A Math

A Math (IGCSE)

E Math

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

Shading and Identifying Set Regions

Foundations of Set Theory: Mastering Union and Intersection Operations

Sets

<p>The Number Line</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760777236067-bvb4a4kmsaf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3NzcyMzYwNjctYnZiNGE0a21zYWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzAzMDU3NSwiZXhwIjoxNzkzMDMwODc1fQ.tyGWYxqqQuo8g86dKqaFlrfxa_BW9wOM6PHDgtH4qwo" width="400" class="aligncenter" /></figure>
<p>Mark the following numbers on the number line:</p>
<p>$4$, $-6$, $-3$, $\frac{5}{7}$, $3\frac{1}{2}$, $-\frac{3}{4}$, $\pi$, $1.\dot{3}$</p>

Mark the following numbers on the number line:

$4$, $-6$, $-3$, $\frac{5}{7}$, $3\frac{1}{2}$, $-\frac{3}{4}$, $\pi$, $1.\dot{3}$

Ordering Real Numbers and the Number Line

Negative and Positive Numbers

Real Numbers

<p>$5-2$ </p>
<p>$5-\left( -2 \right)$ </p>
<p>$-5-\left( -2 \right)$</p>

Addition and Subtraction involving Negative Numbers

Subtraction involving Negative Numbers

<p>If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by </p>
<p style="text-align: center">$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$</p>
<p>This formula for solving quadratic equations is usually used when the <strong>quadratic expression cannot be factorised easily. </strong></p>
<p style="margin-top: 1.5rem"><strong>Tip: </strong></p>
<p>Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.</p>

If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by 

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

This formula for solving quadratic equations is usually used when the **quadratic expression cannot be factorised easily. **

Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.

Solving Quadratic Equations Using the Quadratic Formula

Solving Quadratic Equations by using Formula

Quadratic Functions, Equations and Inequalities

Interpreting and Sketching Motion Graphs

Distance Travelled and Average Speed

Conversion of Speed-Time Graph to Distance-Time Graph

Functions and Graphs

<p>When subtracting two vectors $\overrightarrow{AB}$ and $\overrightarrow{CB}$:</p>
<p style="text-align: center">$\overrightarrow{AB} - \overrightarrow{CB} = \overrightarrow{AC}$</p>
<p>For any two column vectors, $\mathbf{a} = \begin{pmatrix} p \\ q \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} r \\ s \end{pmatrix}$, then</p>
<p style="margin-left: 12.5rem">$\!\!\!\!$ $\mathbf{a} - \mathbf{b} = \begin{pmatrix} p - r \\ q - s \end{pmatrix}$</p>

When subtracting two vectors $\overrightarrow{AB}$ and $\overrightarrow{CB}$:

$\overrightarrow{AB} - \overrightarrow{CB} = \overrightarrow{AC}$

For any two column vectors, $\mathbf{a} = \begin{pmatrix} p \\ q \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} r \\ s \end{pmatrix}$, then

$\!\!\!\!$ $\mathbf{a} - \mathbf{b} = \begin{pmatrix} p - r \\ q - s \end{pmatrix}$

Vector Subtraction

Vectors

The total delivery time $T$, in seconds, follows the distribution $T\sim N(1800,225+\sigma^2)$, where $\sigma>0$. (a) Given that $\sigma=40$, find $P(1680<T<1920)$. (b) The delivery company requires the delivery time to differ from its mean by less than $120$ seconds with probability at least $0.90$. Find the greatest possible value of $\sigma$. (c) The company introduces the additional requirement that $P(T>1900)\leq0.05$. Find the range of values of $\sigma$ for which both requirements are satisfied.

The total delivery time $T$, in seconds, follows the distribution $T\sim N(1800,225+\sigma^2)$, where $\sigma>0$.

Normal Probabilities and Quantiles

Solving Normal Probability Inequalities

Delivery time under two probability requirements

Continuous Random Variables

A curve is defined parametrically by $x=\ln(t+1)$ and $y=\ln(5-t)$, where $t$ takes all real values for which both expressions are defined.

Tangents and Normals to Parametric Curves

Converting Parametric Equations to Cartesian Form

Normal and coordinate ranges of a logarithmic curve

Applications of Differentiation

Graphing Techniques

<p>A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-71" data-list-item-path="p1421-item-71"><div class="list-item-paragraph">Find $(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-73" data-list-item-path="p1421-item-73"><div class="list-item-paragraph">Find $z^3+(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-75" data-list-item-path="p1421-item-75"><div class="list-item-paragraph">Find $(z^*)^6$ in Cartesian form.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.

