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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The curve $C$ is given by the equation $y=\frac{1}{x}+\frac{2}{x^2}, x \neq 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, find the set of values that&nbsp;$y$ can take.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the curve $C$, stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Given that the solution of the inequality $ax^2+bx+c &gt; \frac{1}{x}+\frac{2}{x^2}$ is the set&nbsp;$\{x \in \mathbb{R} : -1.5 &lt; x &lt; -1 \text{ or}\hspace{0.5em} 1 &lt; x &lt; k\}$, find the values of $a$, $b$ and $c$.<span class="inline-marks">[3]</span></p>
<p>Hence find the value of $k$.<span class="inline-marks">[1]</span></p>

The curve $C$ is given by the equation $y=\frac{1}{x}+\frac{2}{x^2}, x \neq 0$.

Without using a calculator, find the set of values that $y$ can take.

Sketch the curve $C$, stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes.

Given that the solution of the inequality $ax^2+bx+c > \frac{1}{x}+\frac{2}{x^2}$ is the set $\{x \in \mathbb{R} : -1.5 < x < -1 \text{ or } 1 < x < k\}$, find the values of $a$, $b$ and $c$.

Sketching Rational Functions

Graphical Method

Inequalities involving Rational Functions

Determining Model Coefficients Using Linear Systems

Asymptotes and Discontinuities

Finding Possible y-values of a Curve

2015 ACJC P1 Q9

Graphing Techniques

Inequalities

<p>A curve has equation $y = ax + b + \frac{c}{x^2 - 1}$, where $a$, $b$ and $c$ are real constants. It is given that the curve crosses the $x$-axis at $x = 2$. The normal to the curve at the point $(0, 3)$ meets the $x$-axis at $x = 7.5$. Find the values of $a$, $b$ and $c$.<span class="inline-marks">[4]</span></p>

A curve has equation $y = ax + b + \frac{c}{x^2 - 1}$, where $a$, $b$ and $c$ are real constants. It is given that the curve crosses the $x$-axis at $x = 2$. The normal to the curve at the point $(0, 3)$ meets the $x$-axis at $x = 7.5$. Find the values of $a$, $b$ and $c$.

Tangents and Normals at a Point

2025 RI P1 Q1 

Applications of Differentiation

<p>Curve $C$ has equation $y = \frac{px^2 + qx + r}{x - 1}$, where&nbsp;$p$, $q$ and $r$ are real numbers. Given that&nbsp;$C$ has a turning point at $(3, 7)$ and an oblique asymptote parallel to the line $y = \frac{3}{2}x + \sqrt{5}$, find $p$, $q$ and $r$.<span class="inline-marks">[4]</span></p>

Curve $C$ has equation $y = \frac{px^2 + qx + r}{x - 1}$, where $p$, $q$ and $r$ are real numbers. Given that $C$ has a turning point at $(3, 7)$ and an oblique asymptote parallel to the line $y = \frac{3}{2}x + \sqrt{5}$, find $p$, $q$ and $r$.

2025 NYJC P1 Q1

<p>A sequence of real numbers, ${{x}_{n}}$, satisfies the recurrence relation</p>
<p style="text-align: center">${{x}_{n+1}}=a{{\left( -3 \right)}^{n}}+b{{n}^{2}}+c{{x}_{n}}$, for $n\in \mathbb{Z}$, $a\in \mathbb{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}$ and $n\ge 1$.</p>
<p>Given that ${{x}_{1}}=1$, ${{x}_{2}}=-8$, ${{x}_{3}}=70$, and ${{x}_{4}}=-377$, find the values of $a$, $b$ and $c$.</p><p style="text-align: right;">[3]</p>
<p>Hence find the values of ${{x}_{10}}$ and ${{x}_{11}}$.</p><p style="text-align: right;">[2]</p>

A sequence of real numbers, ${{x}_{n}}$, satisfies the recurrence relation

${{x}_{n+1}}=a{{\left( -3 \right)}^{n}}+b{{n}^{2}}+c{{x}_{n}}$, for $n\in \mathbb{Z}$, $a\in \mathbb{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}$ and $n\ge 1$.

Given that ${{x}_{1}}=1$, ${{x}_{2}}=-8$, ${{x}_{3}}=70$, and ${{x}_{4}}=-377$, find the values of $a$, $b$ and $c$.

Hence find the values of ${{x}_{10}}$ and ${{x}_{11}}$.

