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PSLE Foundation Mathematics

<p>The curve $y=3\cos\left(x-\dfrac{\pi}{3}\right)$ meets the $y$-axis at $P$ and first meets the positive $x$-axis at $Q(b,\ 0)$. The chord $PQ$ is drawn.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q19-parts-item-1" data-list-item-path="amarea-q19-parts-item-1"><div class="list-item-paragraph">Find the exact value of $b$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q19-parts-item-2" data-list-item-path="amarea-q19-parts-item-2"><div class="list-item-paragraph">Find the equation of the chord $PQ$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q19-parts-item-3" data-list-item-path="amarea-q19-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve and the chord.<span class="inline-marks">[5]</span></div></li></ol>

The curve $y=3\cos\left(x-\dfrac{\pi}{3}\right)$ meets the $y$-axis at $P$ and first meets the positive $x$-axis at $Q(b,\ 0)$. The chord $PQ$ is drawn.

Hence find the exact area enclosed by the curve and the chord.

Area Bounded by a Curve and Straight Lines

Equation of a Straight Line

Shifted cosine curve and its chord

Integration

Coordinate Geometry

<p>The points $A$ and $B$ lie on the curve $y=8+2x-x^2$, where the $x$-coordinates of $A$ and $B$ are $-1$ and $3$, respectively.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q09-parts-item-1" data-list-item-path="amarea-q09-parts-item-1"><div class="list-item-paragraph">Find the equation of the chord $AB$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q09-parts-item-2" data-list-item-path="amarea-q09-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the chord.<span class="inline-marks">[4]</span></div></li></ol>

The points $A$ and $B$ lie on the curve $y=8+2x-x^2$, where the $x$-coordinates of $A$ and $B$ are $-1$ and $3$, respectively.

Hence find the area enclosed by the curve and the chord.

Area between a parabola and a chord

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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

Midpoint of a Line Segment

Perpendicular Bisectors

Perpendicular bisector and the third vertex of a triangle

<p>The diagram shows an isosceles triangle $PQR$, where&nbsp;$PQ=QR$ and the line segment $QR$ is parallel to the $y$-axis. Point $P$ lies on the $y$-axis and the equation of the line $PR$ is $x+3y=42$. Point $Q$ is $(-9,\ k)$, where&nbsp;$k$ is a positive constant.</p>
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<ol class="abc-list"><li data-list-item-id="cg-q11-a" data-list-item-path="cg-q11-a"><div class="list-item-paragraph">Find the coordinates of $P$, $Q$ and $R$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q11-b" data-list-item-path="cg-q11-b"><div class="list-item-paragraph">Given that the point $S$ is such that&nbsp;$PQRS$ is a kite and the line segment $RS$ is parallel to the $x$-axis, find the coordinates of $S$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q11-c" data-list-item-path="cg-q11-c"><div class="list-item-paragraph">Find the area of the kite $PQRS$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle $PQR$, where $PQ=QR$ and the line segment $QR$ is parallel to the $y$-axis. Point $P$ lies on the $y$-axis and the equation of the line $PR$ is $x+3y=42$. Point $Q$ is $(-9,\ k)$, where $k$ is a positive constant.

Find the coordinates of $P$, $Q$ and $R$.

Given that the point $S$ is such that $PQRS$ is a kite and the line segment $RS$ is parallel to the $x$-axis, find the coordinates of $S$.

Length of a Line Segment

Isosceles triangle extended to a kite

<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>
<div class="graph-block" data-graph-config="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" data-width="397" data-height="349" data-proportional style="width:397px;height:349px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:397px;--graph-aspect:397 / 349;"></div>
<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>
<div class="graph-block" data-graph-config="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" data-width="395" data-height="250" data-proportional style="width:395px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:395px;--graph-aspect:395 / 250;"></div>
<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>
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<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>
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<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 $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q04-i" data-list-item-path="cg-q04-i"><div class="list-item-paragraph">the coordinates of $B$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-ii" data-list-item-path="cg-q04-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-iii" data-list-item-path="cg-q04-iii"><div class="list-item-paragraph">the coordinates of $A$, $C$ and $D$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q04-iv" data-list-item-path="cg-q04-iv"><div class="list-item-paragraph">the length of $BC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-v" data-list-item-path="cg-q04-v"><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 $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.

the coordinates of the midpoint of $AC$,

the coordinates of $A$, $C$ and $D$,

Rectangle with a diagonal along the x-axis

<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>
<div class="graph-block" data-graph-config="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" data-width="380" data-height="250" data-proportional style="width:380px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:380px;--graph-aspect:380 / 250;"></div>
<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 $y=x+1$.

