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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

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<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 $b$.

Applications of Differentiation — General

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 $0 < x < 5$.

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

2024 CGSS P1 Q9

<p>A circle has a diameter $AB$. The point $A$ has coordinates $(1, -6)$ and the equation of the tangent to the circle at $B$ is $3x + 4y = k$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that the equation of the normal to the circle at the point $A$ is $4x - 3y = 22$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is also given also that the line $x = -1$ touches the circle at the point $(-1, -2)$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Find the coordinates of the centre and the radius of the circle.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the value of $k$.<span class="inline-marks">[3]</span></div></li></ol>

A circle has a diameter $AB$. The point $A$ has coordinates $(1, -6)$ and the equation of the tangent to the circle at $B$ is $3x + 4y = k$.

It is also given also that the line $x = -1$ touches the circle at the point $(-1, -2)$.

Problems Relating to Circle

2024 CGSS P2 Q8

<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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MTIzMTExMjItcm1vdWV4emFkci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzA3OTMyLCJleHAiOjE3ODU3MDgyMzJ9.eXmDOu3uw6HzwVPWufd7s1_guULKRukcMXVl8E23Qus" 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

<p>Two points $ A(-3,-6) $ and $ B (7,-6) $ lie on a circle centre $ G$. The equation of the tangent to the circle at the point $ Q(1,-12) $ is $x+5y+59=0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773467964823-vim5kux87"><div class="list-item-paragraph">Show that the coordinates of $G$ is $ (2,-7) $<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1773467967295-xi8wvogbp"><div class="list-item-paragraph">Find the equation of the circle in the form of $x^2+y^2+2gx+2fy+c=0$, stating the value of $f$, $g$ and  $c$.<span class="inline-marks">[3]</span></div></li></ol>

Two points $ A(-3,-6) $ and $ B (7,-6) $ lie on a circle centre $ G$. The equation of the tangent to the circle at the point $ Q(1,-12) $ is $x+5y+59=0$.

Further Coordinate Geometry — General

2025 MSHS PRELIMS P2 Q7

<p>The equation of a curve is $y=\frac{ \ln x}{x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773466434665-oon3h3sb1"><div class="list-item-paragraph">Find the coordinates of the stationary point of this curve. <br>Leave your answer in terms of e.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773466436825-f0ta69f8n"><div class="list-item-paragraph">Determine the nature of the stationary point.<span class="inline-marks">[3]</span></div></li></ol>

The equation of a curve is $y=\frac{ \ln x}{x}$.

2025 MSHS PRELIMS P2 Q2

<p>The curve $(x-1)^2+(y-3)^2 =5$ intersects the line $y-3x=5$ at two points. Find the coordinates of these two points.<span class="inline-marks">[5]</span></p>

The curve $(x-1)^2+(y-3)^2 =5$ intersects the line $y-3x=5$ at two points. Find the coordinates of these two points.

2025 MSHS PRELIMS P1 Q1

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></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/1772948612821-4pgfoodk0n5.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzI5NDg2MTI4MjEtNHBnZm9vZGswbjUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwNzkzMiwiZXhwIjoxNzg1NzA4MjMyfQ.x53vPKaxwJikGftybbfUJA1jyH9px_Y3K_Uzi1qyD0M" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a quadrilateral $OABC$ where&nbsp;$O$ is the origin. The point $A$ is $(-1, 5)$ and the point $B$ is $(3, 7)$. The perpendicular bisector of $AB$ meets the $x$-axis at $C$.</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 coordinates of $C$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1772948560928-c4qekwukf" data-list-item-path="item-1772948560928-c4qekwukf"><div class="list-item-paragraph">Find the area of the quadrilateral OABC.<span class="inline-marks">[2]</span></div></li></ol>

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

The diagram shows a quadrilateral $OABC$ where $O$ is the origin. The point $A$ is $(-1, 5)$ and the point $B$ is $(3, 7)$. The perpendicular bisector of $AB$ meets the $x$-axis at $C$.

2025 MGS PRELIMS P1 Q3

<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/1770805603072-58nwvw5dd8k.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDU2MDMwNzItNThud3Z3NWRkOGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwNzkzMiwiZXhwIjoxNzg1NzA4MjMyfQ.2ir1m8Gu-tZWCRfbNvgALFQ-5_ZG1oHObLNA3Q66VXE" style="max-width: min(250px, 100%); height: auto; display: block;" alt="The diagram shows an isosceles triangle $PQR$ on a coordinate plane." class="aligncenter custom-width" /></figure></div>
<p>The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where&nbsp;$h$ and $k$ are positive constants and $h &gt; k$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$h + k = 12$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is now given that&nbsp;$h = 7$.<br>The point $S$ is such that&nbsp;$PQRS$ is a kite. Given that&nbsp;$S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the area of the kite.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where $h$ and $k$ are positive constants and $h > k$.

It is now given that $h = 7$.\nThe point $S$ is such that $PQRS$ is a kite. Given that $S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.

IP3 Class Test (May 2025) Q11 (2023 ACSBR AM Prelim P2 Q5)

<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>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>The line $y + x = 1$ intersects the curve $x^2 + y^2 - 2y = 7$ at $P$ and $Q$. Find the equation of the perpendicular bisector $PQ$.</p>

The line $y + x = 1$ intersects the curve $x^2 + y^2 - 2y = 7$ at $P$ and $Q$. Find the equation of the perpendicular bisector $PQ$.

<p>The line $y + x + 7 = 0$ meets the curve $y^2 + x = 5$ at the points $P$ and $Q$.</p>
<p>Find the equation of the perpendicular bisector of $PQ$.</p>

The line $y + x + 7 = 0$ meets the curve $y^2 + x = 5$ at the points $P$ and $Q$.

Find the 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,

Math Question Bank