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<p>The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.<span class="inline-marks">[3]</span></p>
<p>Deduce that the geometric progression is convergent.<span class="inline-marks">[2]</span></p>
<p>Let&nbsp;$S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.<span class="inline-marks">[4]</span></p>

The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.

Deduce that the geometric progression is convergent.

Let $S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.

VJC P1 Q13

APGP

<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="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjI1MCwiaGVpZ2h0IjoyNTAsImVsZW1lbnRzIjpbeyJ4IjoxLCJ5IjowLCJpZCI6IjZiODUwNTNhLTJhNTgtNDQxZS04ZTdhLTBlNTI0YzM4MDc3NiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImZpeGVkIjpmYWxzZSwibGFiZWwiOiJcXCgxXFwpIiwidmlzaWJsZSI6dHJ1ZSwicG9pbnRTaXplIjoxLjUsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy05LjMsLTEzLjNdfSx7IngiOjMsInkiOjAsImlkIjoiNjgwMWE2YTUtNDdiYi00NTcwLThjZDEtMTc5NWFlMGEwYWFhIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOmZhbHNlLCJsYWJlbCI6IlxcKDNcXCkiLCJ2aXNpYmxlIjpmYWxzZSwicG9pbnRTaXplIjoxLCJwb2ludFN0eWxlIjoiY2lyY2xlIiwibGFiZWxPZmZzZXQiOls2LjksLTEzLjNdfSx7IngiOjQsInkiOjIsImlkIjoiYTY1M2JhMWEtMWNjZi00ZGNjLWE3ODMtMWE3NDJkMzQ3MmE1IiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOnRydWUsImxhYmVsIjoiXFwoKDQsMilcXCkiLCJwb2ludFNpemUiOjEsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy05LjYsMTMuMl19LHsieCI6NSwieSI6LTAuMDMsImlkIjoiYWQxNmVmNTgtNTk3MS00MzJlLTkxMTctMGE4MmE4M2ExMzM4IiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwiZml4ZWQiOmZhbHNlLCJsYWJlbCI6IlxcKDVcXCkiLCJwb2ludFNpemUiOjEsInBvaW50U3R5bGUiOiJjaXJjbGUiLCJsYWJlbE9mZnNldCI6Wy04LjcsLTEyLjRdfSx7IngiOjIsInkiOi0yLCJpZCI6IjEwMzNhMzg4LTE0ZjMtNDFmZi1hZjk0LWIwZmE5YjE0NWIyZCIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImZpeGVkIjp0cnVlLCJsYWJlbCI6IlxcKCgyLC0yKVxcKSIsInBvaW50U2l6ZSI6MSwicG9pbnRTdHlsZSI6ImNpcmNsZSIsImxhYmVsT2Zmc2V0IjpbLTIwLjQsLTEzLjRdfSx7ImlkIjoiN2YxNmI0MzctOTU5Zi00MmQzLTgwZTctMmRlMGJjMjhjODk0IiwiZGFzaCI6MiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlstOS4yOCwtNF0sInBvaW50MiI6WzkuMjQsLTRdLCJ2aXNpYmxlIjp0cnVlLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiOGMyNTY4MDAtODU1Mi00ZWM0LTlhMjgtYjlhZTZmZTg4ZjZiIiwidHlwZSI6InNwbGluZSIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1swLjIyLDEwLjE2XSxbMC41NywzLjE1XSxbMS4wMSwwXSxbMS40LC0xLjQyXSxbMi4wMSwtMl0sWzIuNiwtMS41NF0sWzMsMF0sWzMuMzEsMS4yMV0sWzQuMDEsMl0sWzQuNjEsMS4zMV0sWzUuMDEsMF0sWzUuMjQsLTAuOTldLFs2LjYzLC0xMC45N11dLCJ0ZW5zaW9uIjoxLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiZWYzNzUzN2QtZjM2Ny00MDY1LTg1MDAtY2FmOTc2NjhhNTY0IiwidHlwZSI6InNwbGluZSIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