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

<p>The table below shows experimental values of two variables $x$ and $y$ which are connected by an equation of the form $ax^2 + by = 1$, where&nbsp;$a$ and $b$ are constants.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 490px;"><colgroup><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$5$</th><th style="text-align: center; --cell-justify: center">$6$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$0.47$</td><td style="text-align: center; --cell-justify: center">$0.87$</td><td style="text-align: center; --cell-justify: center">$1.53$</td><td style="text-align: center; --cell-justify: center">$2.47$</td><td style="text-align: center; --cell-justify: center">$3.87$</td><td style="text-align: center; --cell-justify: center">$5.14$</td></tr></tbody></table></div></div>
<p>Using a suitable scale, plot the graph of $y$ against $x^2$. Use your graph to</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">determine which value of $y$ is inaccurate and estimate its correct value,</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">estimate the value of $a$ and of $b$.</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">By drawing a suitable straight line, find the values of $x$ and $y$ which satisfy the following pair of simultaneous equations.</div><div class="list-item-paragraph" style="text-align: center">$ax^2 + by = 1$<br>$2x^2 + y = 3$</div></li></ol>
<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/pasted/1775549390753-ag7pmkj47wl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU1NDkzOTA3NTMtYWc3cG1rajQ3d2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNywiZXhwIjoxNzkzNTQ2OTE3fQ._u5jLMuV-sD9nmsoO9vLrXqLJOOgB6tgaGbrMnmRauQ" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The table below shows experimental values of two variables $x$ and $y$ which are connected by an equation of the form $ax^2 + by = 1$, where $a$ and $b$ are constants.

Using a suitable scale, plot the graph of $y$ against $x^2$. Use your graph to

determine which value of $y$ is inaccurate and estimate its correct value,

estimate the value of $a$ and of $b$.

Plotting Linearised Data and Lines of Best Fit

Solving Equations with Additional Lines on Linearised Graphs

Identifying and Correcting Anomalous Linearised Data

Interpreting Linearised Graphs and Finding Constants

Algebraic Linearisation

Linear Law

<p>The table below shows experimental values of two variables $x$ and $y$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 420px;"><colgroup><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$0.5$</th><th style="text-align: center; --cell-justify: center">$1.5$</th><th style="text-align: center; --cell-justify: center">$2.5$</th><th style="text-align: center; --cell-justify: center">$3.5$</th><th style="text-align: center; --cell-justify: center">$4.5$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$12$</td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$74$</td><td style="text-align: center; --cell-justify: center">$185$</td><td style="text-align: center; --cell-justify: center">$463$</td></tr></tbody></table></div></div>
<p>It is known that&nbsp;$x$ and $y$ are related by the equation of the form $y^2 = \frac{a^x}{b}$, where&nbsp;$a$ and $b$ are constants.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Express this equation in a form suitable for drawing a straight line graph.</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Using a suitable scale, draw the graph for the given data and use it to estimate the value of $a$ and of $b$.</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-content"><span class="list-item-text">Estimate the value of $x$ for which $\lg \sqrt{y} = 0.7$.</span></div></li></ol>
<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/pasted/1775549390753-ag7pmkj47wl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU1NDkzOTA3NTMtYWc3cG1rajQ3d2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNywiZXhwIjoxNzkzNTQ2OTE3fQ._u5jLMuV-sD9nmsoO9vLrXqLJOOgB6tgaGbrMnmRauQ" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Graph paper provided for plotting" class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Graph paper provided for plotting</div></div>

The table below shows experimental values of two variables $x$ and $y$.

It is known that $x$ and $y$ are related by the equation of the form $y^2 = \frac{a^x}{b}$, where $a$ and $b$ are constants.

Express this equation in a form suitable for drawing a straight line graph.

Using a suitable scale, draw the graph for the given data and use it to estimate the value of $a$ and of $b$.

Estimate the value of $x$ for which $\lg \sqrt{y} = 0.7$.

Predictions from Linearised Models

Linearisation Using Logarithms

<p>The variables $x$ and $y$ are related by a certain equation. This equation may be expressed in two forms suitable for drawing line graphs. The two graphs shown have different variables plotted at each axis. Given the coordinates of a point on each line, find the original equation relating $x$ and $y$.</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;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div></div></div>

The variables $x$ and $y$ are related by a certain equation. This equation may be expressed in two forms suitable for drawing line graphs. The two graphs shown have different variables plotted at each axis. Given the coordinates of a point on each line, find the original equation relating $x$ and $y$.

