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PSLE

PSLE Foundation Mathematics

<div class="graph-block" data-graph-config="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" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<p>It is given that&nbsp;$\mathrm{f}(x)$ is a cubic function. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the factorised form.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find the value of $\mathrm{f}(1)$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}(x)$ is a cubic function. 

Forming Polynomials from Roots and Factors

Polynomial Functions and Key Features

Polynomials, Cubic Equations and Partial Fractions

Functions and Graphs

<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

Graphing Techniques

Transformations of Graphs

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the curve $y = x(x - 3)(x - 5)$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the following integrals.</div><ol class="i-list"><li data-list-item-id="subitem-i" data-list-item-path="item-b/subitem-i"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{0}^{3} x^3 - 9x^2 + 18x \, \mathrm{d}x$</div></li><li data-list-item-id="subitem-ii" data-list-item-path="item-b/subitem-ii"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{3}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x$</div></li><li data-list-item-id="subitem-iii" data-list-item-path="item-b/subitem-iii"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{0}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x$</div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Explain why the area between the curve and the $x$-axis in the interval $0 \le x \le 6$ is not the answer to <strong>(b)(iii)</strong>. And find the area between the curve and the $x$-axis for the interval $0 \le x \le 6$.</div></li></ol>
<p>In part (c), use the curve $y=x^3-9x^2+18x=x(x-3)(x-6)$ from part (b). The sketch in part (a) is a separate curve.</p>

Sketch the curve $y = x(x - 3)(x - 5)$.

Explain why the area between the curve and the $x$-axis in the interval $0 \le x \le 6$ is not the answer to **(b)(iii)**. And find the area between the curve and the $x$-axis for the interval $0 \le x \le 6$.

In part (c), use the curve $y=x^3-9x^2+18x=x(x-3)(x-6)$ from part (b). The sketch in part (a) is a separate curve.

Understanding Results of Definite Integrals

Area Enclosed by the Curve and the X-Axis

Definite Integral

Integration

Differentiation

<p>A curve has the equation $y = -2x^3 - 3x^2 - x + 5$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the coordinates of the stationary point(s) on the curve.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Determine the nature of the stationary point(s) by using the second derivative test.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence sketch the curve.</div></li></ol>

A curve has the equation $y = -2x^3 - 3x^2 - x + 5$.

Find the coordinates of the stationary point(s) on the curve.

Determine the nature of the stationary point(s) by using the second derivative test.

Stationary Points and Their Nature

<p>A curve has the equation $y = 2x^3 + 3x^2 - 120x + 4$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the coordinates of the stationary point(s) on the curve.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Determine the nature of the stationary point(s) by using the second derivative test.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence sketch the curve.</div></li></ol>

A curve has the equation $y = 2x^3 + 3x^2 - 120x + 4$.

<p>Find the stationary point of the curve $y=x^3-1$ and determine its nature. Hence, sketch the graph.<br><br></p>

Find the stationary point of the curve $y=x^3-1$ and determine its nature. Hence, sketch the graph.



Stationary Points of Inflection

Applications of Differentiation

<p>Find the coordinates of the stationary points on the curve $y=2x^3+3x^2+x-5$. Determine the nature of each point by using the second derivative test. Hence, sketch the curve.</p>

Find the coordinates of the stationary points on the curve $y=2x^3+3x^2+x-5$. Determine the nature of each point by using the second derivative test. Hence, sketch the curve.

<p>Find the coordinates of the stationary points on the curve $y=2x^3+3x^2-120x+4$. Determine the nature of each point by using the second derivative test. Hence, sketch the curve.</p>

Find the coordinates of the stationary points on the curve $y=2x^3+3x^2-120x+4$. Determine the nature of each point by using the second derivative test. Hence, sketch the curve.

