A Math

A Math (IGCSE)

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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The curve $y=(x-1)^2$ intersects the line $y=3-x$ at the points $P$ and $Q$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q07-parts-item-1" data-list-item-path="amarea-q07-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $P$ and $Q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q07-parts-item-2" data-list-item-path="amarea-q07-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the line.<span class="inline-marks">[4]</span></div></li></ol>

The curve $y=(x-1)^2$ intersects the line $y=3-x$ at the points $P$ and $Q$.

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

Area Bounded by a Curve and Straight Lines

Intersection of a Line and a Curve

Shifted parabola below a descending line

Quadratic Functions, Equations and Inequalities

Integration

<p>The curve $x=(y-2)^2$ intersects the line $x=4-y$ at the points $P$ and $Q$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q25-parts-item-1" data-list-item-path="amarea-q25-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $P$ and $Q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q25-parts-item-2" data-list-item-path="amarea-q25-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the line.<span class="inline-marks">[5]</span></div></li></ol>

The curve $x=(y-2)^2$ intersects the line $x=4-y$ at the points $P$ and $Q$.

Area Enclosed by the Curve and the Y-Axis

Sideways parabola and an oblique line

<p>Given that&nbsp;$\mathrm{f}(x) = x^2 - ax + 3$, where&nbsp;$a$ is a constant,</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">find the range of values of $a$ for which $f(x) &gt; x - 1$, for all real values of $x$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">find the value(s) of $a$ for which the line $y = a + 4$ is a tangent to the curve $y = f(x)$.<span class="inline-marks">[3]</span></div></li></ol>

Given that $\mathrm{f}(x) = x^2 - ax + 3$, where $a$ is a constant,

Nature of Roots of Quadratic Equations

Quadratic Expressions Positive or Negative for All Real Values

2024 FMSS PRELIMS P2 Q4

<p></p>
<p></p>
<ol class="i-list"><li data-list-item-id="item-1781083736022-kfskn68ky"><div class="list-item-paragraph">Given that the line $y = 2x - k$ meets the curve $y = \frac{k}{4}x^2 - 2kx + 1$, find the range of values of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1781083737079-h01r9msxr"><div class="list-item-paragraph">A student claimed that when&nbsp;$k = -\frac{1}{2}$, the curve does not intersect the line. By showing your workings clearly, explain whether the statement is valid.<span class="inline-marks">[2]</span></div></li></ol>

2024 CCHY P1 Q1

<ol class="abc-list"><li data-list-item-id="item-a-7"><div class="list-item-content"><span class="list-item-text">Find the range of values of $p$ for which $y = x^2 - x - 2$ lies entirely above the line $y = p(x + 2)$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1772956600479-dlgb1j5s6"><div class="list-item-paragraph">Hence, deduce, without finding the discriminant, the number of intersection points when&nbsp;$p = 3$.</div></li></ol>

Find the range of values of $p$ for which $y=x^2-x-2$ lies entirely above the line $y=p(x+2)$.

Hence, deduce, without finding the discriminant, the number of intersection points when $p=3$.

2025 AHS PRELIMS P1 Q1

<p>The line $y = x - 4$ and the curve $y = \frac{2}{x - 3}$ intersect at the points $A$ and $B$.</p>
<p>Find the length of $AB$.<span class="inline-marks">[6]</span></p>

The line $y=x-4$ and the curve $y=\frac{2}{x-3}$ intersect at the points $A$ and $B$.

Length of a Line Segment

2025 DSS PRELIMS P1 Q1

Coordinate Geometry

<p>A metal cube is heated to a temperature of $205$ ${}^\circ\mathrm{C}$ before being dropped into a liquid. As the cube cools, its temperature, $T$ ${}^\circ\mathrm{C}$, $t$ minutes after it enters the liquid is given by $T = K + 175\mathrm{e}^{-mt}$, where&nbsp;$K$ and $m$ are constants.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Show that the value of $K$ is $30$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>When&nbsp;$t = 3$, the temperature of the cube reaches $128$ ${}^\circ\mathrm{C}$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Find the value of $m$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1772934012813-bhcrcd5uv" data-list-item-path="item-1772934012813-bhcrcd5uv"><div class="list-item-paragraph">Find the rate at which the temperature of the cube is decreasing at the instant when&nbsp;$t=8$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1772934227228-uph8ba7oq" data-list-item-path="item-1772934227228-uph8ba7oq"><div class="list-item-paragraph">Explain why the temperature of the cube can never fall below $30^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = (4 + p)x^2 - 2x + p$, where $p$ is a constant.

The curve has a maximum point and intersects the line $y = 2x - 1$.

Interpreting Quadratic Graphs

2025 SST PRELIMS P1 Q3

<p>The equation of a curve is $y = (m+1)x^2 - 8x + 3m$, where&nbsp;$m$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">In the case where&nbsp;$m = 1$, show that the line $y = 1 - 4x$ is a tangent to the curve. Find the coordinates of the point of contact.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Find the range of values of $m$ for which the curve is above the line $y = 5$.</span><span class="inline-marks">[5]</span></div></li></ol>

The equation of a curve is $y=(m+1)x^2-8x+3m$, where $m$ is a constant.

In the case where $m=1$, show that the line $y=1-4x$ is a tangent to the curve. Find the coordinates of the point of contact.

Find the range of values of $m$ for which the curve is above the line $y=5$.

2025 SCGS PRELIMS P1 Q5

<p>The straight line $2x + y = 1$ intersects the curve $x^2 - xy - y^2 + 5 = 0$ at the points $A$ and $B$.</p>
<p>Show that the length $AB$ is $5\sqrt{5}$ units.<span class="inline-marks">[5]</span></p>

The straight line $2x + y = 1$ intersects the curve $x^2 - xy - y^2 + 5 = 0$ at the points $A$ and $B$.