<p>The variables $x$ and $y$ are related by the equation $y=a{{b}^{x}}-3$. When&nbsp;$\lg \left( y+3 \right)$ is plotted against $x$, a straight line graph is obtained which passes through $\left( -2,5 \right)$ and $\left( 10,-1 \right)$. Find the value of $a$ and $b$.</p>

The variables $x$ and $y$ are related by the equation $y=a{{b}^{x}}-3$. When $\lg \left( y+3 \right)$ is plotted against $x$, a straight line graph is obtained which passes through $\left( -2,5 \right)$ and $\left( 10,-1 \right)$. Find the value of $a$ and $b$.

<p>The diagram shows part of a straight-line graph drawn to represent the equation $y = nA^x$. Find the value of $n$.</p>
<div class="graph-block" data-graph-config="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" data-width="380" data-height="200" style="width:380px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>

The diagram shows part of a straight-line graph drawn to represent the equation $y = nA^x$. Find the value of $n$.

<p>A straight line graph is drawn for&nbsp;$\lg \frac{1}{y}$ against $x$ to represent the curve $\frac{1}{y} = A^{(x+B)}$. Given that the line passes through the points $(0, 4)$ and $(3, 13)$, find the value of $A$ and $B$. Hence, find the value of $x$ when&nbsp;$\frac{\lg \frac{1}{y}}{x} = 4$.</p>

A straight line graph is drawn for $\lg \frac{1}{y}$ against $x$ to represent the curve $\frac{1}{y} = A^{(x+B)}$. Given that the line passes through the points $(0, 4)$ and $(3, 13)$, find the value of $A$ and $B$. Hence, find the value of $x$ when $\frac{\lg \frac{1}{y}}{x} = 4$.

<p>The variables $x$ and $y$ are connected by the equation $a^y = kx$, where&nbsp;$k$ and $a$ are constants. The straight line passes through the points $(0, 2)$ and $(3, 5)$ and these points are obtained by plotting $y$ against $\ln x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $a$ and $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $y$ when&nbsp;$x = 0.2$.</span></div></li></ol>

The variables $x$ and $y$ are connected by the equation $a^y = kx$, where $k$ and $a$ are constants. The straight line passes through the points $(0, 2)$ and $(3, 5)$ and these points are obtained by plotting $y$ against $\ln x$.

Find the value of $y$ when $x = 0.2$.

<p>The figure shows a straight line graph which is obtained by plotting $\lg y$ against $\lg \frac{1}{x}$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="225" style="width:300px;height:225px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the value of $\theta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Express $\lg y$ in terms of $\lg \frac{1}{x}$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence, express $y^4$ against $\frac{1}{x}$.</div></li></ol>

The figure shows a straight line graph which is obtained by plotting $\lg y$ against $\lg \frac{1}{x}$.

Express $\lg y$ in terms of $\lg \frac{1}{x}$.

Hence, express $y^4$ against $\frac{1}{x}$.

<p>In the figure, $x$ and $y$ are related by the equation $y = 5x - \frac{\sqrt{x}}{2}$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="" style="max-width: min(380px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<div class="graph-block" data-graph-config="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" data-width="350" data-height="200" style="width:350px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the value of $\theta$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the coordinates of $A$ and $B$.</div></li></ol>

In the figure, $x$ and $y$ are related by the equation $y = 5x - \frac{\sqrt{x}}{2}$.

<p>When&nbsp;$\lg \left( y+2 \right)$ is plotted against $\lg x$, a straight line graph is obtained. Given that the line passes through $\left( 2,6 \right)$ and $\left( 7,0 \right)$. Express $y$ in terms of $x$.</p>

When $\lg \left( y+2 \right)$ is plotted against $\lg x$, a straight line graph is obtained. Given that the line passes through $\left( 2,6 \right)$ and $\left( 7,0 \right)$. Express $y$ in terms of $x$.

