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PSLE

PSLE Foundation Mathematics

<p>Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 2x^3 - 5x^2 + x - 2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = 4x^2 e^x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{\ln x}{2x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = |2x - x^2|$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-1 &lt; a &lt; 0$ and $0 &lt; b &lt; 1$;</span></div></li><li><div class="list-item-content"><span class="list-item-text">$a &lt; -1$ and $b &gt; 1$.</span></div></li></ol></li></ol>
<p>What could be said about the graph if&nbsp;$ab + 1 = 0$?</p>

Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.

$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$

$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by

What could be said about the graph if $ab + 1 = 0$?

Sketching Exponential Functions and Solving Equations Graphically

Sketching Rational Functions

Sketching Logarithmic Functions and Solving Equations Graphically

Asymptotes and Discontinuities

Graphs of Modulus Function

Polynomial Functions and Key Features

Graphing Techniques

Transformations of Graphs

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{\ln(3x - 2)}{x}, \, x &gt; \frac{2}{3}\]</p>
<p style="text-align: center">\[\mathrm{g} : x \mapsto 3x^2 - 12x + 13, \, x \in \mathbb{R}\]</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">Sketch the graph of $y = \mathrm{f}(x)$, indicating the equations of asymptotes; coordinates of intersections with the axes and turning points, if any.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{fg}$ exists and find its range.<span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{h}$ is defined as</p>
<p style="text-align: center">\[\mathrm{h} : x \mapsto \mathrm{g}(x), \, x \le k\]</p>
<p>such that&nbsp;$\mathrm{h}^{-1}$ exists.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-content"><span class="list-item-text">State the largest value of $k$. For this value of $k$, find $\mathrm{h}^{-1}$ in a similar form.</span><span class="inline-marks">[4]</span></div></li></ol>

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

\[\mathrm{f} : x \mapsto \frac{\ln(3x - 2)}{x}, \, x > \frac{2}{3}\]

\[\mathrm{g} : x \mapsto 3x^2 - 12x + 13, \, x \in \mathbb{R}\]

Sketch the graph of $y = \mathrm{f}(x)$, indicating the equations of asymptotes; coordinates of intersections with the axes and turning points, if any.

The function $\mathrm{h}$ is defined as

\[\mathrm{h} : x \mapsto \mathrm{g}(x), \, x \le k\]

State the largest value of $k$. For this value of $k$, find $\mathrm{h}^{-1}$ in a similar form.

Composite Functions

Existence of Composite Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Inverse Functions

2024 ACJC Promo Q5

Functions

<ol class="abc-list"><li><div class="list-item-paragraph">Solve $6^x = \frac{10}{3} - 6^{-x}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y = \ln x$, showing any points of intersection with the axes.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">To solve $e^{1-2x} = x^3$, a straight line can be drawn on the same axes as the graph in part <strong>(b)</strong>:</div><ol class="i-list"><li><div class="list-item-paragraph">Determine the equation of the straight line to be drawn.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, state the number of solutions for&nbsp;$e^{1-2x} = x^3$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Sketch the graph of $y = \ln x$, showing any points of intersection with the axes.

Solving Exponential Equations Using Substitution

IP3 Class Test (May 2025) Q10 (2023 Jurong Sec AM Prelim P2 Q6)

Exponential and Logarithmic Functions

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} &gt; 0$ for $x &gt; 1$.</span><span class="inline-marks">[4]</span></div></li></ol>

Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.

Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} > 0$ for $x > 1$.

Solving Logarithm Equations

2023 Ahmad Ibrahim Sec AM S3 EOY Q8

<p>The diagram below shows the graph of $y={{\log }_{3}}x$. By drawing a suitable line on the diagram, solve the equation ${{9}^{x}}=3x$.</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/1766125163835-58vb5dlxot7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxMjUxNjM4MzUtNTh2YjVkbHhvdDcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcxNzI2MSwiZXhwIjoxNzkyNzE3NTYxfQ.gWSe9dBYO4Q6WkFgK_n6BsfDhLyBE34xucTz_0zYo-U" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram below shows the graph of $y={{\log }_{3}}x$. By drawing a suitable line on the diagram, solve the equation ${{9}^{x}}=3x$.

