boat reaches point $Q$ in $7.5$ minutes.</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">the position vector of the boat at $9.30$ p.m.</span></div></li></ol>

At $1800$ hours, a boat sails north-west with a speed of $8\sqrt{2}$ km/h from point $P$ and travels to point $Q$ with position vector $\begin{pmatrix} 10 \\ -12 \end{pmatrix}$ km. Find

the initial position vector at point $P$ if the boat reaches point $Q$ in $7.5$ minutes.

the position vector of the boat at $9.30$ p.m.

Vector Kinematics

Vectors

<p>Application of Vectors: Motion</p>
<p>A rocket was launched at a fireworks display. At time $t$ seconds, the position of the rockets, in metres relative to the launch location (let this be the origin $O$), was $\mathbf{r}=20t\mathbf{i}+\left[ {{(t-4)}^{3}}+64 \right]\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertical unit vectors respectively. The rocket exploded when $t=4$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down the height of the rocket when it exploded.</div></li><li><div class="list-item-paragraph">Find the displacement of the rocket from $t=3$ to $t=4$.</div></li><li><div class="list-item-paragraph">Show that the Cartesian equation representing the path of the rocket was $y={{\left( \frac{x}{20}-4 \right)}^{3}}+64$ and sketch the path of the rocket. Hence state the direction of movement of the rocket when it exploded.</div></li></ol>

A rocket was launched at a fireworks display. At time $t$ seconds, the position of the rockets, in metres relative to the launch location (let this be the origin $O$), was $\mathbf{r}=20t\mathbf{i}+\left[ {{(t-4)}^{3}}+64 \right]\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertical unit vectors respectively. The rocket exploded when $t=4$.

Sketching Parametric Curves

Parametric Equations

Converting Parametric Equations to Cartesian Form

AS Mathematics, Berry J., Fentem R., Francis B., Graham T., Harper Collins 2000, Ex 24. P1 Q6 [Modified]

Graphing Techniques

<p>As part of a structure for a marble race, a plane $p$ is placed at an angle $\theta $ to the horizontal ground. The ground is taken to be the $x-y$ plane and the line of intersection between $p$ and the ground is the $y$-axis (see diagram).</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761360109678-3igsmzvuj9l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjAxMDk2NzgtM2lnc216dnVqOWwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMiwiZXhwIjoxNzkzNTA1MTMyfQ.EfHb9t9vKowkPdpQZq1hgfxDYvVUh6E9BvbhVflGtag" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Given that a point $A$ with coordinates $\left( 4,1,2 \right)$ lies on $p$, find a vector equation of $p$ in the scalar product form. Hence find the value of $\theta $, leaving your answer in degrees correct to one decimal place.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">A straight rod is attached to $p$ from $A$ to a point $B$ on the $y$-axis, as shown in the diagram. Given that $AB=6$ units, find the position vector of $B$.</div><p style="text-align: right;">[4]</p></li></ol>
<p>A marble is released from rest at $A$ and it moves along the rod $AB$ towards $B$. It is observed that at time $t$ seconds after the marble is released, the height of the marble above the ground is $2-k{{t}^{2}}$ units where $k$ is a positive constant less than $1$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find in terms of $k$,</div><ol class="abc-list"><li><div class="list-item-paragraph">the time taken for the marble to reach $B$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">the coordinates of marble when $t=0.5$.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

As part of a structure for a marble race, a plane $p$ is placed at an angle $\theta $ to the horizontal ground. The ground is taken to be the $x-y$ plane and the line of intersection between $p$ and the ground is the $y$-axis (see diagram).

A marble is released from rest at $A$ and it moves along the rod $AB$ towards $B$. It is observed that at time $t$ seconds after the marble is released, the height of the marble above the ground is $2-k{{t}^{2}}$ units where $k$ is a positive constant less than $1$.