<p>It is given that&nbsp;$y = ax^{b}$, where&nbsp;$a$ and $b$ are constants. The diagram below shows the equation by plotting $\lg y$ against $\lg x$. Find the value of $a$ and of $b$.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjMwMCwiaGVpZ2h0IjoyMDAsImVsZW1lbnRzIjpbeyJpZCI6IjU1ZmRlZDkzLWUzNWItNDU0OC04YzE1LTMyMWUwNDJmMGNkYyIsInR5cGUiOiJmdW5jdGlvbmdyYXBoIiwieE1heCI6Ny41LCJ4TWluIjotMC41LCJjb2xvciI6IiMwMDAwMDAiLCJleHByZXNzaW9uIjoiLTUvNyAqeCArIDUiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjAsInkiOjUsImlkIjoiYmNkMzE5MjktZWQwMS00MDljLTg5MTYtZmU4NGM3YWZmZGNlIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCggKDAsNSkgXFwpIiwicG9pbnRTaXplIjoxfSx7IngiOjcsInkiOjAsImlkIjoiNWMxNzYwMmEtMmFjMS00NTUyLWFlY2EtZTM4MjNjZGEzYWM2IiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCggKDcsMCkgXFwpIiwicG9pbnRTaXplIjoxLCJsYWJlbE9mZnNldCI6Wy0yMCwtMTBdfV0sInhBeGlzTGFiZWwiOiJcXChcXGxne3h9XFwpIiwieUF4aXNMYWJlbCI6IlxcKFxcbGd7eX1cXCkiLCJib3VuZGluZ0JveCI6Wy0xLjUsNi41LDEzLC0xXSwib3JpZ2luTGFiZWwiOiIiLCJncmlkU2V0dGluZ3MiOnsibWFqb3JDb2xvciI6IiM5NGEzYjgiLCJtYWpvclN0ZXBYIjoxLCJtYWpvclN0ZXBZIjoxLCJtaW5vckNvbG9yIjoiI2NiZDVlMSIsIm1ham9yT3BhY2l0eSI6MC40NSwibWlub3JPcGFjaXR5IjowLjQ1LCJtaW5vckVsZW1lbnRzWCI6NCwibWlub3JFbGVtZW50c1kiOjQsIm1ham9yU3Ryb2tlV2lkdGgiOjAuOCwibWlub3JTdHJva2VXaWR0aCI6MC41fSwic2hvd0F4aXNMYWJlbHMiOmZhbHNlLCJzaG93TmF2aWdhdGlvbiI6ZmFsc2UsImtlZXBBc3BlY3RSYXRpbyI6ZmFsc2UsInhBeGlzTGFiZWxPZmZzZXQiOlstMTEsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0yNSw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstNSwtMl19" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>

It is given that $y = ax^{b}$, where $a$ and $b$ are constants. The diagram below shows the equation by plotting $\lg y$ against $\lg x$. Find the value of $a$ and of $b$.

<p>Variables $x$ and $y$ are connected by the equation $y={{10}^{k-nx}}$, where&nbsp;$n$ and $k$ are constants. When a graph of $\lg y$ is plotted against $x$, a straight line passing through the points $\left( 2,4 \right)$ and $\left( 7,5 \right)$ is obtained. Find the value of $n$ and $k$.</p>

Variables $x$ and $y$ are connected by the equation $y={{10}^{k-nx}}$, where $n$ and $k$ are constants. When a graph of $\lg y$ is plotted against $x$, a straight line passing through the points $\left( 2,4 \right)$ and $\left( 7,5 \right)$ is obtained. Find the value of $n$ and $k$.

<p>When the graph of $\frac{{{y}^{2}}}{x}$ against $\ln x$ is drawn, a straight line of $\ln x=\frac{{{y}^{2}}}{kx}+\frac{n}{3}$ is obtained. Given that this line passes through the point $\left( 2,5 \right)$ and has a gradient $3$, find the values of $k$ and $n$.</p>

When the graph of $\frac{{{y}^{2}}}{x}$ against $\ln x$ is drawn, a straight line of $\ln x=\frac{{{y}^{2}}}{kx}+\frac{n}{3}$ is obtained. Given that this line passes through the point $\left( 2,5 \right)$ and has a gradient $3$, find the values of $k$ and $n$.

<p>Variables $x$ and $y$ are related in such a way such that, when&nbsp;$\frac{x}{y}$ is plotted against $e^{x}$, a straight line is produced. The line passes through the points $(2, 4)$ and $(7, 19)$ as shown in the diagram. Find $y$ in terms of $x$.</p>
<div class="graph-block" data-graph-config="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" data-width="250" data-height="210" style="width:250px;height:210px;max-width:100%;display:block;margin:0 auto;"></div>

Variables $x$ and $y$ are related in such a way such that, when $\frac{x}{y}$ is plotted against $e^{x}$, a straight line is produced. The line passes through the points $(2, 4)$ and $(7, 19)$ as shown in the diagram. Find $y$ in terms of $x$.

