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H1 Math

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IP Math

<p>A sequence \(u_1, u_2, u_3, \ldots\) is defined by</p>
<p style="margin-left: 4rem">\(u_1=a,\)<br>\(u_{r+1}=3u_r+5,  r\geqslant 1,\)</p>
<p>where&nbsp;\(a\) is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find expressions for&nbsp;\(u_2\) and \(u_3\) in terms of \(a\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">It is given that&nbsp;\(\displaystyle\sum_{r=1}^{4}u_r=110\). Find the value of \(a\).<span class="inline-marks">[3]</span></div></li></ol>

A sequence \(u_1, u_2, u_3, \ldots\) is defined by

\(u_1=a,\)
\(u_{r+1}=3u_r+5,  r\geqslant 1,\)

Find expressions for \(u_2\) and \(u_3\) in terms of \(a\).

It is given that \(\displaystyle\sum_{r=1}^{4}u_r=110\). Find the value of \(a\).

Sigma Notation

N2025 P1 Q1

Recurrence Relations and Summation

Prove by mathematical induction that _r=1n 1(3r-2)(3r+1)= n3n+1 for every n in Z+.

Prove by mathematical induction that \(\displaystyle\sum_{r=1}^{n}\dfrac{1}{(3r-2)(3r+1)}=\dfrac{n}{3n+1}\) for every \(n\in\mathbb Z^+\).

A Telescoping Reciprocal Sum

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph"><p>The sum of the first \(20\) terms of an arithmetic sequence is \(65\). The sum of the next \(8\) terms is \(-30\).</p><p>Find the first term and the common difference of this sequence.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph"><p>A sequence has&nbsp;\(n\)th term given by \(u_n=e^{k(n+1)}\), where&nbsp;\(k\) is a constant and \(k\ne 0\).</p><p>Show that this sequence is geometric.</p><span class="inline-marks">[2]</span></div></li></ol>

Proving the progression

Block Sums

Geometric Progression

Arithmetic Progression

N2025 P1 Q2

APGP

<p>The function \(y=\mathrm{f}(x)\) is defined by \(\mathrm{f}(x)=x^2-2x+6\), with \(x\leq1\).</p>
<ol class="abc-list"><li data-list-item-id="p1q4-a" data-list-item-path="p1q4-a"><div class="list-item-paragraph">Find \(\mathrm{f}^{-1}(x)\) and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q4-b" data-list-item-path="p1q4-b"><div class="list-item-paragraph">Find the range of the function \(\mathrm{g}(x)=3+\sqrt{\mathrm{f}(x)}\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(y=\mathrm{f}(x)\) is defined by \(\mathrm{f}(x)=x^2-2x+6\), with \(x\leq1\).

Find \(\mathrm{f}^{-1}(x)\) and state its domain.

Find the range of the function \(\mathrm{g}(x)=3+\sqrt{\mathrm{f}(x)}\).