Magnitude of a Vector

Three-Dimensional Geometry

Equations of Planes

Angle Between Two Planes

2022 TJC Promo Q12

<p>Coordinates axes $Oxyz$ are set up with the origin $O$ at the base of an airport control tower. The $x$-axis is due East, the $y$-axis due North and the $z$-axis vertical. The units of distances are kilometres. An airplane $A$ takes off from the point $X$. For the first $4$ minutes, the position vector of $A$ at time $t$ minutes after take-off, is given by</p>
<p style="text-align: center">$\mathbf{r}=\left( 2+t \right)\mathbf{i}+\left( 1+2t \right)\mathbf{j}+3t\mathbf{k}$,  $0\le t\le 4$.</p>
<ol class="i-list"><li data-list-item-id="item-1761358551577-2lu7xrtly" data-list-item-path="item-1761358551577-2lu7xrtly"><div class="list-item-paragraph">State the coordinates of $X$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1761358552298-u10get01e" data-list-item-path="item-1761358552298-u10get01e"><div class="list-item-paragraph">Find the acute angle the flight path makes with the horizontal.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761358553775-dg9rsj706" data-list-item-path="item-1761358553775-dg9rsj706"><div class="list-item-paragraph">The airplane enters a cloud at a height of $5$ km. Find the coordinates of the point where it enters the cloud.<span class="inline-marks">[2]</span></div></li></ol>
<p>A second airplane $B$ takes off from the point $\left( -2,-1,0 \right)$ at the same time as the first airplane $A$ and is traveling at a constant speed in a straight line for the first $4$ minutes. Two minutes after take-off, $B$ is at the point $\left( 1,5,\alpha \right)$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1761358579515-8hq68d133" data-list-item-path="item-1761358579515-8hq68d133"><div class="list-item-paragraph">Find in terms of $\alpha $, the position vector of $B$ after $t$ minutes where&nbsp;$0\le t\le 4$. Explain if it is possible for the two airplanes to collide in the first $4$ minutes.<span class="inline-marks">[4]</span></div></li></ol>

Coordinates axes $Oxyz$ are set up with the origin $O$ at the base of an airport control tower. The $x$-axis is due East, the $y$-axis due North and the $z$-axis vertical. The units of distances are kilometres. An airplane $A$ takes off from the point $X$. For the first $4$ minutes, the position vector of $A$ at time $t$ minutes after take-off, is given by

$\mathbf{r}=\left( 2+t \right)\mathbf{i}+\left( 1+2t \right)\mathbf{j}+3t\mathbf{k}$,  $0\le t\le 4$.

A second airplane $B$ takes off from the point $\left( -2,-1,0 \right)$ at the same time as the first airplane $A$ and is traveling at a constant speed in a straight line for the first $4$ minutes. Two minutes after take-off, $B$ is at the point $\left( 1,5,\alpha \right)$.

Angle Between a Line and a Plane

Intersection of a Line and a Plane

2021 TJC P1 Q11

<p>In this question, the distance is measured in metres and time is in seconds. </p>
<p>A radio-controlled airplane takes off from ground level and is assumed to be travelling at a steady speed in a straight line. The position vector of the airplane $t$ seconds after it takes off is given by $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where&nbsp;$\mathbf{a}$ refers to the position vector of the point where it departs and $\mathbf{b}$ is known as its velocity vector.</p>
<ol class="i-list"><li data-list-item-id="item-1761356496487-2h4j6nfbx" data-list-item-path="item-1761356496487-2h4j6nfbx"><div class="list-item-paragraph">Given that the airplane reaches the point $P$ with coordinates $\left( -9,\,\,4,\,\,6 \right)$ after $6$ seconds and its velocity vector is $-2\mathbf{i}+\mathbf{j}+\mathbf{k}$, find the coordinates of the point where it departs.<span class="inline-marks">[1]</span></div></li></ol>
<p>Paul stands at the point $C$ with coordinates $\left( -7,\,\,5,\,\,2 \right)$ to observe the airplane.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1761356516947-fhyxvmo11" data-list-item-path="item-1761356516947-fhyxvmo11"><div class="list-item-paragraph">Find the shortest distance from Paul’s position to the flight path of the airplane.<span class="inline-marks">[3]</span></div></li></ol>
<p>While the airplane is in the air, a drone is seen flying at a steady speed in a straight line with equation</p>
<p style="text-align: center">$\frac{2x-1}{-6}=\frac{y+7}{4}=\frac{z-10}{k}$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1761356540309-1rbrnn3jh" data-list-item-path="item-1761356540309-1rbrnn3jh"><div class="list-item-paragraph">Show that the equation of the flight path of the drone can be written as&nbsp;$\mathbf{r}=\mathbf{p}+\lambda \mathbf{q}$, where&nbsp;$\lambda $ is a real parameter and $\mathbf{p}$ and $\mathbf{q}$ are vectors to be determined, leaving your answer in terms of $k$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1761356575432-rf6bb2k14" data-list-item-path="item-1761356575432-rf6bb2k14"><div class="list-item-paragraph">Given that the flight paths of the radio-controlled airplane and the drone intersect, find $k$.<span class="inline-marks">[3]</span></div></li></ol>
<p>At $P$, the airplane suddenly changes its speed and direction. The position vector of the airplane $s$ seconds after it leaves $P$ is given by</p>
<p>\[\mathbf{r} = \begin{pmatrix} -9 \\ 4 \\ 6 \end{pmatrix} + s \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}, \text{ where}\hspace{0.5em} s \in \mathbb{R}, s \geq 0.\]</p>
<p>It travels at a steady speed in a straight line towards an inclined slope, which is assumed to be a plane with equation</p>
<p style="text-align: center">$x-y+7z=2$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1761356744994-u7j0w247r" data-list-item-path="item-1761356744994-u7j0w247r"><div class="list-item-paragraph">Determine if the new flight path is perpendicular to the inclined slope.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761356746400-7m17e0any" data-list-item-path="item-1761356746400-7m17e0any"><div class="list-item-paragraph">The airplane eventually collides with the slope. Find the coordinates of the point of collision.<span class="inline-marks">[3]</span></div></li></ol>
<p>TGM correction to part (iii): the symmetric equation describes a full line, so use $\lambda\in\mathbb R$. The source paper says “non-negative constant”, but gives no starting point or direction that would restrict the drone to a ray. The parameter is not assumed to be elapsed time.</p>

In this question, the distance is measured in metres and time is in seconds. 