<p>The diagram below shows a straight-line graph where&nbsp;$xy$ is plotted against $x^{2} + y$. Express $y$ in terms of $x$.</p>
<div class="graph-block" data-graph-config="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" data-width="370" data-height="210" style="width:370px;height:210px;max-width:100%;display:block;margin:0 auto;"></div>

The diagram below shows a straight-line graph where $xy$ is plotted against $x^{2} + y$. Express $y$ in terms of $x$.

<p>A straight line passing through $R(1, -7)$, $S(5, k)$ and $T(9, 9)$ is obtained when $\lg y$ is plotted against $x$, where $x$ and $y$ are variables.</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/pasted/1768305062719-su27zvdz8go.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgzMDUwNjI3MTktc3UyN3p2ZHo4Z28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNywiZXhwIjoxNzkzNTQ2OTE3fQ.Gv0xvnrG37VIpTxSCe5CkGywfRlTE3VeYyUwA5LKeWA" style="max-width: min(170px, 100%); height: auto; display: block;" alt="Graph of lg y against x" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $y$ in terms of $x$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that $k = 1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the value of $y$ at point $S$.</span><span class="inline-marks">[1]</span></div></li></ol>

A straight line passing through $R(1, -7)$, $S(5, k)$ and $T(9, 9)$ is obtained when $\lg y$ is plotted against $x$, where $x$ and $y$ are variables.

Hence, find the value of $y$ at point $S$.

2023 CSS AM S3 EOY Q2

<p>The figure shows part of a straight line graph obtained by plotting values of the variables $\frac{y-1}{x}$ against $x$. Express $y$ in terms of $x$.</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/pasted/1775556653561-hq2pa2jzrc4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU1NTY2NTM1NjEtaHEycGEyanpyYzQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNywiZXhwIjoxNzkzNTQ2OTE3fQ.sN_TLZ6390N26HkyrDXS1HsEN754Mn_muCUspqVsssI" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows part of a straight line graph obtained by plotting values of the variables $\frac{y-1}{x}$ against $x$. Express $y$ in terms of $x$.

<p>Variables $x$ and $y$ are related by the equation $y = ax^b - 5$. When&nbsp;$\lg(y + 5)$ is plotted against $\lg x$, a straight line graph is obtained which passes through $(2, -4)$ and $(-2, 8)$, as shown in the diagram. Find the value of $a$ and $b$.</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/pasted/1768291381239-sruedq55s4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyOTEzODEyMzktc3J1ZWRxNTVzNC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQ2NjE3LCJleHAiOjE3OTM1NDY5MTd9.ruLuzuAbyItvlfsbW8WO2EM1HyKUNf7wuAg3tKdvGZ4" style="max-width: min(270px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Variables $x$ and $y$ are related by the equation $y = ax^b - 5$. When $\lg(y + 5)$ is plotted against $\lg x$, a straight line graph is obtained which passes through $(2, -4)$ and $(-2, 8)$, as shown in the diagram. Find the value of $a$ and $b$.

<p>The diagram shows part of a straight-line graph drawn to represent the equation $y = Ax^n$. Find the value of $A$ and $n$.</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/pasted/1768291352606-u11q230jw6p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyOTEzNTI2MDYtdTExcTIzMGp3NnAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNywiZXhwIjoxNzkzNTQ2OTE3fQ.vGn-zdz83RPxqrrJQWtuaIIXxYFet2oV5KChv7XOxYE" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram shows part of a straight-line graph drawn to represent the equation $y = Ax^n$. Find the value of $A$ and $n$.

<p>Variables $x$ and $y$ are related by the equation of the form $y = 10^{a(x+b)}$, where $a$ and $b$ are constants. When $\lg y$ is plotted against $x$, a straight line which passes through the point $(1, -2)$ and $(5, 4)$ is obtained. Find the value of $a$ and of $b$. Hence, find the value of $\lg y^{(2x)}$ when $x = 3$.</p>

Variables $x$ and $y$ are related by the equation of the form $y = 10^{a(x+b)}$, where $a$ and $b$ are constants. When $\lg y$ is plotted against $x$, a straight line which passes through the point $(1, -2)$ and $(5, 4)$ is obtained. Find the value of $a$ and of $b$. Hence, find the value of $\lg y^{(2x)}$ when $x = 3$.

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