1stOS42MywtNC4zOV0sWy00LjI4LC01LjI4XSxbLTEuMTYsLTcuMjldLFstMC4zMywtMTIuMDRdXSwidGVuc2lvbiI6MSwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjowLjU3LCJ5IjotNy4yOSwiaWQiOiI5ZDkyZTNlNS1jNmM3LTQyYmUtOTA5My03NjRjMjgwMDEzMDciLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeD0wXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOi02Ljc0LCJ5IjotMy4xMiwiaWQiOiJhMTc0NWRhZi01NTJmLTQ0NzMtYjhmYi1iZWViZTU3NDc2NGIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT00XFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjEuNCwieSI6NS42MSwiaWQiOiI5ZjBiMmY0Yy1iOGE5LTQ0ZDYtYmZjZS0wMzc4NjcwMDlmMjgiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT1cXG1hdGhybXtmfSh4KVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJpZCI6ImRiY2JkZWNjLWVmMmQtNGMwNi05OTAzLWU3MDMxNjliMDIyNSIsImRhc2giOjMsInR5cGUiOiJwcm9qZWN0aW9uIiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiIiLCJwb2ludDEiOlsxLDBdLCJsYWJlbE9mZnNldCI6WzAsLTE0XSwic3Ryb2tlV2lkdGgiOjEsInByb2plY3Rpb25BeGlzIjoieCIsInByb2plY3Rpb25Tb3VyY2VJZCI6ImE5NTY1N2ZjLWI5MWUtNGQ5Ni1iNmE4LTI1YjFlNTZmODhiZSIsInByb2plY3Rpb25Tb3VyY2VGaWVsZCI6Inh5In0seyJ4IjozLCJ5IjowLCJpZCI6ImJhODcyNjc5LTM0YjAtNGNhNy05ZjI4LTEzYzgwNDFmY2UxYyIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiIiwicG9pbnRTaXplIjoxLCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4IjozLCJ5IjotMC42LCJpZCI6IjFiYzgzNmQ2LTNmOGItNDZhMy04YTFiLTEyZDlmNTVhMmI0ZSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCgzXFwpIiwidGV4dFJvdGF0aW9uIjowfV0sInNob3dYQXhpcyI6dHJ1ZSwic2hvd1lBeGlzIjp0cnVlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstOCw4LjUsNywtOV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtYWpvckNvbG9yIjoiIzk0YTNiOCIsIm1ham9yU3RlcFgiOjEsIm1ham9yU3RlcFkiOjEsIm1pbm9yQ29sb3IiOiIjY2JkNWUxIiwibWFqb3JPcGFjaXR5IjowLjU1LCJtaW5vck9wYWNpdHkiOjAuMzUsIm1pbm9yRWxlbWVudHNYIjo0LCJtaW5vckVsZW1lbnRzWSI6NCwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjpmYWxzZSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xMy4zLDMuOV0sIm9yaWdpbkxhYmVsT2Zmc2V0IjpbLTUsLTJdfQ==" 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;"><p>Given that&nbsp;$(3,0)$ is a point of inflection of $y=\mathrm{f}\left( x \right)$. State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721008576-2k1azs74n" data-list-item-path="item-1782721008576-2k1azs74n"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721009392-41xoixgky" data-list-item-path="item-1782721009392-41xoixgky"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721009848-wlrux4odu" data-list-item-path="item-1782721009848-wlrux4odu"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721010480-idm7y5b1r" data-list-item-path="item-1782721010480-idm7y5b1r"><div class="list-item-paragraph">concave downwards.</div></li></ol></div></div></div>