Applying Complex Conjugate Identities

Using conjugates of powers

Complex Numbers

<p>Let&nbsp;$z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-91" data-list-item-path="p1421-item-91"><div class="list-item-paragraph">Show that&nbsp;$A^*=z^*w$, and hence that&nbsp;$zw^*+z^*w$ is real.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-93" data-list-item-path="p1421-item-93"><div class="list-item-paragraph">Show that&nbsp;$zw^*-z^*w$ has zero real part. Explain why it need not be a nonzero purely imaginary number.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-95" data-list-item-path="p1421-item-95"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{zw^*-z^*w}{i}$ is real.<span class="inline-marks">[2]</span></div></li></ol>

Let $z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.

Properties of Complex Conjugates

Recognising real and imaginary expressions

<p>The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-42" data-list-item-path="p1421-item-42"><div class="list-item-paragraph">Find $z^*w^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-44" data-list-item-path="p1421-item-44"><div class="list-item-paragraph">Find $(zz^*)(ww^*)$, explaining how you regroup the factors.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-46" data-list-item-path="p1421-item-46"><div class="list-item-paragraph"><p>Find $\operatorname{Re}(zw)$ and $\operatorname{Im}(zw)$.</p><span class="inline-marks">[2]</span></div></li></ol>

The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.

Using a product and its conjugate

<p>Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-59" data-list-item-path="p1421-item-59"><div class="list-item-paragraph">$\dfrac{w^*}{z^*}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-61" data-list-item-path="p1421-item-61"><div class="list-item-paragraph">$\left(\dfrac{z+w}{z-w}\right)^*$. Explain why the denominator is nonzero.<span class="inline-marks">[4]</span></div></li></ol>

Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.

Conjugating ratios without finding the numbers

<p>The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-7" data-list-item-path="p1421-item-7"><div class="list-item-paragraph">$(3z-2w^*)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-9" data-list-item-path="p1421-item-9"><div class="list-item-paragraph">$(z^*+w)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-11" data-list-item-path="p1421-item-11"><div class="list-item-paragraph">$(iz)^*$ and $iz^*$. Explain why these values differ.<span class="inline-marks">[3]</span></div></li></ol>

The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.

Conjugating a linear combination

<p>A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-26" data-list-item-path="p1421-item-26"><div class="list-item-paragraph">Find $z$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-28" data-list-item-path="p1421-item-28"><div class="list-item-paragraph">Find $z-z^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-30" data-list-item-path="p1421-item-30"><div class="list-item-paragraph">Use the given sum and product to find $\dfrac1z+\dfrac1{z^*}$, without evaluating the two reciprocals separately.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)>0$.

Solving Equations Involving Complex Conjugates

Recovering a number from its sum and product

<p>Evaluate $\displaystyle\int_0^{\pi/4}\cos^4y\sin y\,\mathrm{d}y$.</p>

Evaluate $\displaystyle\int_0^{\pi/4}\cos^4y\sin y\,\mathrm{d}y$.

Integrable Forms of Trigonometric Functions

Integration by Substitution

A Definite Trigonometric Integral

Techniques of Integration

<p>Using $x=4\sin\theta$, where&nbsp;$0&lt;\theta&lt;\dfrac{\pi}{2}$, find $\displaystyle\int\dfrac{\sqrt{16-x^2}}{x^2}\,\mathrm{d}x$ for&nbsp;$0&lt;x&lt;4$.</p>

Using $x=4\sin\theta$, where $0<\theta<\dfrac{\pi}{2}$, find $\displaystyle\int\dfrac{\sqrt{16-x^2}}{x^2}\,\mathrm{d}x$ for $0<x<4$.