Rectangle from a vertex pair and a diagonal

<p>The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="200" data-proportional style="width:400px;height:200px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:400px;--graph-aspect:400 / 200;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q01-i" data-list-item-path="cg-q01-i"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q01-ii" data-list-item-path="cg-q01-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iii" data-list-item-path="cg-q01-iii"><div class="list-item-paragraph">the gradient of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iv" data-list-item-path="cg-q01-iv"><div class="list-item-paragraph">the equation of $AB$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.

the coordinates of the midpoint of $AB$,

Gradient of a Straight Line

Length, midpoint, gradient and equation from two points

<p>A straight line passes through $P(-2,-1)$ and $Q(3,3)$. Find its equation in the form $ax+by=c$, where&nbsp;$a$, $b$ and $c$ are integers.</p>

A straight line passes through $P(-2,-1)$ and $Q(3,3)$. Find its equation in the form $ax+by=c$, where $a$, $b$ and $c$ are integers.

Sec 3/4 E Math T04-Q02 - Finding the equation of a straight line

<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>It is given that&nbsp;$y = \ln \left(\frac{2x}{5 - x} \right)$, where&nbsp;$0 &lt; x &lt; 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">If&nbsp;$y$ is decreasing at a rate of $10$ units/s, find the rate of change of $x$ when&nbsp;$y = \ln 8$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">The equation of the normal to the curve at point $A$ is $8y = 8\ln 2 + 25 - 10x$. Find the $x$-coordinate of point $A$.<span class="inline-marks">[4]</span></div></li></ol>

It is given that $y = \ln \left(\frac{2x}{5 - x} \right)$, where $0 < x < 5$.

Derivatives of Natural Logarithmic Functions

Connected Rates of Change

2024 CGSS P1 Q9

<p>The diagram shows a line $DE$ which cuts the $y$-axis at $P$ and a line through $F$ meets the $y$-axis at $Q$ such that angle $FPQ =$ angle $FQP$.</p>
<p>The coordinates of $D$ and $E$ are $(-4, 0)$ and $(10, 7)$ respectively.</p>
<p>$F$ is a point on the line $DE$ such that&nbsp;$DF : FE = 4 : 3$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782712311122-rmouexzadr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MTIzMTExMjItcm1vdWV4emFkci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODA1LCJleHAiOjE3OTI2OTYxMDV9.GGZ64634p3-jROS3yhgJwuQ3a5pDvnsTciIBlO8gazg" style="max-width: min(380px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find the coordinates of $F$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Find the equation of the straight line $FQ$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the area of the triangle $QFE$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a line $DE$ which cuts the $y$-axis at $P$ and a line through $F$ meets the $y$-axis at $Q$ such that angle $FPQ =$ angle $FQP$.

The coordinates of $D$ and $E$ are $(-4, 0)$ and $(10, 7)$ respectively.

$F$ is a point on the line $DE$ such that $DF : FE = 4 : 3$.

2024 CGSS P2 Q5

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782643154194-60z17dgtnt8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI2NDMxNTQxOTQtNjB6MTdkZ3RudDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTgwNSwiZXhwIjoxNzkyNjk2MTA1fQ.e1pyQCw5Y6SSlrC0FoCpxHE_vCGzrs7u_IvzoEuTs4M" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a triangle $PQR$ with vertices $P(9,10)$ and $Q(1,-2)$. The point $R$ lies on the perpendicular bisector of $PQ$ and the equation of the line $QR$ is $y = 8x - 10$.</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">Find the equation of the perpendicular bisector of $PQ$.</span><span class="inline-marks">[4]</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">Find the coordinates of $R$.</span><span class="inline-marks">[2]</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">Find the area of triangle $PQR$.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $PQR$ with vertices $P(9,10)$ and $Q(1,-2)$. The point $R$ lies on the perpendicular bisector of $PQ$ and the equation of the line $QR$ is $y=8x-10$.

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

2025 DSS PRELIMS P1 Q12

<p>The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.<br>Find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">the equation of the straight line in the form of $y = mx + c$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">the coordinates of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">the exact area of triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770806346981-qa5ypeh48l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDYzNDY5ODEtcWE1eXBlaDQ4bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODA1LCJleHAiOjE3OTI2OTYxMDV9.i1L_AVUOoGROzKXCMh7RjRmjaGPiiO1UXOY1hRF5BZI" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.
Find

Intersection of a Line and a Curve

Nature of Roots of Quadratic Equations

IP3 Class Test (May 2025) Bonus Question

Quadratic Functions, Equations and Inequalities

<p>The point $P$ has the coordinates of $(-3, 8)$ and the line joining the points $P$ and $Q$ has a gradient of $-\frac{3}{4}$.</p>
<p>The perpendicular bisector of $PQ$ passes through the point $(8, 6)$.</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the equation of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the coordinates of $Q$.</span></div></li></ol>

The point $P$ has the coordinates of $(-3, 8)$ and the line joining the points $P$ and $Q$ has a gradient of $-\frac{3}{4}$.

The perpendicular bisector of $PQ$ passes through the point $(8, 6)$.

<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 $3x+5y-21=0$.

Math Question Bank