Concavity and Gradients

Transformations of Graphs

<p>A function is defined by $f(x)=\dfrac{5x-2}{2x-3}$, where&nbsp;$x\ne \frac{3}{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the $x$- and $y$-intercepts of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\dfrac{5x-2}{2x-3}$, where $x\ne \frac{3}{2}$.

<ol class="i-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">Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where&nbsp;$r &gt; 0$ and $y \le r$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A hemispherical bowl of fixed radius $r\text{ cm}$ is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate.</p>
<p>Use your graph in part <strong>(i)</strong> to show that when the depth of water is $h\text{ cm}$ (where&nbsp;$h \le r$), the volume of water in the bowl is given by</p>
<p style="text-align: center">\[V = \frac{\pi h^2}{3}(3r - h),\]</p>
<p style="text-align: right">[4]</p>
<ol class="i-list" start="2"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Given that a full bowl of water would become empty in $24\text{ s}$, find the rate of decrease, in terms of $r$ and $h$, of the depth of water in the bowl at the instant when the depth of water is $h\text{ cm}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Without any differentiation, determine, in terms of $r$, the slowest rate at which the depth of water is decreasing.<span class="inline-marks">[2]</span></div></li></ol>

Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where $r > 0$ and $y \le r$.

A hemispherical bowl of fixed radius $r\text{ cm}$ is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate.

Use your graph in part **(i)** to show that when the depth of water is $h\text{ cm}$ (where $h \le r$), the volume of water in the bowl is given by

Given that a full bowl of water would become empty in $24\text{ s}$, find the rate of decrease, in terms of $r$ and $h$, of the depth of water in the bowl at the instant when the depth of water is $h\text{ cm}$.

Without any differentiation, determine, in terms of $r$, the slowest rate at which the depth of water is decreasing.

2016 VJC P1 Q9

Applications of Differentiation

<p>A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where&nbsp;$x\ne -4$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the $x$- and $y$-intercepts of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where $x\ne -4$.

Transformations of a rational function and modulus

<p>$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$</p>

$\displaystyle\int{\frac{1}{\sqrt{7+10x-{{x}^{2}}}}}\mathrm{ d}x$

Integration of ${\frac{1}{\sqrt{a^2 - x^2}}, \frac{1}{a^2 + x^2}, \frac{1}{a^2 - x^2}, \frac{1}{x^2 - a^2}}$

Techniques of  Integration

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the curve with equation $y=\left|\frac{4}{p-x}\right|$, where&nbsp;$p$ is a positive constant. State, in terms of $p$, the equations of the asymptotes and the coordinates of any intersections with the coordinate axes. On the same diagram, sketch the line with equation $y=q(x-p)$, where&nbsp;$q$ is a positive constant.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find, in terms of $p$ and $q$, the root of the equation $\left|\frac{4}{p-x}\right|=q(x-p)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence solve the inequality $\left|\frac{4}{p-x}\right|&gt;q(x-p)$.<span class="inline-marks">[2]</span></div></li></ol>

Sketching of Rational Functions

Inequalities involving Rational Functions

Inequalities involving Modulus Functions

Modulus Rational Function and Linear Intersection

Graphing Techniques

Inequalities

<p>Without the use of a calculator, show that</p>
<p style="text-align: center">\[\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\,\mathrm{d}x=p\ln 2+q\pi,\]</p>
<p>where&nbsp;$p$ and $q$ are exact real constants to be determined.<span class="inline-marks">[6]</span></p>

Without the use of a calculator, show that

\[\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\,\mathrm{d}x=p\ln 2+q\pi,\]

where $p$ and $q$ are exact real constants to be determined.

Integration of Algebraic Fractions

Exact Definite Integral with a Modulus

<ol class="abc-list" start="2"><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">Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.<span class="inline-marks">[3]</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">Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.

Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.

2016 DHS C1BT Q2

<p>A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:</p>
<ol class="ABC-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">A translation by $2$ units in the negative $x$-direction.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A scaling by a scale factor of $3$ parallel to the $x$-axis.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A reflection about the $y$-axis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A translation by $5$ units in the positive $y$-direction.</div></li></ol>
<p>The equation of the resulting curve is $y=12-\dfrac{4}{3}x+e^{\frac{2}{3}x-4}$.</p>
<p>Find the equation of the original curve $y=f(x)$.<span class="inline-marks">[5]</span></p>

A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:

The equation of the resulting curve is $y=12-\dfrac{4}{3}x+e^{\frac{2}{3}x-4}$.

Find the equation of the original curve $y=f(x)$.