A Supplied Sine Substitution

<p>A particle moves back and forth in a horizontal line between two end points. A fixed origin $O$ is the midpoint between these end points. Its displacement $x$ metres from $O$ at time $t$ seconds is $x=4\sin(2t+\frac\pi3)$, where&nbsp;$0\leq t\leq\pi$.</p>
<ol class="abc-list"><li data-list-item-id="p3-tzc-q02-item01" data-list-item-path="p3-tzc-q02-item01"><div class="list-item-paragraph">Determine</div><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub01-item01" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item01"><div class="list-item-paragraph">the amplitude;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item02" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item02"><div class="list-item-paragraph">the initial displacement;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item03" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item03"><div class="list-item-paragraph">the value of $t$ when the particle first passes through $O$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item02" data-list-item-path="p3-tzc-q02-item02"><div class="list-item-paragraph">The acceleration $a$ at displacement $s$ satisfies $a=-n^2s$, where&nbsp;$n&gt;0$.</div><div class="list-item-paragraph">The displacement is $s=A\sin(nt+b)$, where&nbsp;$t\geq0$, $A,n&gt;0$ and $-\pi\leq b\leq\pi$.</div><div class="list-item-paragraph">By finding expressions for&nbsp;$\frac{\mathrm{d}s}{\mathrm{d}t}$ and $\frac{\mathrm{d}^2s}{\mathrm{d}t^2}$, verify that&nbsp;$s=A\sin(nt+b)$ satisfies $a=-n^2s$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-item03" data-list-item-path="p3-tzc-q02-item03"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub03-item01" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item01"><div class="list-item-paragraph">Use the chain rule to show that&nbsp;$a=v\frac{\mathrm{d}v}{\mathrm{d}s}$, where&nbsp;$v$ is velocity.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item02" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item02"><div class="list-item-paragraph">By solving $v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s$, show that&nbsp;$v^2=n^2(A^2-s^2)$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item03" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item03"><div class="list-item-paragraph">Hence, or otherwise, find the particle’s maximum speed.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item04" data-list-item-path="p3-tzc-q02-item04"><div class="list-item-paragraph">The continuous random variable $S$ denotes the particle’s displacement $s$ from $O$ at time $t$.</div><div class="list-item-paragraph">\[f(s)=\begin{cases}\frac1{\pi\sqrt{A^2-s^2}},&-A&lt;s&lt;A,\\0,&\text{otherwise}.\end{cases}\]</div><div class="list-item-paragraph">Show that&nbsp;$P(-\frac A2\leq S\leq\frac A2)=\frac13$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p3-tzc-q02-item05" data-list-item-path="p3-tzc-q02-item05"><div class="list-item-paragraph">For $-A&lt;s&lt;A$, $f(s)$ can be expressed as $\frac{k}{|v(s)|}$, where $k&gt;0$ and $v(s)$ is the particle’s velocity at displacement $s$ from $O$.</div><div class="list-item-paragraph">Find the value of $k$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p3-tzc-q02-item06" data-list-item-path="p3-tzc-q02-item06"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub06-item01" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item01"><div class="list-item-paragraph">Determine $E(S)$, justifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub06-item02" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item02"><div class="list-item-paragraph">Interpret the result in the context of the particle’s motion.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A particle moves back and forth in a horizontal line between two end points. A fixed origin $O$ is the midpoint between these end points. Its displacement $x$ metres from $O$ at time $t$ seconds is $x=4\sin(2t+\frac\pi3)$, where $0\leq t\leq\pi$.

Solving Differential Equations by Separation of Variables

Given the Displacement Function

Derivatives of Trigonometric Functions

Probability Density Functions

Expectation and Variance of Continuous Random Variables

Second and Higher Derivatives

Product, Quotient and Chain Rules

Probabilities by Integrating a Density Function

IB AA HL May 2026 P3 TZC Q2

Techniques of Differentiation

Differential Equations

Kinematics

<p>Rowan is investigating the spread of an infection through a population of rabbits.</p>
<p>He models $N$, the number (in thousands) of infected rabbits in the population, at time $t$ days.</p>
<p>Rowan models the spread of the infection by the differential equation $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where&nbsp;$t\geq0$ and $k\in\mathbb{R}^+$.</p>
<p>Initially, $1500$ rabbits are infected.</p>
<ol class="abc-list"><li data-list-item-id="p1-tza-q12-item01" data-list-item-path="p1-tza-q12-item01"><div class="list-item-paragraph">Express $\frac1{-3+4x-x^2}$ in the form $\frac{A}{x-1}+\frac{B}{3-x}$, where&nbsp;$A,B\in\mathbb{R}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1-tza-q12-item02" data-list-item-path="p1-tza-q12-item02"><div class="list-item-paragraph">Hence, show that&nbsp;$\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p$, where&nbsp;$p\in\mathbb{R}^+$ is a constant to be determined.<span class="inline-marks">[8]</span></div></li><li data-list-item-id="p1-tza-q12-item03" data-list-item-path="p1-tza-q12-item03"><div class="list-item-paragraph">Find an expression for&nbsp;$N$ in the form $N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}$, where&nbsp;$a,b,c\in\mathbb{Z}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p1-tza-q12-item04" data-list-item-path="p1-tza-q12-item04"><div class="list-item-paragraph">According to Rowan’s model, the number of infected rabbits approaches a limit, $L$, over the long term.</div><div class="list-item-paragraph">Find the value of $L$.<span class="inline-marks">[2]</span></div></li></ol>

Rowan is investigating the spread of an infection through a population of rabbits.

