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PSLE

PSLE Foundation Mathematics

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<p>The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.

Find the equation of the perpendicular bisector of $AB$.

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Intersection of Two Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular Bisectors

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<p>The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q10-i" data-list-item-path="cg-q10-i"><div class="list-item-paragraph">Given that&nbsp;$k&lt;5$, find the value of $k$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-ii" data-list-item-path="cg-q10-ii"><div class="list-item-paragraph">Find the coordinates of $P$ and show that&nbsp;$P$ is the midpoint of $AB$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-iii" data-list-item-path="cg-q10-iii"><div class="list-item-paragraph">Hence find the coordinates of $Q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-iv" data-list-item-path="cg-q10-iv"><div class="list-item-paragraph">The point $D$ is such that&nbsp;$ABCD$ is a rectangle. Find the coordinates of $D$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-v" data-list-item-path="cg-q10-v"><div class="list-item-paragraph">Find the area of the rectangle $ABCD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-vi" data-list-item-path="cg-q10-vi"><div class="list-item-paragraph">The point $F$ lies on $BC$ such that&nbsp;$BF : FC = 1 : 3$. Find the coordinates of $F$.<span class="inline-marks">[2]</span></div></li></ol>

The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.

Given that $k<5$, find the value of $k$.

Find the coordinates of $P$ and show that $P$ is the midpoint of $AB$.

The point $D$ is such that $ABCD$ is a rectangle. Find the coordinates of $D$.

The point $F$ lies on $BC$ such that $BF : FC = 1 : 3$. Find the coordinates of $F$.

Ratio Theorem and Section Formula

Right-angled triangle completed to a rectangle

<p>The diagram shows a trapezium $ABCD$ in which $A$ is $(1,\ 2)$, $B$ is $(5,\ 10)$ and $D$ is $(9,\ 3)$. Angle $ABC$ is $90^\circ$ and $AB$ is parallel to $DC$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q09-i" data-list-item-path="cg-q09-i"><div class="list-item-paragraph">Find the equation of $CD$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-ii" data-list-item-path="cg-q09-ii"><div class="list-item-paragraph">Find the equation of $BC$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iii" data-list-item-path="cg-q09-iii"><div class="list-item-paragraph">Hence find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iv" data-list-item-path="cg-q09-iv"><div class="list-item-paragraph">Find the area of the trapezium $ABCD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q09-v" data-list-item-path="cg-q09-v"><div class="list-item-paragraph">The point $E$ lies on $AC$ such that&nbsp;$AE : EC = 3 : 2$. Find the coordinates of $E$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a trapezium $ABCD$ in which $A$ is $(1,\ 2)$, $B$ is $(5,\ 10)$ and $D$ is $(9,\ 3)$. Angle $ABC$ is $90^\circ$ and $AB$ is parallel to $DC$.

The point $E$ lies on $AC$ such that $AE : EC = 3 : 2$. Find the coordinates of $E$.

Right-angled trapezium and a point dividing a diagonal

<p>$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="250" data-proportional style="width:400px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:400px;--graph-aspect:400 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q07-i" data-list-item-path="cg-q07-i"><div class="list-item-paragraph">Find the equations of $AC$ and $BD$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-ii" data-list-item-path="cg-q07-ii"><div class="list-item-paragraph">Calculate the coordinates of $T$, $C$ and $D$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-iii" data-list-item-path="cg-q07-iii"><div class="list-item-paragraph">Find the area of the rhombus $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.

Calculate the coordinates of $T$, $C$ and $D$.

Diagonals of a rhombus

<p>The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="250" data-proportional style="width:440px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:440px;--graph-aspect:440 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q06-i" data-list-item-path="cg-q06-i"><div class="list-item-paragraph">It is given that&nbsp;$B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.</div><ol class="abc-list"><li data-list-item-id="cg-q06-i-a" data-list-item-path="cg-q06-i/cg-q06-i-a"><div class="list-item-paragraph">Find the coordinates of $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q06-i-b" data-list-item-path="cg-q06-i/cg-q06-i-b"><div class="list-item-paragraph">Find the equation of the diagonal $BD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q06-i-c" data-list-item-path="cg-q06-i/cg-q06-i-c"><div class="list-item-paragraph">Hence find the coordinates of $T$ and of $D$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="cg-q06-ii" data-list-item-path="cg-q06-ii"><div class="list-item-paragraph">Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.

It is given that $B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.

Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.

Kite with a side parallel to the x-axis

<p>The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="280" style="width:440px;height:280px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q05-i" data-list-item-path="cg-q05-i"><div class="list-item-paragraph">the coordinates of $M$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q05-ii" data-list-item-path="cg-q05-ii"><div class="list-item-paragraph">the equation of the perpendicular bisector of $AC$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q05-iii" data-list-item-path="cg-q05-iii"><div class="list-item-paragraph">the value of $a$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q05-iv" data-list-item-path="cg-q05-iv"><div class="list-item-paragraph">the area of the triangle $ABC$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.

the equation of the perpendicular bisector of $AC$,

Perpendicular bisector of a side of a triangle

<p>The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q03-i" data-list-item-path="cg-q03-i"><div class="list-item-paragraph">the equation of $CD$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q03-ii" data-list-item-path="cg-q03-ii"><div class="list-item-paragraph">the coordinates of $C$ and of $B$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q03-iii" data-list-item-path="cg-q03-iii"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q03-iv" data-list-item-path="cg-q03-iv"><div class="list-item-paragraph">the area of the rectangle $ABCD$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.