Recovering a function after successive graph transformations

<p>An upright vessel is in the shape of a right circular cone with vertical height $12\text{ cm}$ and base radius $5\text{ cm}$. The vessel is initially completely filled with water. Water then leaks from a small hole at the base at a constant rate of $20\pi\text{ cm}^3\text{/s}$. At time $t$ seconds, the water has depth $h\text{ cm}$ and its surface has radius $r\text{ cm}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph"><p>Using similar triangles, show that the volume $V\text{ cm}^3$ of water remaining in the vessel is given by $V=25\pi \left(h- \frac{h^2}{12}+\frac{h^3}{432} \right)$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the rate at which the depth of the water is decreasing at the instant when&nbsp;$h=6$.<span class="inline-marks">[3]</span></div></li></ol>

An upright vessel is in the shape of a right circular cone with vertical height $12\text{ cm}$ and base radius $5\text{ cm}$. The vessel is initially completely filled with water. Water then leaks from a small hole at the base at a constant rate of $20\pi\text{ cm}^3\text{/s}$. At time $t$ seconds, the water has depth $h\text{ cm}$ and its surface has radius $r\text{ cm}$.

Connected Rates of Changes

Water Leaking from an Upright Conical Vessel

<p>The probability distribution function of a discrete random variable, $X$, is given as</p>
<p style="text-align: center">\[\mathrm{P}(X = x) = \frac{k}{x!},\text{ for}\hspace{0.5em} x = 0, 1, 2, 3, \dots,\]</p>
<p>where&nbsp;$k$ is a positive constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">By considering an expansion found in the list of formulae (MF27), show that&nbsp;$k = \frac{1}{\mathrm{e}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the value of $\mathrm{E}(X)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{Var}(X) = \mathrm{E}(X)$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the probability that, in a random sample of 50 observations of $X$, the mean of the sample is at least 0.95.<span class="inline-marks">[3]</span></div></li></ol>

The probability distribution function of a discrete random variable, $X$, is given as

\[\mathrm{P}(X = x) = \frac{k}{x!},\text{ for } x = 0, 1, 2, 3, \dots,\]

It is given that $\mathrm{Var}(X) = \mathrm{E}(X)$.

Discrete Random Variables (DRV)

<p>Consider a function $g$ defined as&nbsp;$g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.</p>
<ol class="abc-list"><li data-list-item-id="inverse-restrict-q3a" data-list-item-path="inverse-restrict-q3a"><div class="list-item-paragraph">Sketch the graph of $g$ and hence explain why $g$ cannot have an inverse.</div></li><li data-list-item-id="inverse-restrict-q3b" data-list-item-path="inverse-restrict-q3b"><div class="list-item-paragraph">If&nbsp;$g:x\mapsto (x+2)^2-3,\ x\ge q$ has an inverse, write down the smallest value $q$ can take.</div></li><li data-list-item-id="inverse-restrict-q3c" data-list-item-path="inverse-restrict-q3c"><div class="list-item-paragraph">For this value of $q$, find the function $g^{-1}$.</div></li><li data-list-item-id="inverse-restrict-q3d" data-list-item-path="inverse-restrict-q3d"><div class="list-item-paragraph">Sketch the graphs of $g$ and $g^{-1}$ on the same axes, clearly showing the relationship between them.</div></li><li data-list-item-id="inverse-restrict-q3e" data-list-item-path="inverse-restrict-q3e"><div class="list-item-paragraph">Write down the range of $g^{-1}$.</div></li><li data-list-item-id="item-1783676614199-a1m0ed6xb" data-list-item-path="item-1783676614199-a1m0ed6xb"><div class="list-item-paragraph">Find the exact solution of the equation $g (x) =g^{-1}(x)$.</div></li></ol>

Consider a function $g$ defined as $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$.