Show that the length $AB$ is $5\sqrt{5}$ units.

2025 SCGS PRELIMS P1 Q2

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDYzNDY5ODEtcWE1eXBlaDQ4bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODY3NzY2LCJleHAiOjE3OTI4NjgwNjZ9.EWMR1sb93UGozVZdaGde9dP7G8LIvP21iBm3t59ZhJI" 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

Area of Rectilinear Figures

Equation of a Straight Line

IP3 Class Test (May 2025) Bonus Question

<p>Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.<span class="inline-marks">[4]</span></p>

Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.

IP3 Class Test (May 2025) Q1

<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 $PQ$.

Perpendicular Bisectors

<p>The line $y = x + 6$ meets the curve $y^2 = x + 12$ at the points $A$ and $B$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the coordinates of $A$ and $B$,</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></ol>

The line $y = x + 6$ meets the curve $y^2 = x + 12$ at the points $A$ and $B$. Find

the equation of the perpendicular bisector of $AB$.

<p>Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.<span class="inline-marks">[5]</span></p>

Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.

2023 CSS AM S3 EOY Q3

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the graph of $y = -4x^2 + 4kx - (k^2 + 1)$ lies entirely below the $x$-axis.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $k$ for which the line $y = kx + 2$ meets the curve $x^2 + y^2 = 2$.</span><span class="inline-marks">[4]</span></div></li></ol>

Explain why the graph of $y = -4x^2 + 4kx - (k^2 + 1)$ lies entirely below the $x$-axis.

Find the range of values of $k$ for which the line $y = kx + 2$ meets the curve $x^2 + y^2 = 2$.

2023 ACSBR AM S3 EOY Q7

<p>The diagram shows part of the curve $x=y^2-3y$ and the line $x+y=3$.</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/1768008012336-hpxi3pcmgut.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDgwMTIzMzYtaHB4aTNwY21ndXQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2Nzc2NiwiZXhwIjoxNzkyODY4MDY2fQ.TzfFiDk02OdCSS6LupqCbeKm_tpxPIX0zzcBeS7DiJM" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>If the line and the curve intersect at $P$ and $Q$, find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">coordinates of $P$ and $Q$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">area of the shaded region.</span></div></li></ol>

The diagram shows part of the curve $x=y^2-3y$ and the line $x+y=3$.

If the line and the curve intersect at $P$ and $Q$, find the

<p>The diagram shows parts of the curve $y = 3 - \sqrt{2x}$ and the line $2y = x$. The curve and the line intersect at the point $A$. The curve $y = 3 - \sqrt{2x}$ cuts the axes at $(0, 3)$ and $(4.5, 0)$.</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/1768006753617-db8aayzxqc5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDY3NTM2MTctZGI4YWF5enhxYzUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2Nzc2NiwiZXhwIjoxNzkyODY4MDY2fQ.ApUCBYWrMkn9nR2pViXEW-PDhUOVWG8iAEWliM4edoQ" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that the area bounded by the curve $y = 3 - \sqrt{2x}$, the $x$-axis and the line $x = 9$ can be expressed as&nbsp;$(a\sqrt{2} + b)$ square units, where&nbsp;$a$ and $b$ are constants.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the coordinates of point $A$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the area bounded by the straight line $2y = x$, the curve $y = 3 - \sqrt{2x}$ and the $y$-axis.</div></li></ol>
<p>The coefficients in the requested exact form are rational numbers.</p>

The diagram shows parts of the curve $y = 3 - \sqrt{2x}$ and the line $2y = x$. The curve and the line intersect at the point $A$. The curve $y = 3 - \sqrt{2x}$ cuts the axes at $(0, 3)$ and $(4.5, 0)$.

Show that the area bounded by the curve $y = 3 - \sqrt{2x}$, the $x$-axis and the line $x = 9$ can be expressed as $(a\sqrt{2} + b)$ square units, where $a$ and $b$ are constants.

Find the area bounded by the straight line $2y = x$, the curve $y = 3 - \sqrt{2x}$ and the $y$-axis.

The coefficients in the requested exact form are rational numbers.

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}\left( x \right)=27{{x}^{3}}-18{{x}^{2}}+6nx+2n+3$, show that&nbsp;$x=-1$ is a solution to $\mathrm{f}\left( \frac{x}{3} \right)=0$ for all values of $n$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Show that for any real number $m$ the curve $y={{x}^{3}}+5{{x}^{2}}-m$ and the line $y=6-mx$ meet at the point where&nbsp;$x=1$.</div></li><li><div class="list-item-paragraph">Find the range of values of $m$ such that the curve $y={{x}^{3}}+5{{x}^{2}}-m$ and the line $y=6-mx$ meet only at the point $x=1$.</div></li></ol></li></ol>

Cubic Expressions and Equations

Remainder and Factor Theorems

Polynomials, Cubic Equations and Partial Fractions

<p>Find the range of values of $k$ for which the line $y=3x+k$ will not intersect the curve $y=6+5x-{{x}^{2}}$.</p>

Find the range of values of $k$ for which the line $y=3x+k$ will not intersect the curve $y=6+5x-{{x}^{2}}$.

<p>Find the value of $p$ for which the line $y+4x=p$ is a tangent to the curve $y={{x}^{2}}-p$.</p>

Find the value of $p$ for which the line $y+4x=p$ is a tangent to the curve $y={{x}^{2}}-p$.

Math Question Bank