Function Notation, Domain and Range

Inverse Functions

Range of a Function

IB Math HL January 2016 P1 Q4

Functions

<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjQyMCwiaGVpZ2h0IjozMDAsImVsZW1lbnRzIjpbeyJpZCI6ImltcG9ydGVkLXBhcmFtZXRyaWMtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMCIsInRNYXgiOjIuMDk0Mzk1MTAyMzkzMTk1MywidE1pbiI6LTIuMDk0Mzk1MTAyMzkzMTk1MywidHlwZSI6InBhcmFtZXRyaWMiLCJjb2xvciI6IiMwMDAwMDAiLCJleHByZXNzaW9uWCI6IjI4Kih0KmNvcyh0KSsxKSIsImV4cHJlc3Npb25ZIjoiMjAqKDItdCpzaW4odCkpIiwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6ImltcG9ydGVkLXNlZ21lbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMjgsOC41ODQwNzM0NjQxMDIwNjldLCJwb2ludDIiOlsyOCw0MF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTIiLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzEyLjI5MiwyNi45NTZdLCJwb2ludDIiOls0My43MDgsMjYuOTU2XSwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjoyNywieSI6NDIsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0zIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEFcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjI3LCJ5Ijo2LjUsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC00IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEJcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjkuNSwieSI6MjYuMywiaWQiOiJpbXBvcnRlZC10ZXh0LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTUiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoQ1xcKSIsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6NDUsInkiOjI2LjMsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC02IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKERcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjI4LjgsInkiOjI4LjQsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC03IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEVcXCkiLCJzdHJva2VXaWR0aCI6MS41fV0sInNob3dYQXhpcyI6dHJ1ZSwic2hvd1lBeGlzIjp0cnVlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInhBeGlzUmFuZ2UiOlstOC41LDY0XSwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJ5QXhpc1JhbmdlIjpbLTUsNDhdLCJib3VuZGluZ0JveCI6Wy01LDUwLDY1LC01XSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwiZ3JpZFNldHRpbmdzIjp7Im1ham9yQ29sb3IiOiIjOTRhM2I4IiwibWFqb3JTdGVwWCI6MSwibWFqb3JTdGVwWSI6MSwibWlub3JDb2xvciI6IiNjYmQ1ZTEiLCJtYWpvck9wYWNpdHkiOjAuNDUsIm1pbm9yT3BhY2l0eSI6MC40NSwibWlub3JFbGVtZW50c1giOjQsIm1pbm9yRWxlbWVudHNZIjo0LCJtYWpvclN0cm9rZVdpZHRoIjowLjgsIm1pbm9yU3Ryb2tlV2lkdGgiOjAuNSwic2hvd01ham9yR3JpZExpbmVzIjp0cnVlLCJzaG93TWlub3JHcmlkTGluZXMiOnRydWV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJ4VGlja0xhYmVsRXZlcnkiOjEsInlUaWNrTGFiZWxFdmVyeSI6MSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xNCw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstNSwtMl19" data-width="420" data-height="300" style="width:420px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations</p>
<p style="text-align: center">\(x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),\)</p>
<p>for&nbsp;\(-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), where&nbsp;\(x\) and \(y\) are measured in metres. The \(x\)-axis models the horizontal ground.</p>
<p>The maximum point on the curve is at \(A\), and the line \(AB\) is vertical. Points \(C\) and \(D\) are where the tangent to the curve is parallel to the \(y\)-axis. Point \(E\) is the intersection of the lines \(AB\) and \(CD\).</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><ol class="i-list"><li data-list-item-id="imported-item-block-3-item-1-sub-0" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-0"><div class="list-item-paragraph">Find the values of \(\theta\) at points \(C\) and \(D\), giving your answers correct to \(2\) significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1-sub-1" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-1"><div class="list-item-paragraph">Hence find the greatest width, \(CD\), of the loop.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>The \(x\)-coordinate of point \(A\) is \(28\).</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-5-item-0-sub-0" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-0"><div class="list-item-paragraph">Find the value of \(\theta\) at point \(A\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-1" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-1"><div class="list-item-paragraph">Find the \(2\) possible values of \(\theta\) at point \(B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-2" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-2"><div class="list-item-paragraph">Hence find the distance \(AB\).<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>Engineers know that, for safety reasons, the distance \(AE\) should be within \(10\%\) of the distance \(\frac{1}{2}CD\).</p>
<ol class="abc-list" start="4"><li data-list-item-id="imported-item-block-7-item-0" data-list-item-path="imported-item-block-7-item-0"><div class="list-item-paragraph">Determine whether the design of this section of track satisfies this condition.<span class="inline-marks">[2]</span></div></li></ol>

Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations

\(x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),\)

for \(-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), where \(x\) and \(y\) are measured in metres. The \(x\)-axis models the horizontal ground.

The maximum point on the curve is at \(A\), and the line \(AB\) is vertical. Points \(C\) and \(D\) are where the tangent to the curve is parallel to the \(y\)-axis. Point \(E\) is the intersection of the lines \(AB\) and \(CD\).

Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta\).

