A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<p>Expand each of the following expressions.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{\left( x+7 \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( 3x+5y \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( 9-x \right)}^{2}}$</div></li><li><div class="list-item-paragraph">${{\left( 3x-8y \right)}^{2}}$</div></li><li><div class="list-item-paragraph">$\left( x-10 \right)\left( x+10 \right)$</div></li><li><div class="list-item-paragraph">$\left( 2x-7y \right)\left( 2x+7y \right)$</div></li></ol>

Expand each of the following expressions.

Expansion and Simplification of Quadratic Expressions

Expanding Expressions Using Standard Identities

Quadratic Functions, Equations and Inequalities

<ol class="i-list"><li><div class="list-item-paragraph">Express ${{x}^{2}}-6x-13$ in the form ${{(x-a)}^{2}}-b$, where&nbsp;$a$ and $b$ are integers.</div></li><li><div class="list-item-paragraph">Hence, solve the equation ${{x}^{2}}-6x-13=3$ giving your answers correct to $2$ decimal places.</div></li></ol>

Solving Quadratic Equation by Completing the Square

Completing the Square to Solve Quadratic Equations

<p>Expand and simplify each of the following expressions.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$9x\times \left( -2y \right)$</div></li><li><div class="list-item-paragraph">$2x{{y}^{3}}\times 6x{{y}^{2}}$</div></li><li><div class="list-item-paragraph">$x\left( 5y+9 \right)$</div></li><li><div class="list-item-paragraph">$-4x\left( -x+3y \right)$</div></li><li><div class="list-item-paragraph">$3z\left( 6x+y \right)+8z\left( x-5y \right)$</div></li><li><div class="list-item-paragraph">$-x\left( 5y+2z \right)-7x\left( y-4z \right)$</div></li></ol>

Expand and simplify each of the following expressions.

Algebraic Expansion with Variables

<p>The diagram below shows the distance-time graph of an object $A$ from point $Q$ during a one-minute interval.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760432006168-i1lmddo89ik.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MzIwMDYxNjgtaTFsbWRkbzg5aWsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNjk2MywiZXhwIjoxNzg5NjI3MjYzfQ.o8WSpV0BjJtpjG4JcAqA2H8cgXEN7C4N-gmFPQ4Xod0" style="max-width: 400px; height: auto; display: block;" class="aligncenter custom-width" /></figure>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the average speed of the object during the $1$ minute interval.</div></li><li><div class="list-item-paragraph">A second object $B$travels at a constant speed of $5$ m/s, starting at point $P$. On the same axes above, draw the travel graph for this journey. Use your graph to answer the following questions:</div><ol class="i-list"><li><div class="list-item-paragraph">After how long does object $A$ meet object $B$?</div></li><li><div class="list-item-paragraph">How long does it take object $B$ to cover the same total distance as object $A$?</div></li></ol></li></ol>

The diagram below shows the distance-time graph of an object $A$ from point $Q$ during a one-minute interval.

Quadratic Functions - General

Distance–Time Graph of an Object’s Motion Over One Minute

<p>Solve the following equations, giving your answers to 3 significant figures where necessary.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( 3x-2 \right)}^{2}}=16$</div></li><li><div class="list-item-paragraph">${{\left( 5-x \right)}^{2}}=7$</div></li><li><div class="list-item-paragraph">${{\left( \frac{1}{2}x-4 \right)}^{2}}=81$ </div></li><li><div class="list-item-paragraph">${{(2x+5)}^{2}}=45$</div></li><li><div class="list-item-paragraph">${{\left( x-5 \right)}^{2}}=\frac{9}{16}$</div></li><li><div class="list-item-paragraph">${{\left( 2x+\frac{3}{2} \right)}^{2}}=\frac{81}{100}$</div></li></ol>

Solve the following equations, giving your answers to 3 significant figures where necessary.

Solving Quadratic Equations

Solving Equations Involving Squared Expressions

<p>Express the following expressions in partial fractions:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{2x+1}{{{(x+1)}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{x}{{{(2x+1)}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{{{x}^{2}}}{{{\left( 1-x \right)}^{2}}}$</div></li></ol>

Express the following expressions in partial fractions:

Question 114068

<ol class="abc-list"><li><div class="list-item-paragraph">Factorise ${{x}^{2}}+x-6$.</div></li><li><div class="list-item-paragraph">Hence, factorise ${{\left( {{h}^{2}}+1 \right)}^{2}}+\left( {{h}^{2}}+1 \right)-6$ completely.</div></li></ol>

Applying Quadratic Factorisation to Composite Expressions

<p>Express the following expressions in partial fractions:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{{{x}^{2}}+8x-1}{(x-3){{(x-1)}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{{{x}^{2}}-x+1}{\left( 1-x \right){{(2x-1)}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{1-2x+{{x}^{2}}}{\left( {{x}^{2}}-4 \right)(2x+3)}$</div></li></ol>

