A Math

A Math (IGCSE)

E Math

H2 Math

IB HL Math

IP Math

<p>Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">It is given that the expression $b^2 - 4ac$ is called the <strong>discriminant</strong> of the equation $ax^2 + bx + c = 0$.</div><div class="list-item-paragraph">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the discriminant.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely above the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely below the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ and the $x$-axis have at least one common point.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>Note that&nbsp;$a \neq 0$ throughout this question.</p>

Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.

Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.

It is given that the expression $b^2 - 4ac$ is called the **discriminant** of the equation $ax^2 + bx + c = 0$.\n\nState the relationship between the number of distinct **real roots** of the equation $ax^2 + bx + c = 0$ and the discriminant.

State the relationship between the number of distinct **real roots** of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.

Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that

Note that $a \neq 0$ throughout this question.

IP2 Class Test (May 2025) Bonus Question

Trigonometric Ratios

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769518939767-rz5dgqo1yi.png" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In the diagram, $AB = 16$ cm, $CD = 18$ cm, $\angle ABC = 90^\circ$, $\cos \angle BAC = \frac{4}{5}$ and $BCDE$ is a straight line.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $AC$,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\tan \angle ACB$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $AD$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\angle ADE$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the shortest distance from $C$ to $AD$.</span><span class="inline-marks">[3]</span></div></li></ol>

In the diagram, $AB = 16$ cm, $CD = 18$ cm, $\angle ABC = 90^\circ$, $\cos \angle BAC = \frac{4}{5}$ and $BCDE$ is a straight line.

Find the shortest distance from $C$ to $AD$.

IP2 Class Test (May 2025) Q13

<p>Sketch the graph of $y = (x - 1)(x + 5)$.</p>
<p>Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.<span class="inline-marks">[5]</span></p>
<p>Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.<span class="inline-marks">[1]</span></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769517826384-pn2smbhjeen.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Sketch the graph of $y = (x - 1)(x + 5)$.

Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.

Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.

IP2 Class Test (May 2025) Q12

Quadratic Expressions, Equations, and Functions

<p>A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">As a result of the three members quitting, the amount each member had to pay was \$$5$ more than the initial amount.</div><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $x^2 + 25x - 900 = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Solve the equation $x^2 + 25x - 900 = 0$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the amount of money that each member had to pay initially.<span class="inline-marks">[2]</span></div></li></ol>

A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.

Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.

Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.

Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.

As a result of the three members quitting, the amount each member had to pay was \$$5$ more than the initial amount.\n\nForm an equation in $x$ and show that it reduces to $x^2 + 25x - 900 = 0$.

IP2 Class Test (May 2025) Q11

<p>The diagram shows the graph of $y = x^2 - x - 5$ for&nbsp;$-3 \le x \le 4$.<br>Use the graph to answer the following questions. Present your solutions on the next page.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769516562543-yh7ojfj8sr.png" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph"><strong>Using the above graph,</strong></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find the value of $y$ when&nbsp;$x = 3$</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the coordinates of the turning point,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the equation of the line of symmetry,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">estimate the solutions of the equation $x^2 - x - 5 = 0$.</span><span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State the equation of the straight line to be drawn.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain how the straight line in <strong>(ii)(a)</strong> drawn on the graph can lead to an estimation of the solution(s) to the equation $x^2 - 3x - 2 = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $x^2 - 3x - 2 = 0$ graphically.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The diagram shows the graph of $y = x^2 - x - 5$ for $-3 \le x \le 4$.
Use the graph to answer the following questions. Present your solutions on the next page.

A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.

IP2 Class Test (May 2025) Q10 (2024 RGS Y2 CBA1 Q5)

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the inequality $-2(3 + x) \le \frac{2x - 3}{2} &lt; \frac{17 - 7x}{6}$ and represent the solution on a number line.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, state the minimum value of $x^2$, which satisfy the above inequality.</span><span class="inline-marks">[1]</span></div></li></ol>

Solve the inequality $-2(3 + x) \le \frac{2x - 3}{2} < \frac{17 - 7x}{6}$ and represent the solution on a number line.

Hence, state the minimum value of $x^2$, which satisfy the above inequality.