The \(x\)-coordinate of point \(A\) is \(28\).

Engineers know that, for safety reasons, the distance \(AE\) should be within \(10\%\) of the distance \(\frac{1}{2}CD\).

Determine whether the design of this section of track satisfies this condition.

Tangent and Normal Parallel to the Axes

Problems Involving Maximisation and Minimisation

N2025 P1 Q11

Applications of Differentiation

<p>Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.</p>

Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.

Permutation and Combination

<p>Simplify \(\sum_{r=1}^{n}(2^{-r}-4)\) in terms of \(n\).<span class="inline-marks">[4]</span></p>

Simplify \(\sum_{r=1}^{n}(2^{-r}-4)\) in terms of \(n\).

Geometric and mixed sums

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;\(\sin 2 nx = \sin((2n + 1)x) \cos x - \cos((2n + 1)x) \sin x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph"><p><strong>Hence</strong> prove, by induction, that \[\cos x + \cos 3x + \cos 5x + ... + \cos((2n - 1)x) = \frac{\sin 2nx}{2 \sin x},\] for all \(n \in \mathbb{Z}^+\), \(\sin x \neq 0\).</p><span class="inline-marks">[12]</span></div></li></ol>

Show that \(\sin 2 nx = \sin((2n + 1)x) \cos x - \cos((2n + 1)x) \sin x\).

Mathematical Induction

<ol class="abc-list"><li data-list-item-id="t06-p1-q06-a" data-list-item-path="t06-p1-q06-a"><div class="list-item-paragraph">Show that&nbsp;\(\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="t06-p1-q06-b" data-list-item-path="t06-p1-q06-b"><div class="list-item-paragraph">Hence prove by mathematical induction that&nbsp;\(\sin x+\sin3x+\sin5x+\cdots+\sin((2n-1)x)=\frac{1-\cos2nx}{2\sin x}\) for all \(n\in\mathbb Z^+\), \(\sin x\ne0\).<span class="inline-marks">[7]</span></div></li><li data-list-item-id="t06-p1-q06-c" data-list-item-path="t06-p1-q06-c"><div class="list-item-paragraph">Hence solve \(\sin x+\sin3x=\cos x\) for&nbsp;\(0&lt;x&lt;\pi\).<span class="inline-marks">[5]</span></div></li></ol>

Show that \(\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x\).

Hence prove by mathematical induction that \(\sin x+\sin3x+\sin5x+\cdots+\sin((2n-1)x)=\frac{1-\cos2nx}{2\sin x}\) for all \(n\in\mathbb Z^+\), \(\sin x\ne0\).

Hence solve \(\sin x+\sin3x=\cos x\) for \(0<x<\pi\).

Compound and Double-Angle Identities

Trigonometric Equations and Identities - General

Proof by Mathematical Induction

Trigonometry

<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

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x\).<span class="inline-marks">[2]</span></div></li></ol>
<p>A curve \(C\) is given by the equation \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm{e}\).</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Using the result in part <strong>(a)</strong>, find the exact volume generated when&nbsp;\(R\) is rotated \(360^\circ\) about the \(x\)-axis.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find the volume generated when&nbsp;\(R\) is rotated \(360^\circ\) about the \(y\)-axis, giving your answer correct to \(2\) decimal places.<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x\).

A curve \(C\) is given by the equation \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm{e}\).

Find the volume generated when \(R\) is rotated \(360^\circ\) about the \(y\)-axis, giving your answer correct to \(2\) decimal places.

Volume of Revolution about the y-axis

Volume of Revolution about the x-axis

N2025 P1 Q7

Applications of Integration

<p>Prove, by induction, \((1+a)^n \ge 1+na\), where&nbsp;\(a \in \mathbb{R}^+\), for all positive integers of \(n\).<span class="inline-marks">[5]</span></p>

Prove, by induction, \((1+a)^n \ge 1+na\), where \(a \in \mathbb{R}^+\), for all positive integers of \(n\).