He models $N$, the number (in thousands) of infected rabbits in the population, at time $t$ days.

Rowan models the spread of the infection by the differential equation $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where $t\geq0$ and $k\in\mathbb{R}^+$.

Initially, $1500$ rabbits are infected.

Express $\frac1{-3+4x-x^2}$ in the form $\frac{A}{x-1}+\frac{B}{3-x}$, where $A,B\in\mathbb{R}$.

Hence, show that $\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p$, where $p\in\mathbb{R}^+$ is a constant to be determined.

Find an expression for $N$ in the form $N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}$, where $a,b,c\in\mathbb{Z}$.

Integration Using Partial Fractions

Partial Fractions

IB AA HL May 2026 P1 TZA Q12

Polynomials, Cubic Equations and Partial Fractions

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="ri-p2-q4-2-content-items-1" data-list-item-path="ri-p2-q4-2-content-items-1"><div class="list-item-paragraph">The curve $y=\mathrm{f}(x)$ passes through the origin and has gradient given by</div><div class="list-item-paragraph">\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x+y}{1+xy}.\]</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\displaystyle(1+xy)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2+(y-1)\dfrac{\mathrm{d}y}{\mathrm{d}x}=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-2"><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)(i)</strong>, show that</div><div class="list-item-paragraph">\[(1+xy)\frac{\mathrm{d}^3y}{\mathrm{d}x^3}+\mathrm{p}(x)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)+\mathrm{q}(y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+r\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=0,\]</div><div class="list-item-paragraph">where the functions $\mathrm{p}(x)$, $\mathrm{q}(y)$ and the constant $r$ are to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-3" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-3"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="ri-p2-q4-2-content-items-2" data-list-item-path="ri-p2-q4-2-content-items-2"><div class="list-item-paragraph">The equation</div><div class="list-item-paragraph">\[2\sqrt{2}\cos\left(x-\frac\pi4\right)+3\sin2x-3=0\]</div><div class="list-item-paragraph">has a root $\theta$, which is close to zero.</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\theta^2+a\theta+b\approx0$, where&nbsp;$a$ and $b$ are integers to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-2"><div class="list-item-paragraph">Hence find an approximation for&nbsp;$\theta$ in surd form.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

**Do not use a calculator in answering this question.**

Small Angle Approximations

Approximations Using Power Series

Maclaurin Series from Differential Equations

Higher Derivatives by Implicit Differentiation

2026 RI P2 Q4

Maclaurin Series and Power Series

<p>The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where&nbsp;$x,y&gt;0$. It is given that&nbsp;$y=2$ when&nbsp;$x=1$.</p>
<ol class="abc-list"><li data-list-item-id="p2-tzc-q12-item01" data-list-item-path="p2-tzc-q12-item01"><div class="list-item-paragraph">Use Euler’s method with step size $0.01$ to find an approximation for&nbsp;$y$ when&nbsp;$x=1.05$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item02" data-list-item-path="p2-tzc-q12-item02"><div class="list-item-paragraph">The substitution $\frac1{y^2}=-2z$ is used.</div><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}z}{\mathrm{d}x}-\frac{2z}{x}=1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p2-tzc-q12-item03" data-list-item-path="p2-tzc-q12-item03"><div class="list-item-paragraph">Hence solve the differential equation, giving $y$ as a function of $x$.<span class="inline-marks">[7]</span></div></li><li data-list-item-id="p2-tzc-q12-item04" data-list-item-path="p2-tzc-q12-item04"><div class="list-item-paragraph">State the domain and range of the solution.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item05" data-list-item-path="p2-tzc-q12-item05"><div class="list-item-paragraph"><p>By considering the shape of the graph of the solution, state whether the approximation in part <strong>(a)</strong> is an overestimate or an underestimate, giving a reason.</p><span class="inline-marks">[1]</span></div></li></ol>

The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}$, where $x,y>0$. It is given that $y=2$ when $x=1$.

Use Euler’s method with step size $0.01$ to find an approximation for $y$ when $x=1.05$.

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