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IB AA HL Math

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N(A) E Math

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PSLE Foundation Mathematics

<p>The complex numbers \(z\) and \(w\) satisfy the simultaneous equations \(\vert{}z\vert{} - w^* = -3 - \sqrt{2}\mathrm{i}\) and \(w^* + w + 5z = 1 + 20\mathrm{i}\), where&nbsp;\(w^*\) is the complex conjugate of \(w\). Find the value of \(z\) and the corresponding value of \(w\).<span class="inline-marks">[4]</span></p>

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations \(\vert{}z\vert{} - w^* = -3 - \sqrt{2}\mathrm{i}\) and \(w^* + w + 5z = 1 + 20\mathrm{i}\), where \(w^*\) is the complex conjugate of \(w\). Find the value of \(z\) and the corresponding value of \(w\).

Complex Numbers

<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 \(0<x<4\).

Integration by Substitution

A Supplied Sine Substitution

Techniques of Integration

<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

A Definite Trigonometric Integral

<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

Continuous Random Variables

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>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 \(x=1.05\).

Hence solve the differential equation, giving \(y\) as a function of \(x\).

State the domain and range of the solution.

Euler's Method

Concavity of Graphs

Function Notation, Domain and Range

Solving Differential Equations by Substitution

Linear First-Order Differential Equations and Integrating Factors

IB AA HL May 2026 P2 TZC Q12

Functions

Applications of Differentiation

<p>Consider the cubic equation \(x^3-12x^2+qx-8=0\), where&nbsp;\(q\) is real. In each of the following cases, its three roots are positive and distinct.</p>
<ol class="abc-list"><li data-list-item-id="compare-a" data-list-item-path="compare-a"><div class="list-item-paragraph">The roots form an arithmetic sequence. Find the roots and the value of \(q\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="compare-b" data-list-item-path="compare-b"><div class="list-item-paragraph">In a different case the roots form an increasing geometric sequence. Find the roots and the value of \(q\).<span class="inline-marks">[5]</span></div></li></ol>

Consider the cubic equation \(x^3-12x^2+qx-8=0\), where \(q\) is real. In each of the following cases, its three roots are positive and distinct.

The roots form an arithmetic sequence. Find the roots and the value of \(q\).

In a different case the roots form an increasing geometric sequence. Find the roots and the value of \(q\).

Cubic Expressions and Equations

Geometric Progression

Arithmetic Progression

Sum and Product of Polynomial Roots

Two sequence patterns for roots of a cubic

Sequence and Series

<p>Given&nbsp;\(h(x)=1-|x-2|\), where&nbsp;\(0\le x\le4\), and \(H(x)=\dfrac{1}{h(x)}\), the range of \(H(x)\) is the set&nbsp;\(R\) such that&nbsp;\(R=\{y:y\ge a\text{ or}\hspace{0.5em}y\le b\}\) where&nbsp;\(a,b\in\mathbb Z\).</p>
<p>Determine the values of \(a\) and \(b\).</p>

Given \(h(x)=1-|x-2|\), where \(0\le x\le4\), and \(H(x)=\dfrac{1}{h(x)}\), the range of \(H(x)\) is the set \(R\) such that \(R=\{y:y\ge a\text{ or }y\le b\}\) where \(a,b\in\mathbb Z\).

Modulus Functions

Range of a Function

Reciprocal Graph, \(y=\frac{1}{f(x)}\)

Graphs of Modulus Function

Two branches in a reciprocal modulus range

Modulus Function

Transformations of Graphs

<p>For&nbsp;\(-2\le x\le3\), define \(p(x)=|2x-1|-3\) and \(q(x)=\dfrac{1}{p(x)}\) wherever \(p(x)\ne0\). The range of \(q\) is \(\{y:y\le a\text{ or}\hspace{0.5em}y\ge b\}\). Find the exact values of \(a\) and \(b\).<span class="inline-marks">[4]</span></p>

For \(-2\le x\le3\), define \(p(x)=|2x-1|-3\) and \(q(x)=\dfrac{1}{p(x)}\) wherever \(p(x)\ne0\). The range of \(q\) is \(\{y:y\le a\text{ or }y\ge b\}\). Find the exact values of \(a\) and \(b\).