A radio-controlled airplane takes off from ground level and is assumed to be travelling at a steady speed in a straight line. The position vector of the airplane $t$ seconds after it takes off is given by $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where $\mathbf{a}$ refers to the position vector of the point where it departs and $\mathbf{b}$ is known as its velocity vector.

Paul stands at the point $C$ with coordinates $\left( -7,\,\,5,\,\,2 \right)$ to observe the airplane.

While the airplane is in the air, a drone is seen flying at a steady speed in a straight line with equation

$\frac{2x-1}{-6}=\frac{y+7}{4}=\frac{z-10}{k}$.

At $P$, the airplane suddenly changes its speed and direction. The position vector of the airplane $s$ seconds after it leaves $P$ is given by

\[\mathbf{r} = \begin{pmatrix} -9 \\ 4 \\ 6 \end{pmatrix} + s \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}, \text{ where } s \in \mathbb{R}, s \geq 0.\]

It travels at a steady speed in a straight line towards an inclined slope, which is assumed to be a plane with equation

TGM correction to part (iii): the symmetric equation describes a full line, so use $\lambda\in\mathbb R$. The source paper says “non-negative constant”, but gives no starting point or direction that would restrict the drone to a ray. The parameter is not assumed to be elapsed time.

Intersection of Two Lines

Equations of Lines

Relationships Between a Line and a Plane

Distance from a Point to a Line

2019 MI P1 Q11

<p>Distances in this question are in metres.</p>
<p>Harry and Tom’s model airplanes are taking off from the horizontal ground, which is the $x-y$ plane. Tom’s airplane takes off after Harry’s. The position of Harry’s airplane $t$ seconds after it takes off is given by $\mathbf{r}=\left( 5\mathbf{i}+6\mathbf{j} \right)+t\left( -4\mathbf{i}+2\mathbf{j}+4\mathbf{k} \right)$. The position of Tom’s airplane $s$ seconds after it takes off is given by $\mathbf{r}=\left( -39\mathbf{i}+44\mathbf{j} \right)+s\left( 4\mathbf{i}-6\mathbf{j}+7\mathbf{k} \right)$.<br></br></p>
<p>Extension question:</p>
<p>Interpret the geometrical meaning of the vector $-4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ in the equation of the position of Harry's airplane.<br></br></p>
<ol class="i-list"><li data-list-item-id="item-1761307969308-gcwgecob2" data-list-item-path="item-1761307969308-gcwgecob2"><div class="list-item-paragraph">State the height of Harry’s airplane two seconds after it takes off and find its distance travelled in the two seconds.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761307969878-ut9fdfflx" data-list-item-path="item-1761307969878-ut9fdfflx"><div class="list-item-paragraph">Find the acute angle between the path of Harry’s airplane and the ground.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761307970793-o0iaoi0t0" data-list-item-path="item-1761307970793-o0iaoi0t0"><div class="list-item-paragraph">Show that the paths of the airplanes are perpendicular.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1761307971881-l5ykk5giy" data-list-item-path="item-1761307971881-l5ykk5giy"><div class="list-item-paragraph">Given that the two airplanes collide, find the coordinates of the point of collision. How long after Harry’s airplane takes off does Tom’s airplane take off?<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761307972766-oirioc0w4" data-list-item-path="item-1761307972766-oirioc0w4"><div class="list-item-paragraph">Find the cartesian equation of the plane in which both paths of the airplanes lie.<span class="inline-marks">[3]</span></div></li></ol>

Distances in this question are in metres.

Harry and Tom’s model airplanes are taking off from the horizontal ground, which is the $x-y$ plane. Tom’s airplane takes off after Harry’s. The position of Harry’s airplane $t$ seconds after it takes off is given by $\mathbf{r}=\left( 5\mathbf{i}+6\mathbf{j} \right)+t\left( -4\mathbf{i}+2\mathbf{j}+4\mathbf{k} \right)$. The position of Tom’s airplane $s$ seconds after it takes off is given by $\mathbf{r}=\left( -39\mathbf{i}+44\mathbf{j} \right)+s\left( 4\mathbf{i}-6\mathbf{j}+7\mathbf{k} \right)$.<br></br>

Interpret the geometrical meaning of the vector $-4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ in the equation of the position of Harry's airplane.<br></br>

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