IP2 Class Test (May 2025) Q9 (2025 RGS Y3 CBA1 Q2)

Linear Equations and Simple Inequalities

<p>$p$ is inversely proportional to the square of $q$, where&nbsp;$p$, $q$ are positive. It is known that&nbsp;$p={{p}_{0}}$ when&nbsp;$q={{q}_{0}}$.  Find $q$ in terms of ${{q}_{0}}$, when&nbsp;$2p=3{{p}_{0}}$.	<span class="inline-marks">[3]</span></p>

$p$ is inversely proportional to the square of $q$, where $p$, $q$ are positive. It is known that $p={{p}_{0}}$ when $q={{q}_{0}}$.  Find $q$ in terms of ${{q}_{0}}$, when $2p=3{{p}_{0}}$.	

IP2 Class Test (May 2025) Q8

Direct and Inverse Proportion

<p>Given that&nbsp;$x$ and $y$ are integers such that&nbsp;$-6 \le x &lt; 2$ and $-5 &lt; y \le -2$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the smallest value of $x^2 + y^2$,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the largest value of $\frac{y}{x}$,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the largest value of $x + xy$.</span><span class="inline-marks">[1]</span></div></li></ol>

Given that $x$ and $y$ are integers such that $-6 \le x < 2$ and $-5 < y \le -2$, find

IP2 Class Test (May 2025) Q7

<p>In a physics experiment, a formula is given by $F = mg \sin(\theta) - \mu mg \cos(\theta)$, where&nbsp;$\mu$, $g$ are constants and $m$, $\theta$ may vary. It is given that&nbsp;$g = 9.8$, $\mu = 0.5$, $m &gt; 0$ and $0 &lt; \theta &lt; 90{}^\circ$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Evaluate $F$ given that&nbsp;$m = 15$ and $\theta = 30{}^\circ$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the value of $\theta$ such that&nbsp;$F = 0$. Note that&nbsp;$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Another variable, $a$, is defined by $a = \frac{F}{m}$. Express $a$ in terms of $\mu$, $g$ and $\theta$.</div><div class="list-item-paragraph">Hence find the range of values of $\theta$ such that&nbsp;$a &gt; 0$.</div><div class="list-item-paragraph">You may assume the fact that&nbsp;$\tan(\theta) &gt; p \iff \theta &gt; \tan^{-1}(p)$ for&nbsp;$0 &lt; \theta &lt; 90{}^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

In a physics experiment, a formula is given by $F = mg \sin(\theta) - \mu mg \cos(\theta)$, where $\mu$, $g$ are constants and $m$, $\theta$ may vary. It is given that $g = 9.8$, $\mu = 0.5$, $m > 0$ and $0 < \theta < 90{}^\circ$.

Evaluate $F$ given that $m = 15$ and $\theta = 30{}^\circ$.

Find the value of $\theta$ such that $F = 0$. Note that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Another variable, $a$, is defined by $a = \frac{F}{m}$. Express $a$ in terms of $\mu$, $g$ and $\theta$.

IP2 Class Test (May 2025) Q6

<ol class="i-list"><li><div class="list-item-paragraph">Make $a$ the subject of the formula $\frac{1}{3b} = \frac{2}{3a} + \frac{1}{5c}$<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Make $x$ the subject of the formula $\sqrt{\frac{2}{y}} = \sqrt{\frac{5x + 3}{x - 5}}$<span class="inline-marks">[2]</span></div></li></ol>

Make $a$ the subject of the formula \[\frac{1}{3b} = \frac{2}{3a} + \frac{1}{5c}\]

Make $x$ the subject of the formula \[\sqrt{\frac{2}{y}} = \sqrt{\frac{5x + 3}{x - 5}}\]

Algebraic Fractions

Algebraic Formulae

IP2 Class Test (May 2025) Q5

Algebraic Expressions, Equations, and Formulae

<ol class="i-list"><li><div class="list-item-paragraph">Simplify and express $\frac{2x}{2x^2 - x - 3} - \frac{3}{x + 1}$ as a single fraction in the simplest form.</div><div class="list-item-paragraph">Hence solve the equation $\frac{2x}{2x^2 - x - 3} - \frac{3}{x + 1} = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Simplify and express $\frac{1}{3 + 5x - 2x^2} - \frac{x + 3}{x - 3}$ as a single fraction in the simplest form.</div><div class="list-item-paragraph">Hence solve the equation $\frac{1}{3 + 5x - 2x^2} - \frac{x + 3}{x - 3} = 0$.<span class="inline-marks">[5]</span></div></li></ol>

Simplify and express $\frac{2x}{2x^2 - x - 3} - \frac{3}{x + 1}$ as a single fraction in the simplest form.

Simplify and express $\frac{1}{3 + 5x - 2x^2} - \frac{x + 3}{x - 3}$ as a single fraction in the simplest form.