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\lg x$.</div></li><li><div class="list-item-paragraph">In order to solve the equation ${{10}^{x}}=\frac{10\,000}{{{x}^{3}}}$, a suitable straight line has to be drawn on the same set of axes as the graph 	of $y=\lg x$. Find the equation of the straight line and the number of solutions.</div></li></ol>

<p>Sketch the graph of $y=\ln \sqrt{3x-1}$ for&nbsp;$x&gt;\frac{1}{3}$. Find the equation of the straight line which must be added on the graph of $y=\ln \sqrt{3x-1}$ to solve the equation ${{\mathrm{e}}^{3x}}+{{\mathrm{e}}^{4}}=3x{{\mathrm{e}}^{4}}$. <br><br></p>

Sketch the graph of $y=\ln \sqrt{3x-1}$ for $x>\frac{1}{3}$. Find the equation of the straight line which must be added on the graph of $y=\ln \sqrt{3x-1}$ to solve the equation ${{\mathrm{e}}^{3x}}+{{\mathrm{e}}^{4}}=3x{{\mathrm{e}}^{4}}$. 



<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\ln \left( 2+x \right)$. Find the equation of the straight line which must be added on the graph of 	$y=\ln \left( 2+x \right)$ to solve the equation ${{\left( 2+x \right)}^{2}}{{\mathrm{e}}^{-x}}=2$.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y=\ln \sqrt{\frac{1}{x-1}}$ for&nbsp;$x&gt;1$. Insert in your sketch the additional graph required to illustrate how a graphical solution to the equation $x={{\mathrm{e}}^{6\left( 2-x \right)}}+1$ may be obtained.</div></li></ol>

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\ln x$.</div></li><li><div class="list-item-paragraph">In order to solve the equation ${{\mathrm{e}}^{x-3}}={{x}^{2}}$, a suitable straight line has to be drawn on the same set of axes as the graph of $y=\ln x$. Find the equation of the straight line and the number of solutions.<br><br></div></li></ol>

<p>The diagram shows two curves ${{C}_{1}}$ and ${{C}_{2}}$ which represent $y=\ln \left( x+2 \right)$ and $y=\lg \left( x+2 \right)$  respectively, for&nbsp;$x&gt;a$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the value of $a$.</div></li><li><div class="list-item-paragraph">Explain briefly if&nbsp;${{C}_{1}}$ represents the graph of $y=\ln \left( x+2 \right)$ .</div></li><li><div class="list-item-paragraph">Find the solution in which $\lg \left( x+2 \right)=\ln \left( x+2 \right)$.</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/1766124187344-kj2vz1yep3a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxMjQxODczNDQta2oydnoxeWVwM2EucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcxNzI2MSwiZXhwIjoxNzkyNzE3NTYxfQ.9oKzQtxaYev6C2ggtSsZNeM5kbcXMKEMEYpc-6Cw0vo" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram shows two curves ${{C}_{1}}$ and ${{C}_{2}}$ which represent $y=\ln \left( x+2 \right)$ and $y=\lg \left( x+2 \right)$  respectively, for $x>a$.

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\ln x$.</div></li><li><div class="list-item-paragraph">In order to solve the equation ${{x}^{2}}={{\operatorname{e}}^{2x-3}}$, a suitable straight line has to be drawn on the same diagram as the graph of $y=\ln x$. Find the equation of the straight line and the number of solutions to the equation.</div></li></ol>

Solving Equation Graphically

<p>The curve of $y={{\log }_{a}}\left( 2x+b \right)+1$ passes through $\left( \frac{3}{2},2 \right)$ and $\left( \frac{93}{2},3 \right)$. Find the values of $a$ and $b$.</p>
<p>Hence, sketch the graphs of $y={{\log }_{a}}\left( 2x+b \right)+1$, labelling the coordinates of axial intercepts and equations of asymptotes.</p>

The curve of $y={{\log }_{a}}\left( 2x+b \right)+1$ passes through $\left( \frac{3}{2},2 \right)$ and $\left( \frac{93}{2},3 \right)$. Find the values of $a$ and $b$.

Hence, sketch the graphs of $y={{\log }_{a}}\left( 2x+b \right)+1$, labelling the coordinates of axial intercepts and equations of asymptotes.

<p>The curve of $y=a\ln \left( 2x-\operatorname{e} \right)+1$ passes through $\left(\operatorname{e},4 \right)$. Find the value of $a$.</p>
<p>Hence, sketch the graphs of $y=a\ln \left( 2x+\operatorname{e} \right)+1$, labelling the coordinates of axial intercepts and equations of asymptotes.</p>

The curve of $y=a\ln \left( 2x-\operatorname{e} \right)+1$ passes through $\left(\operatorname{e},4 \right)$. Find the value of $a$.