<p>It is given that&nbsp;\(y = \sin^{-1} px\), where&nbsp;\(p\) is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(a)</strong> find, in terms of \(p\), the Maclaurin series for&nbsp;\(y\) up to and including the term in \(x^3\).</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph"><p>Use your answer to part <strong>(b)</strong> to find the series expansion of \(\frac{1}{\sqrt{1 - 9x^2}}\), up to and including the term in \(x^2\).</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that \(y = \sin^{-1} px\), where \(p\) is a constant.

Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y\).

Maclaurin Series by Successive Differentiation

Problems involving Calculus

N2025 P1 Q9

Maclaurin Series and Power Series

<ol class="abc-list list-columns-2" data-list-columns="2" data-list-flow="column"><li data-list-item-id="item-1787074918782-eg1jkwaa5" data-list-item-path="item-1787074918782-eg1jkwaa5" style="grid-column: 1; grid-row: 1;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{{{x}^{2}}{{({{x}^{3}}+3)}^{3}}}\,\mathrm{d}x\)</p></div></li><li data-list-item-id="item-1787074955422-xdz0eoruj" data-list-item-path="item-1787074955422-xdz0eoruj" style="grid-column: 1; grid-row: 2;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{(5{{x}^{4}}+4x)({{x}^{5}}+2{{x}^{2}}}+1{{)}^{3}}\,\mathrm{d}x\)</p></div></li><li data-list-item-id="item-1787074966558-y8nv8u0g3" data-list-item-path="item-1787074966558-y8nv8u0g3" style="grid-column: 1; grid-row: 3;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{\frac{3x-2}{{{\left( 3{{x}^{2}}-4x-1 \right)}^{3}}}}\,\mathrm{d}x\)</p></div></li><li data-list-item-id="item-1787075040359-5v8laelox" data-list-item-path="item-1787075040359-5v8laelox" style="grid-column: 1; grid-row: 4;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{2}}}{\sqrt{{{x}^{3}}+1}}\,\mathrm{d}x}\) </p></div></li><li data-list-item-id="item-1787075051473-jvo81iutj" data-list-item-path="item-1787075051473-jvo81iutj" style="grid-column: 1; grid-row: 5;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{\frac{21{{x}^{2}}+14}{\sqrt{{{x}^{3}}+2x+1}}\,\mathrm{d}x}\)</p></div></li><li data-list-item-id="item-1787075064040-viw4ca6vv" data-list-item-path="item-1787075064040-viw4ca6vv" style="grid-column: 2; grid-row: 1;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{\frac{x}{{{\sqrt{1-2{{x}^{2}}}}^{5}}}\,\mathrm{d}x}\)</p></div></li><li data-list-item-id="item-1787075092632-q54lic0m0" data-list-item-path="item-1787075092632-q54lic0m0" style="grid-column: 2; grid-row: 2;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{{{\left( \frac{x}{\sqrt{3{{x}^{4}}-1}} \right)}^{3}}\,\mathrm{d}x}\)</p></div></li><li data-list-item-id="item-1787075106784-v2pxwt1ua" data-list-item-path="item-1787075106784-v2pxwt1ua" style="grid-column: 2; grid-row: 3;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{3{{x}^{3}}{{(2{{x}^{4}}-5)}^{8}}}\,\mathrm{d}x\)</p></div></li><li data-list-item-id="item-1787075114848-bpmjbbn6z" data-list-item-path="item-1787075114848-bpmjbbn6z" style="grid-column: 2; grid-row: 4;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{{{e}^{3x}}{{({{e}^{3x}}+5)}^{11}}}\,\mathrm{d}x\)</p></div></li><li data-list-item-id="item-1787075124273-jeym71jxc" data-list-item-path="item-1787075124273-jeym71jxc" style="grid-column: 2; grid-row: 5;"><div class="list-item-paragraph"><p>\(\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{3}}}{{{x}^{4}}+3}\text{}\hspace{0.5em}}\mathrm{d}x\)</p></div></li></ol>