Second Derivative Test

<p>Find the coordinates of the stationary points on the curve $y=x^3-3x+2$. Determine each of its nature by using the first derivative test. Hence, sketch the curve.<br><br></p>

Find the coordinates of the stationary points on the curve $y=x^3-3x+2$. Determine each of its nature by using the first derivative test. Hence, sketch the curve.



<p>Find the coordinates of the stationary points on the curve $y=12x^3+3x^2-6x+1$. Determine the nature of each point by using the first derivative test. Hence, sketch the curve.</p>

Find the coordinates of the stationary points on the curve $y=12x^3+3x^2-6x+1$. Determine the nature of each point by using the first derivative test. Hence, sketch the curve.

First Derivative Test

<ol class="abc-list"><li><div class="list-item-paragraph">Find $\mathrm{Q}\left( x \right)$ if&nbsp;${{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x+3=\left( {{x}^{2}}-5x+3 \right)\mathrm{Q}\left( x \right)+9$.</div></li><li><div class="list-item-paragraph">Solve ${{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x-6=0$. Hence sketch the curve $y={{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x-6$ and state the range of values of 	$x$ such that</div><ol class="i-list"><li><div class="list-item-paragraph">$y&gt;0$,</div></li><li><div class="list-item-paragraph">$\frac{1}{y}&lt;0$.</div></li></ol></li></ol>

Polynomial Inequalities and Sign Charts

Polynomial Division and Remainders

Equations Reducible to Quadratics

Quadratic Functions, Equations and Inequalities

<p>The graph of $y={{x}^{3}}-4{{x}^{2}}+8x+c$ is shown.</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">State the value of $c$.</div></li><li><div class="list-item-paragraph">Explain if&nbsp;${{x}^{3}}-4{{x}^{2}}+8x+c$ can be expressed as a product of three linear factors.</div></li><li><div class="list-item-paragraph">Find the linear factor of ${{x}^{3}}-4{{x}^{2}}+8x+c$ and hence find the value of $x$-intercept of the curve. </div></li><li><div class="list-item-paragraph">Find the coordinates of the intersection points of the curve $y={{x}^{3}}-4{{x}^{2}}+8x+c$ and the line $y=7x-21$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766070081795-14sk0uy8wqj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYwNzAwODE3OTUtMTRzazB1eTh3cWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1NjQwOSwiZXhwIjoxNzkzNTU2NzA5fQ.AV6ckyv52Yo5QObOxXOd23ZVL8ItoIEe7zW9NdH6FEc" style="max-width: min(130px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The graph of $y={{x}^{3}}-4{{x}^{2}}+8x+c$ is shown.

Cubic Expressions and Equations

Linear and Non-Linear Simultaneous Equations

Remainder and Factor Theorems

<ol class="abc-list"><li data-list-item-id="item-1761383465097-lhs8n3c5h" data-list-item-path="item-1761383465097-lhs8n3c5h"><div class="list-item-paragraph">On the same diagram, sketch the graphs $y=\left| 2x-1 \right|$ and $y={{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383465907-xaflj7mr2" data-list-item-path="item-1761383465907-xaflj7mr2"><div class="list-item-paragraph">Hence find the set of values of $x$ for which $\left| 2x-1 \right|&lt;{{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383466491-iqg4ujtg8" data-list-item-path="item-1761383466491-iqg4ujtg8"><div class="list-item-paragraph">Without using a calculator, find the range of values of $k$ for which $\left| 2x-1 \right|={{x}^{2}}-4x-2+k$ has only positive real roots.<span class="inline-marks">[4]</span></div></li></ol>