Reciprocal range of a stretched modulus trough

<ol class="abc-list"><li data-list-item-id="ib2012mhp1tz2q13-part-a" data-list-item-path="ib2012mhp1tz2q13-part-a"><div class="list-item-paragraph">Using the definition of a derivative as&nbsp;\(\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)\), show that the derivative of \(\frac{1}{2x+1}\) is \(\frac{-2}{(2x+1)^2}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ib2012mhp1tz2q13-part-b" data-list-item-path="ib2012mhp1tz2q13-part-b"><div class="list-item-paragraph">Prove by induction that the \(n^{\mathrm{th}}\) derivative of \((2x+1)^{-1}\) is \((-1)^n\frac{2^n n!}{(2x+1)^{n+1}}\).<span class="inline-marks">[9]</span></div></li></ol>

Using the definition of a derivative as \(\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)\), show that the derivative of \(\frac{1}{2x+1}\) is \(\frac{-2}{(2x+1)^2}\).

Prove by induction that the \(n^{\mathrm{th}}\) derivative of \((2x+1)^{-1}\) is \((-1)^n\frac{2^n n!}{(2x+1)^{n+1}}\).

Proof by Mathematical Induction

Differentiation by First Principles

IB Mathematics HL May 2012 P1 TZ2 Q13

Mathematical Induction

<p>Let&nbsp;\(\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x\).</p>
<ol class="abc-list"><li data-list-item-id="quadraticexp-part-a" data-list-item-path="quadraticexp-part-a"><div class="list-item-paragraph">Using the definition \(\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}\), show from first principles that the derivative of \(\mathrm{g}(x)=x^2+2x\) is \(2x+2\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="quadraticexp-part-b" data-list-item-path="quadraticexp-part-b"><div class="list-item-paragraph">Prove by induction that, for every positive integer \(n\), the \(n^{\mathrm{th}}\) derivative of \(\mathrm{f}\) is \(\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]\).<span class="inline-marks">[9]</span></div></li></ol>

Let \(\mathrm{f}(x)=(x^2+2x)\mathrm{e}^x\).

Using the definition \(\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}\), show from first principles that the derivative of \(\mathrm{g}(x)=x^2+2x\) is \(2x+2\).

Prove by induction that, for every positive integer \(n\), the \(n^{\mathrm{th}}\) derivative of \(\mathrm{f}\) is \(\mathrm{f}^{(n)}(x)=\mathrm{e}^x[x^2+2(n+1)x+n(n+1)]\).

Derivatives of Exponential Functions

Quadratic exponential derivative pattern

<p>The function \(\mathrm{f}\) is defined by</p>
<p style="text-align: center">\[\mathrm{f}(x)=x^2+x+3,\ x\geq a\]</p>
<p>where&nbsp;\(a\) is a real constant.</p>
<ol class="abc-list"><li data-list-item-id="q9-a" data-list-item-path="q9-a"><div class="list-item-paragraph">State the least value of \(a\) for which \(\mathrm{f}^{-1}\) exists.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="q9-b" data-list-item-path="q9-b"><div class="list-item-paragraph">Using this value of \(a\), find \(\mathrm{f}^{-1}(x)\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q9-c" data-list-item-path="q9-c"><div class="list-item-paragraph">Find the domain of \(\mathrm{f}^{-1}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q9-d" data-list-item-path="q9-d"><div class="list-item-paragraph">Sketch, on the same axes, the graphs of \(\mathrm{f}\) and \(\mathrm{f}^{-1}\), labelling clearly the coordinates of any endpoints.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=x^2+x+3\).</p>
<ol class="abc-list" start="5"><li data-list-item-id="q9-e" data-list-item-path="q9-e"><div class="list-item-paragraph">Find two possible linear functions \(\mathrm{h}\) such that&nbsp;\((\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15\).<span class="inline-marks">[6]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by

State the least value of \(a\) for which \(\mathrm{f}^{-1}\) exists.