Cancelling Method of MF 26 Integrals

Techniques of Integration

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<p>The diagram shows a hemispherical bowl, of inside radius \(15\text{ cm}\), fixed so its circular rim is horizontal. When the depth of water in the bowl is \(h\text{ cm}\), the volume, \(V\text{ cm}^3\), of water in the bowl is given by \(V=\dfrac{1}{3}\pi h^2(45-h).\)</p>
<p>The bowl is initially empty, and water is poured into the bowl at the constant rate of \(10\pi\text{ cm}^3\) per second.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Find the time taken to fill the bowl.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Find the exact rate at which the depth of water is increasing when the depth of water is \(12\text{ cm}\).<span class="inline-marks">[4]</span></div></li></ol>
<p>The bowl is emptied, and water is poured into the bowl at the variable rate of \(10\pi t\text{ cm}^3\) per second, where&nbsp;\(t\) seconds is the time from when pouring started.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1787122464331-8w1rq71ge" data-list-item-path="item-1787122464331-8w1rq71ge"><div class="list-item-paragraph"><p>Find the time taken to pour \(972\pi\text{ cm}^3\) of water into the bowl.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1787122465108-w2kgz23s2" data-list-item-path="item-1787122465108-w2kgz23s2"><div class="list-item-paragraph"><p>Find the rate at which the depth of water is increasing when the volume of water in the bowl is \(972\pi\text{ cm}^3\).</p><span class="inline-marks">[4]</span></div></li></ol>

The diagram shows a hemispherical bowl, of inside radius \(15\text{ cm}\), fixed so its circular rim is horizontal. When the depth of water in the bowl is \(h\text{ cm}\), the volume, \(V\text{ cm}^3\), of water in the bowl is given by \(V=\dfrac{1}{3}\pi h^2(45-h).\)

The bowl is initially empty, and water is poured into the bowl at the constant rate of \(10\pi\text{ cm}^3\) per second.

Find the exact rate at which the depth of water is increasing when the depth of water is \(12\text{ cm}\).

The bowl is emptied, and water is poured into the bowl at the variable rate of \(10\pi t\text{ cm}^3\) per second, where \(t\) seconds is the time from when pouring started.

Connected Rates of Changes

N2025 P1 Q4

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Without using a calculator, solve the inequality \(\frac{4}{x + 2} \geqslant \frac{x - 3}{x}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">Hence, solve the inequality \(\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}\).<span class="inline-marks">[2]</span></div></li></ol>

Without using a calculator, solve the inequality \(\frac{4}{x + 2} \geqslant \frac{x - 3}{x}\).

Hence, solve the inequality \(\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}\).

Inequalities involving Substitution

Inequalities involving Rational Functions

Inequalities involving Modulus Functions

N2024 P1 Q4

Inequalities

<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 \(n \in \mathbb{Z}^+\).

<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x&gt;-1\).</p>
<ol class="i-list"><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-0" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}\) has an inverse.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-1" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-1"><div class="list-item-paragraph">Find \(\mathrm{f}^{-1}(x)\) and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-2" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-2"><div class="list-item-paragraph">State the range of \(\mathrm{f}^{-1}\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-3" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-3"><div class="list-item-paragraph">State the equation of the line in which the graph of \(y=\mathrm{f}(x)\) must be reflected in order to obtain the graph of \(y=\mathrm{f}^{-1}(x)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-4" data-list-item-path="h2-2025-mi-promo-q9-mi-q9-item-block-1-item-4"><div class="list-item-paragraph">Hence, or otherwise, find the exact solutions of the equation \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\).<span class="inline-marks">[3]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x>-1\).

State the equation of the line in which the graph of \(y=\mathrm{f}(x)\) must be reflected in order to obtain the graph of \(y=\mathrm{f}^{-1}(x)\).

Hence, or otherwise, find the exact solutions of the equation \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\).

Horizontal Line Test

Self-inverse Functions: \( \mathrm{f} = \mathrm{f}^{-1}\)

2025 MI PU1 Promo Q9

<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.

IB AA HL Math Question Bank