Hence, sketch the graphs of $y=a\ln \left( 2x+\operatorname{e} \right)+1$, labelling the coordinates of axial intercepts and equations of asymptotes.

<p>The curve of $y=a{{\log }_{2}}\left( bx+1 \right)+4$ passes through $\left( 1,6 \right)$ and $\left( 3,8 \right)$. Find the values of $a$ and $b$.</p>
<p>Hence, sketch the graphs of $y=a{{\log }_{2}}\left( bx+1 \right)+4$, labelling the coordinates of axial intercepts and equations of asymptotes.</p>

The curve of $y=a{{\log }_{2}}\left( bx+1 \right)+4$ passes through $\left( 1,6 \right)$ and $\left( 3,8 \right)$. Find the values of $a$ and $b$.

Hence, sketch the graphs of $y=a{{\log }_{2}}\left( bx+1 \right)+4$, labelling the coordinates of axial intercepts and equations of asymptotes.

Finding Unknowns of a Logarithmic Function

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y={{\log }_{\frac{1}{3}}}\left( 2x-1 \right)+2$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y=2\ln \left( 2-3x \right)$</div></li></ol></div></div></div>

Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y=\ln \left( 2-x \right)+1$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y={{\log }_{{{\mathrm{e}}^{-1}}}}\left( 2x-3 \right)$</div></li></ol></div></div></div>

<p>Sketch the graphs of each of the following functions, labelling the coordinates of axial intercepts and equations of asymptotes.</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;"><ol class="abc-list"><li><div class="list-item-paragraph">$y=\ln \left( x+1 \right)+3$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="list-item-paragraph">$y={{\log }_{\frac{1}{2}}}\sqrt{x+3}$</div></li></ol></div></div></div>

Graphs of Logarithmic Functions

<ol class="abc-list"><li><div class="list-item-paragraph">Solve the equation ${\log_e{(x^2 + 4x - 1)} = 2 + 2\log_e{(3x + 2)}}$<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Sketch the graphs of ${y = \log_e{x}}$ and ${y = e^{-x}}$ on the same axes. Use your graphs to</div><div class="list-item-paragraph">explain why ${\log_e{x} + e^{-x} > 0}$ for ${x > 1}$.<span class="inline-marks">[4]</span></div></li></ol>

2023 AISS AM S3 EOY Q8

<ol class="i-list"><li data-list-item-id="item-1760424465495-p0zuh823u" data-list-item-path="item-1760424465495-p0zuh823u"><div class="list-item-paragraph">Sketch, on the same diagram, the graphs of</div><div class="list-item-paragraph" style="text-align: center">$y=\frac{2\ln x}{\cos x} \quad$ and $ \quad y=3\cos x\ln x \quad $ for&nbsp;$ \quad \frac{\pi }{2} &lt; x &lt; \frac{3\pi }{2}$,</div><div class="list-item-paragraph">showing clearly the equations of any asymptotes and the coordinates of any axial intercepts.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760424466275-z8ztuo33r" data-list-item-path="item-1760424466275-z8ztuo33r"><div class="list-item-paragraph">The region $R$ is bounded by the two curves in part <strong>(i)</strong>. Find the area of $R$, giving your answer to $2$ decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760424467318-aifm62j8f" data-list-item-path="item-1760424467318-aifm62j8f"><div class="list-item-paragraph">The region $R$ in part <strong>(ii)</strong> is rotated $2\pi $ radians about the $x$-axis to form a solid $S$. Find the volume of $S$.<span class="inline-marks">[2]</span></div></li></ol>

Volume of Revolution about the x-axis

Area bounded by 2 curves

2023 MI P1 Q5

Applications of Integration

<p>On the same axes, sketch the graphs of $y=2+{{\mathrm{e}}^{-{{x}^{2}}}}$ and $y=\left| \ln \left( x-1 \right) \right|$. Label clearly the equations of any asymptotes and any points of intersection with the axes.</p><p style="text-align: right;">[4]</p>
<p>Hence solve the inequality $2+{{\mathrm{e}}^{-{{x}^{2}}}}\le \left| \ln \left( x-1 \right) \right|$.</p><p style="text-align: right;">[2]</p>

On the same axes, sketch the graphs of $y=2+{{\mathrm{e}}^{-{{x}^{2}}}}$ and $y=\left| \ln \left( x-1 \right) \right|$. Label clearly the equations of any asymptotes and any points of intersection with the axes.

Hence solve the inequality $2+{{\mathrm{e}}^{-{{x}^{2}}}}\le \left| \ln \left( x-1 \right) \right|$.

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