Using this value of \(a\), find \(\mathrm{f}^{-1}(x)\).

Sketch, on the same axes, the graphs of \(\mathrm{f}\) and \(\mathrm{f}^{-1}\), labelling clearly the coordinates of any endpoints.

The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=x^2+x+3\).

Find two possible linear functions \(\mathrm{h}\) such that \((\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15\).

Composite Functions

Inverse Functions

Sketching Quadratic Graphs

IB AA HL Q9: Quadratic inverse and composition (paper unspecified)

Quadratic Functions, Equations and Inequalities

<p>Let&nbsp;\(x\) and \(y\) be positive real numbers such that&nbsp;\(x\ne y\).</p>
<ol class="abc-list"><li data-list-item-id="source-a" data-list-item-path="source-a"><div class="list-item-paragraph">Prove by contradiction that&nbsp;\(\frac{x}{y}+\frac{y}{x}&gt;2\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="source-b" data-list-item-path="source-b"><div class="list-item-paragraph">Give a counterexample to show that this inequality may not hold if&nbsp;\(x\) and \(y\) are not both positive.<span class="inline-marks">[2]</span></div></li></ol>

Let \(x\) and \(y\) be positive real numbers such that \(x\ne y\).

Prove by contradiction that \(\frac{x}{y}+\frac{y}{x}>2\).

Give a counterexample to show that this inequality may not hold if \(x\) and \(y\) are not both positive.

Proof by Contradiction

Disproof by Counterexample

Reciprocal inequality for distinct positive numbers

<p>Let&nbsp;\(a\) and \(b\) be distinct positive real numbers.</p>
<ol class="abc-list"><li data-list-item-id="reciprocal-a" data-list-item-path="reciprocal-a"><div class="list-item-paragraph">Prove by contradiction that&nbsp;\(\frac{1}{a}+\frac{1}{b}&gt;\frac{4}{a+b}\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="reciprocal-b" data-list-item-path="reciprocal-b"><div class="list-item-paragraph">Give a counterexample to show that this inequality need not hold when&nbsp;\(a\) and \(b\) are nonzero but not both positive.<span class="inline-marks">[2]</span></div></li></ol>

Let \(a\) and \(b\) be distinct positive real numbers.

Prove by contradiction that \(\frac{1}{a}+\frac{1}{b}>\frac{4}{a+b}\).

Give a counterexample to show that this inequality need not hold when \(a\) and \(b\) are nonzero but not both positive.

Reciprocal sum bounded by the arithmetic mean

<p>Differentiate \(y=x^{2x}\) for&nbsp;\(x&gt;0\).<span class="inline-marks">[4]</span></p>

Implicit Differentiation

Logarithmic differentiation of a variable power

<p>The curve \(x^2+xy+y^2=7\) passes through \((1,2)\). Find the gradient of the curve at this point.<span class="inline-marks">[4]</span></p>

The curve \(x^2+xy+y^2=7\) passes through \((1,2)\). Find the gradient of the curve at this point.

Implicit derivative at a point

<p>Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.<span class="inline-marks">[7]</span></p>

Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.

General Binomial Theorem

Combining two validity conditions

Binomial Theorem

<p>Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.<span class="inline-marks">[5]</span></p>

Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.

A negative integer power

<p>For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where&nbsp;\(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.<span class="inline-marks">[7]</span></p>

For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where \(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.

Recovering a rational exponent

<p>Prove by induction that&nbsp;\(n! \le n^n\) for&nbsp;\(n \in \mathbb{Z}^+\).<span class="inline-marks">[7]</span></p>

Prove by induction that \(n! \le n^n\) for \(n \in \mathbb{Z}^+\).

IB AA HL Math Question Bank