IP2 Class Test (May 2025) Q4

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:

Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:

IP2 Class Test (May 2025) Q3

<p>Factorise each of the following expressions completely.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-10x^2 - 19x - 6$</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$18 - 2(a + b)^2$</span><span class="inline-marks">[2]</span></div></li></ol>

Factorise each of the following expressions completely.

IP2 Class Test (May 2025) Q2

<p>Expand and simplify each of the following expressions.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( \frac{2x}{3}-\frac{1}{2} \right)}^{2}}$<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\left( x+6 \right)\left( 2y+3 \right)-\left( y+2 \right)\left( x-2 \right)$<span class="inline-marks">[2]</span></div></li></ol>

Expand and simplify each of the following expressions.

IP2 Class Test (May 2025) Q1

<p>The distance between a pair of parallel lines, $a$ and $b$, is $4$ cm. The distance from point $A$ to line $a$ is $2$ cm, The distance from point $B$ to line $b$ is $3$ cm, and $AB=2\sqrt{30}$.</p>
<p>Find a point $M$ on line $a$ and a point $N$ on line $b$ such that&nbsp;$MN \perp a$ and the length of $AM + MN + NB$ is the shortest. Hence find the shortest length of $AM + NB$</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769504623407-vx6tpgp144e.png" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The distance between a pair of parallel lines, $a$ and $b$, is $4$ cm. The distance from point $A$ to line $a$ is $2$ cm, The distance from point $B$ to line $b$ is $3$ cm, and $AB=2\sqrt{30}$.

Find a point $M$ on line $a$ and a point $N$ on line $b$ such that $MN \perp a$ and the length of $AM + MN + NB$ is the shortest. Hence find the shortest length of $AM + NB$

Applications of Pythagoras’ Theorem

IP1U13 SWII (Pythagoras) Q24

Pythagoras’ Theorem

<p>In triangle $ABC$, $\angle B=90{}^\circ $, $AB=BC=6$ cm. $\Delta AEF$ is folded along $EF$, such that&nbsp;$A$ collides with $D$ and $BD:DC=1:2$. Find $EC$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769503806061-pwdjevzpvqs.png" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In triangle $ABC$, $\angle B=90{}^\circ $, $AB=BC=6$ cm. $\Delta AEF$ is folded along $EF$, such that $A$ collides with $D$ and $BD:DC=1:2$. Find $EC$.

IP1U13 SWII (Pythagoras) Q22

<p>$AD$ is a median of triangle $ABC$, $\angle ADC=45{}^\circ$, $\Delta ADC$ is folded along $AD$ such that&nbsp;$C$ is mapped onto $C'$. Given that&nbsp;$BC=2$ cm, find $BC'$.</p>
<p>Note: The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769503418493-w422o75v20p.png" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

$AD$ is a median of triangle $ABC$, $\angle ADC=45{}^\circ$, $\Delta ADC$ is folded along $AD$ such that $C$ is mapped onto $C'$. Given that $BC=2$ cm, find $BC'$.

Note: The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.

IP1U13 SWII (Pythagoras) Q21

<p>In square $ABCD$, $E$ is a point on $BC$. $\Delta DCE$ is folded along $DE$ onto $DB$, such that&nbsp;$C$ collides with $F$. Find $EB:CE$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769502110590-m1coqsyyhin.png" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In square $ABCD$, $E$ is a point on $BC$. $\Delta DCE$ is folded along $DE$ onto $DB$, such that $C$ collides with $F$. Find $EB:CE$.

IP1U13 SWII (Pythagoras) Q20

<p>In right-angled triangle $ABC$, $AC=6$ cm, $BC=8$ cm. $\Delta ACD$ is folded along $AD$ onto $AB$ such that&nbsp;$C$ collides with $E$. Find the length of $CD$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769501375287-owwcm6vmk9.png" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In right-angled triangle $ABC$, $AC=6$ cm, $BC=8$ cm. $\Delta ACD$ is folded along $AD$ onto $AB$ such that $C$ collides with $E$. Find the length of $CD$.

IP1U13 SWII (Pythagoras) Q19

<p>A rectangle with dimensions $4$ cm and $8$ cm is folded along $EF$, such that&nbsp;$C$ collides with $A$. Find the length of $EB$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769500106202-hlcu4sh5fo9.png" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A rectangle with dimensions $4$ cm and $8$ cm is folded along $EF$, such that $C$ collides with $A$. Find the length of $EB$.

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