Functions

<p>A curve $C$ is given by the equation</p>
<p style="text-align: center">\[e^{\frac{x^2}{2}}y=\ln x^2,\]</p>
<p>where&nbsp;$x\ne0$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$x\ln x^2+e^{\frac{x^2}{2}}\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point $P$ where&nbsp;$x=1$. Given that the tangent at $P$ meets $C$ again at the point $Q$, find the coordinates of $Q$, giving your answer correct to 5 significant figures.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Determine the acute angle between the tangents to $C$ at the points $P$ and $Q$.<span class="inline-marks">[3]</span></div></li></ol>

Implicit Curve Tangents and Angle of Intersection

<p>The curve whose equation is $y=3-e^x$ undergoes, in succession, the following transformations:</p>
<p>A: Translation by $2$ units in the positive $x$-direction.</p>
<p>B: Scaling by a scale factor of $\frac{1}{2}$ parallel to the $x$-axis.</p>
<p>C: Translation by $1$ unit in the positive $y$-direction.</p>
<p>D: Scaling by a scale factor of $2$ parallel to the $y$-axis.</p>
<p>Find the equation of the resulting curve.<span class="inline-marks">[5]</span></p>

The curve whose equation is $y=3-e^x$ undergoes, in succession, the following transformations:

A: Translation by $2$ units in the positive $x$-direction.

B: Scaling by a scale factor of $\frac{1}{2}$ parallel to the $x$-axis.

C: Translation by $1$ unit in the positive $y$-direction.

D: Scaling by a scale factor of $2$ parallel to the $y$-axis.

Find the equation of the resulting curve.

Successive transformations of an exponential curve

<p>A function is defined by $f(x)=\ln(x-2)$, where&nbsp;$x\ne 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\ln(x-2)$, where $x\ne 2$.

<p>Sand is poured onto a horizontal surface to form a right circular conical pile. The radius of the base of the pile is always $\frac{3}{4}$ of its vertical height. The volume of the pile is increasing at a constant rate of $10\text{ cm}^3\text{/s}$. Find the rate at which the curved surface area of the pile is increasing at the instant when its height is $4\text{ cm}$.<span class="inline-marks">[5]</span></p>

Sand is poured onto a horizontal surface to form a right circular conical pile. The radius of the base of the pile is always $\frac{3}{4}$ of its vertical height. The volume of the pile is increasing at a constant rate of $10\text{ cm}^3\text{/s}$. Find the rate at which the curved surface area of the pile is increasing at the instant when its height is $4\text{ cm}$.

Connected Rates for a Conical Sand Pile

<p>The curve whose equation is $y=x^3+2$ undergoes, in succession, the following transformations:</p>
<p>A: Translation by $3$ units in the positive $x$-direction.</p>
<p>B: Scaling by a scale factor of $2$ parallel to the $x$-axis.</p>
<p>C: Translation by $4$ units in the negative $y$-direction.</p>
<p>D: Scaling by a scale factor of $3$ parallel to the $y$-axis.</p>
<p>Find the equation of the resulting curve.<span class="inline-marks">[5]</span></p>

The curve whose equation is $y=x^3+2$ undergoes, in succession, the following transformations:

A: Translation by $3$ units in the positive $x$-direction.

B: Scaling by a scale factor of $2$ parallel to the $x$-axis.

C: Translation by $4$ units in the negative $y$-direction.

D: Scaling by a scale factor of $3$ parallel to the $y$-axis.

Successive transformations of a cubic curve

<p>A curve, $y=f(x)$, undergoes the transformations $A$, $B$, $C$ and $D$ in succession:</p>
<ol class="ABC-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">A translation by $5$ units in the negative $x$-direction.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A scaling by a scale factor of $3$ parallel to the $x$-axis.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A reflection about the $y$-axis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A translation by $2$ units in the negative $y$-direction.</div></li></ol>
<p>The equation of the resulting curve is $y=9-x+2\ln\left(5-\dfrac{x}{3}\right)$, where&nbsp;$x&lt;15$.</p>
<p>Find the equation of the original curve $y=f(x)$, and state its domain.<span class="inline-marks">[6]</span></p>

The equation of the resulting curve is $y=9-x+2\ln\left(5-\dfrac{x}{3}\right)$, where $x<15$.

Find the equation of the original curve $y=f(x)$, and state its domain.

Recovering a logarithmic function after graph transformations

<p>$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$</p>

$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$

H2 Math Question Bank