Question 114072

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong><br>$A$, $B$ and $C$ are the points $\left( -10,15 \right)$, $\left( -10,-10 \right)$ and $\left( 10,-20 \right)$.</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 data-list-item-id="item-1773733620577-f2bplw13j"><div class="list-item-paragraph">Find the area of triangle $ABC$.</div></li><li data-list-item-id="item-1773733621100-6wlziru6j"><div class="list-item-paragraph">$ABCD$ is a parallelogram. Find the coordinates of the point $D$.</div></li><li data-list-item-id="item-1773733621620-mlysy8cor"><div class="list-item-paragraph">$ABCE$ is a trapezium with $AB$ parallel to $CE$. The area of the trapezium is $650$ units$^{2}$.<br>Find the coordinates of the point $E$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/converted/1760169362834-kz0oiwizouf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNjkzNjI4MzQta3owb2l3aXpvdWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNjk2MywiZXhwIjoxNzg5NjI3MjYzfQ.OQgVTqfgKNxS2S2klA0NDB91dbL7I_CzKjikbHOFqk4" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

**Solutions to this question by accurate drawing will not be accepted.**
$A$, $B$ and $C$ are the points $\left( -10,15 \right)$, $\left( -10,-10 \right)$ and $\left( 10,-20 \right)$.

Area of Rectilinear Figures

Advanced coordinate geometry: Using Coordinate Geometry to Find Missing Points

Coordinate Geometry

<p>Express the following expressions in partial fractions:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{13-2x}{(x-1)({{x}^{2}}+10)}$</div></li><li><div class="list-item-paragraph">$\frac{3-x-{{x}^{2}}}{{{x}^{3}}+2x}$</div></li><li><div class="list-item-paragraph">$\frac{5{{x}^{2}}+9x-10}{\left( x+1 \right)\left( {{x}^{2}}+3 \right)}$</div></li></ol>

Question 114075

<p>Factorise ${{x}^{2}}y-9xy+20y$ fully.</p>

Factorisation of Algebraic Expressions

<p>Express $\frac{1}{(x+1)(x+2)}$ in partial fractions.</p>
<p>Hence,</p>
<ol class="i-list"><li><div class="list-item-paragraph">Evaluate $\frac{1}{(2)(3)}+\frac{1}{(3)(4)}+\frac{1}{(4)(5)}+........+\frac{1}{(19)(20)}$.</div></li><li><div class="list-item-paragraph">Find the value of $n$ if $\frac{1}{(1)(2)}+\frac{1}{(2)(3)}+\frac{1}{(3)(4)}+........+\frac{1}{(n-1)(n)}=\frac{99}{100}$.</div></li></ol>

Express $\frac{1}{(x+1)(x+2)}$ in partial fractions.

Question 114081

<p>Factorise completely $4ax+12bx-a-3b$.</p>

Factorisation by Grouping and Common Factors

<p>The graphs below show the charges of two taxi companies – $\mathrm{M}2$ Taxi and $\mathrm{D5}$ Taxi.</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/converted/1760431664254-pnjxhk030p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MzE2NjQyNTQtcG5qeGhrMDMwcC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjI2OTYzLCJleHAiOjE3ODk2MjcyNjN9.olvVwjPFIA6ez5vVbaFBZ8M3nKUqcTJX-BldGUYc6_c" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">What is the flagdown fare for both taxi companies?</div></li><li><div class="list-item-paragraph">Hariq travelled $6$km on an $\mathrm{M}2$ Taxi. How much does Hariq need to pay?</div></li><li><div class="list-item-paragraph">Tyson wants to travel $11$ km. Which taxi company should he hire? Why?</div></li><li><div class="list-item-paragraph">At what rate was $\mathrm{M}2$ Taxi charging after the first $2$ km?</div></li></ol>

The graphs below show the charges of two taxi companies – $\mathrm{M}2$ Taxi and $\mathrm{D5}$ Taxi.

Taxi Fare Charges Compared by Distance Travelled

<ol class="i-list"><li><div class="list-item-paragraph">Express ${{x}^{2}}+4x-6$ in the form ${{\left( x+a \right)}^{2}}+b$ where&nbsp;$a$ and $b$ are integers</div></li><li><div class="list-item-paragraph">Hence, solve the equation ${{x}^{2}}+4x-6=0$.</div></li></ol>

Completing the Square (Integer Form)

<p>Express the following expressions in partial fractions:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{3x+1}{(x-2)(x+5)}$</div></li><li><div class="list-item-paragraph">$\frac{2x-1}{\left( 2x+3 \right)\left( x-1 \right)}$</div></li><li><div class="list-item-paragraph">$\frac{1-2x}{{{x}^{2}}-25}$</div></li></ol>

Question 114053

<p>Factorise $36{{x}^{2}}-4{{y}^{2}}$ completely.</p>

Factorise $36{{x}^{2}}-4{{y}^{2}}$ completely.

Complete Factorisation of Algebraic Expressions

<p>Expand and simplify each of the following expressions.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$3a\left( 6+a \right)-2\left( 5a-4 \right)$</div></li><li><div class="list-item-paragraph">$-x\left( 9-3x \right)+5x\left( x-9 \right)$</div></li><li><div class="list-item-paragraph">$2{{x}^{2}}+3x-8-\left( -12{{x}^{2}} \right)-6x-2$</div></li><li><div class="list-item-paragraph">$x+\left( -3{{x}^{2}} \right)+3\left( 4{{x}^{2}}-7x \right)$</div></li><li><div class="list-item-paragraph">$2\left( 2-8{{x}^{2}}-9x \right)-5\left( {{x}^{2}}-3x-6 \right)$</div></li></ol>

Math Question Bank