Parameter Conditions for Graphs and Equations

Graphical Method

Inequalities involving Modulus Functions

2025 YIJC P1 Q7

Inequalities

<p>It is given that</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x\le 5$,</p>
<p style="text-align: center">$\mathrm{g}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x < 3$,</p>
<p style="text-align: center">$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
\ln x\,\mathrm{for}0&lt;x\le 1, \\
{{x}^{2}}-6x+5\mathrm{for}1&lt;x\le 5, \\
\end{matrix} \right.$
</p>
<p>and that $\mathrm{h}\left( x \right)=\mathrm{h}\left( x-5 \right)$ for all real values of $x$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the range of $\mathrm{f}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Show that ${{\mathrm{g}}^{-1}}$ exists.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{g}}^{-1}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact value of $\mathrm{h}\left( \frac{11}{2} \right)+\mathrm{h}\left( -2 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{h}\left( x \right)$ for $-4\le x\le 6$, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.</div><p style="text-align: right;">[3]</p></li></ol>

$\mathrm{f}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x\le 5$,

$\mathrm{g}\left( x \right)={{x}^{2}}-6x+5$, for $1 < x < 3$,

$\mathrm{h}\left( x \right)=\left\{ \begin{matrix}
\ln x\,\mathrm{for}0&lt;x\le 1, \\
{{x}^{2}}-6x+5\mathrm{for}1&lt;x\le 5, \\
\end{matrix} \right.$


and that $\mathrm{h}\left( x \right)=\mathrm{h}\left( x-5 \right)$ for all real values of $x$.

One-to-One Functions and Inverse Existence

Piecewise-Defined Functions

Periodic Functions

Inverse Functions

Range of a Function

2022 ACJC Promo Q9

Functions

<ol class="abc-list"><li><div class="list-item-paragraph">The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-4$ for&nbsp;$x\in \mathbb{R}$, $x&lt;-2$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why ${{\mathrm{f}}^{-1}}$ exists.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Sketch on the same diagram, the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\,\mathrm{f}\left( x \right)$, stating the coordinates of endpoints and equations of asymptotes, if any.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Describe the relationship between the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.<span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \ln \left( x+10 \right)$, where&nbsp;$x&gt;-10$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that the composite function $\mathrm{gf}$ exists. Find the range of $\mathrm{gf}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Solve $\mathrm{f}\left( x \right)={{\mathrm{g}}^{-1}}\left( \ln \left( 6 \right) \right)$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Sketching a Function and Its Inverse

Existence of Composite Functions

Sketching $y=f(f^{-1}(x))$ and $y=f^{-1}(f(x))$

Range of a Composite Function

2022 SAJC Promo Q6

<p>A function $\mathrm{f}$ has the equation $\mathrm{f}\left( x \right)=\frac{1}{5}{{x}^{5}}+2{{x}^{3}}+10x-25$.</p>
<ol class="abc-list"><li data-list-item-id="item-1759833652403-ep9su9lxm" data-list-item-path="item-1759833652403-ep9su9lxm"><div class="list-item-paragraph">Without using a calculator, and by considering $\mathrm{f}'\left( x \right)$, explain why the equation $\mathrm{f}\left( x \right)=0$ has only one real root between $0$ and $2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759833653821-grwscm0zp" data-list-item-path="item-1759833653821-grwscm0zp"><div class="list-item-paragraph">Find the root, giving your answer to $3$ decimal places.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759833655088-ek5duwdrs" data-list-item-path="item-1759833655088-ek5duwdrs"><div class="list-item-paragraph">Using a non-graphical method, find the range of values of $x$ for which $\mathrm{f}$ is concave downward.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759833656686-ep0ndi0f0" data-list-item-path="item-1759833656686-ep0ndi0f0"><div class="list-item-paragraph">By considering a sketch of $y=\mathrm{f}\left( x \right)$ and another suitable curve on the same diagram, find the range of values of $k$ such that the equation $\frac{1}{5}{{x}^{5}}+2{{x}^{3}}+\left( 10+k \right)x=0$ has&nbsp;$3$ distinct real roots.<span class="inline-marks">[3]</span></div></li></ol>

A function $\mathrm{f}$ has the equation $\mathrm{f}\left( x \right)=\frac{1}{5}{{x}^{5}}+2{{x}^{3}}+10x-25$.

Math Question Bank