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<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU1NDkzOTA3NTMtYWc3cG1rajQ3d2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNSwiZXhwIjoxNzkzNTQ2OTE1fQ.SH1KE-epTLAQpkJY2N3f1z0BICDcRicozTwBZp9Z3mY" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU1NDkzOTA3NTMtYWc3cG1rajQ3d2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0NjYxNSwiZXhwIjoxNzkzNTQ2OTE1fQ.SH1KE-epTLAQpkJY2N3f1z0BICDcRicozTwBZp9Z3mY" 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 \(\lg \sqrt{y} = 0.7\).

Math Question Bank