Length of a Line Segment

Rectangle from a vertex pair and a diagonal

<p>In the diagram, the points $P(-4,\ -3)$, $Q(1,\ 9)$ and $R(13,\ 4)$ form a triangle.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" data-proportional style="width:300px;height:250px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q02-i" data-list-item-path="cg-q02-i"><div class="list-item-paragraph">Show that&nbsp;$PQ$ is perpendicular to $QR$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q02-ii" data-list-item-path="cg-q02-ii"><div class="list-item-paragraph">Calculate the area of the triangle $PQR$.<span class="inline-marks">[3]</span></div></li></ol>

In the diagram, the points $P(-4,\ -3)$, $Q(1,\ 9)$ and $R(13,\ 4)$ form a triangle.

Show that $PQ$ is perpendicular to $QR$.

Calculate the area of the triangle $PQR$.

Perpendicular sides and the area of a triangle

<p>The equation of a curve is $y = ax^3 + b$, where&nbsp;$a$ and $b$ are constants. The equation of the normal to the curve at the point where&nbsp;$x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.<span class="inline-marks">[6]</span></p>

The equation of a curve is $y = ax^3 + b$, where $a$ and $b$ are constants. The equation of the normal to the curve at the point where $x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.

Tangents and Normals at a Point

2024 FMSS PRELIMS P2 Q2

Differentiation

<p>The line passing through the points $P(-2, 5)$ and $Q(q, 8)$ is parallel to the line $3x + 4y + 4 = 0$.</p>
<p>Find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">value of $q$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">distance $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">equation of the perpendicular bisector of $PQ$.</span></div></li></ol>

The line passing through the points $P(-2, 5)$ and $Q(q, 8)$ is parallel to the line $3x + 4y + 4 = 0$.

equation of the perpendicular bisector of $PQ$.

<p>Find the equation of the line that bisects the line joining $A\left( 14,-10 \right)$ and $B\left( 3,-7 \right)$ and is parallel to the line $3x+5y-21=0$.</p>

Find the equation of the line that bisects the line joining $A\left( 14,-10 \right)$ and $B\left( 3,-7 \right)$ and is parallel to the line $3x+5y-21=0$.

<p>Find the line equation which is parallel to the line $y+x=5$ and meets the line $4y=11x-6$ on the $y$-axis. </p>

Find the line equation which is parallel to the line $y+x=5$ and meets the line $4y=11x-6$ on the $y$-axis. 

<p>The line $y=mx+c$ is perpendicular to the line $2y+5x=1$ and meets the line $-2y+5x=12$ on the $x$-axis. Find the values of $m$ and $c$.</p>

The line $y=mx+c$ is perpendicular to the line $2y+5x=1$ and meets the line $-2y+5x=12$ on the $x$-axis. Find the values of $m$ and $c$.

<p>The line $y=mx+c$ is perpendicular to the line $y-3x=2$ and meets the line $3y+x=24$ on the $x$-axis. Find the values of $m$ and $c$.</p>

The line $y=mx+c$ is perpendicular to the line $y-3x=2$ and meets the line $3y+x=24$ on the $x$-axis. Find the values of $m$ and $c$.

<p>The vertices of triangle $ABC$ are $A(8,-5)$, $B(11,-4)$ and $C(1,k)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ if angle $ACB$ is $90$ degrees,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area of triangle $ABC$.</span></div></li></ol>

The vertices of triangle $ABC$ are $A(8,-5)$, $B(11,-4)$ and $C(1,k)$. Find

the value of $k$ if angle $ACB$ is $90$ degrees,

<p>Find the equation of the line passing through the midpoint of $A\left( -2,5 \right)$ and $B\left( 8,-10 \right)$, and parallel to the line joining the points $C\left( 2,-7 \right)$ and $D\left( 0,-5 \right)$.</p>

Find the equation of the line passing through the midpoint of $A\left( -2,5 \right)$ and $B\left( 8,-10 \right)$, and parallel to the line joining the points $C\left( 2,-7 \right)$ and $D\left( 0,-5 \right)$.

<p>Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $m$ if&nbsp;$PQ$ is perpendicular to $PR$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $PQ$ and $PR$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $PQR$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find

the value of $m$ if $PQ$ is perpendicular to $PR$,

the equation of the perpendicular bisector of $PQ$,

<p>Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ if&nbsp;$AC$ is perpendicular to $BC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $AB$ and $BC$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $ABC$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find

the value of $k$ if $AC$ is perpendicular to $BC$,

the equation of the perpendicular bisector of $AB$,

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></p>
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<p>In the diagram above, we have a trapezium $ABCD$.</p>
<p>The line $BC$ has equation $2y + 8x = 6$ and $D$ has coordinates $(-4, 0)$. $B$ cuts the $y$-axis.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$DC$ is perpendicular to $BC$, find $C$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find $A$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Calculate the area of $ABCD$.</div></li></ol>

**Solutions to this question by accurate drawing will not be accepted.**

In the diagram above, we have a trapezium $ABCD$.

The line $BC$ has equation $2y + 8x = 6$ and $D$ has coordinates $(-4, 0)$. $B$ cuts the $y$-axis.

Given that $DC$ is perpendicular to $BC$, find $C$.

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></p>
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<p>The diagram shows a rhombus $ABCD$ with vertices $A$ and $C$ at the points $(0, 2)$ and $(-6, 4)$ respectively. $B$ lies on the $x$-axis and the line $DC$ produced intersects with the $x$-axis at $E$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of $B$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Explain if&nbsp;$ABCD$ is a square.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the coordinates of $E$.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the area of $ABED$.</div></li></ol>

The diagram shows a rhombus $ABCD$ with vertices $A$ and $C$ at the points $(0, 2)$ and $(-6, 4)$ respectively. $B$ lies on the $x$-axis and the line $DC$ produced intersects with the $x$-axis at $E$.

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