Generating Terms from Recurrence Relations

2025 HCI P1 Q2

Sequence and Series

System of Linear Equations

<p>A sequence ${{t}_{1}},{{t}_{2}},{{t}_{3}},{{t}_{4}},\ldots $ is given by</p>
<p style="text-align: center">${{t}_{n+1}}=a+b{{t}_{n}}$,</p>
<p>where&nbsp;$a$ and $b$ are non-zero constants.</p>
<p>It is given that&nbsp;${{t}_{3}}=320$, ${{t}_{4}}=240$ and ${{t}_{5}}=200$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870146720-s1k6gagvx" data-list-item-path="item-1760870146720-s1k6gagvx"><div class="list-item-paragraph">Determine the value of $a$ and the value of $b$.</div></li><li data-list-item-id="item-1760870147342-aqwsh8ard" data-list-item-path="item-1760870147342-aqwsh8ard"><div class="list-item-paragraph">Find the value of ${{t}_{6}}$.</div></li><li data-list-item-id="item-1760870148010-7e2lh3d3k" data-list-item-path="item-1760870148010-7e2lh3d3k"><div class="list-item-paragraph">Show clearly that&nbsp;${{t}_{1}}=800$.</div></li></ol>
<p>The sequence converges to a limit $L$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760870172066-7t3qt9pe6" data-list-item-path="item-1760870172066-7t3qt9pe6"><div class="list-item-paragraph">Determine the value of $L$.</div></li></ol>

A sequence ${{t}_{1}},{{t}_{2}},{{t}_{3}},{{t}_{4}},\ldots $ is given by

where $a$ and $b$ are non-zero constants.

It is given that ${{t}_{3}}=320$, ${{t}_{4}}=240$ and ${{t}_{5}}=200$.

Limit of a Sequence and Behaviour of a Sequence

Reverse Iteration of Recurrence Relations

T.Madas - Recurrence Relations Q37

<p>During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at one-hour intervals following the initial administration of the trial drug to a patient.</p>
<p>The following readings were obtained:</p>
<p style="text-align: center">${{U}_{3}}=88$, ${{U}_{4}}=76$ and ${{U}_{5}}=70$,</p>
<p>where&nbsp;${{U}_{t}}$ denotes the reading $t$ hours after the drug was first administered.</p>
<p>It is believed that&nbsp;$U$ satisfies the relationship</p>
<p style="text-align: center">${{U}_{t+1}}=p+q{{U}_{t}}$, $t\ge 0$,</p>
<p>where&nbsp;$p$ and $q$ are constants.</p>
<ol class="abc-list"><li data-list-item-id="item-1760867603859-k9t1qqbhq" data-list-item-path="item-1760867603859-k9t1qqbhq"><div class="list-item-paragraph">Find the values of $p$ and $q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760867604504-menwt8l4h" data-list-item-path="item-1760867604504-menwt8l4h"><div class="list-item-paragraph">Determine the initial concentration of the blood agent when the drug was initially administered.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760867605591-rovuqf55s" data-list-item-path="item-1760867605591-rovuqf55s"><div class="list-item-paragraph">Describe how the concentration behaves over a long period of time.<span class="inline-marks">[1]</span></div></li></ol>

During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at one-hour intervals following the initial administration of the trial drug to a patient.

${{U}_{3}}=88$, ${{U}_{4}}=76$ and ${{U}_{5}}=70$,

where ${{U}_{t}}$ denotes the reading $t$ hours after the drug was first administered.

It is believed that $U$ satisfies the relationship

2024 NYJC Promo Q4

<p>The graph of $y=\frac{x+1}{a{{x}^{2}}+bx+c}$, where&nbsp;$a$, $b$ and $c$ are non-zero constants, has an asymptote at $x=-\frac{1}{2}$.</p>
<p>The graph also has a turning point at $\left( -2,\ -\frac{1}{9} \right)$.</p>
<p>Find the values of $a$, $b$ and $c$.<span class="inline-marks">[4]</span></p>

The graph of $y=\frac{x+1}{a{{x}^{2}}+bx+c}$, where $a$, $b$ and $c$ are non-zero constants, has an asymptote at $x=-\frac{1}{2}$.

The graph also has a turning point at $\left( -2,\ -\frac{1}{9} \right)$.

Quotient Rule

Stationary Points and Their Nature

N2024 P1 Q1

Differentiation

Techniques of Differentiation

Limits and Continuity

<p>The sum of the first $n$ terms of a sequence, ${{S}_{n}}$, is given by ${{S}_{n}}=a{{n}^{3}}+b{{n}^{2}}+cn+d$, where&nbsp;$a$, $b$, $c$ and $d$ are constants. It is given that&nbsp;${{S}_{1}}=5$, ${{S}_{2}}=20$, ${{S}_{3}}=57$ and ${{S}_{4}}=128$. Find ${{S}_{n}}$.<span class="inline-marks">[4]</span></p>

The sum of the first $n$ terms of a sequence, ${{S}_{n}}$, is given by ${{S}_{n}}=a{{n}^{3}}+b{{n}^{2}}+cn+d$, where $a$, $b$, $c$ and $d$ are constants. It is given that ${{S}_{1}}=5$, ${{S}_{2}}=20$, ${{S}_{3}}=57$ and ${{S}_{4}}=128$. Find ${{S}_{n}}$.

2023 SAJC Promo Q1

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that&nbsp;${{u}_{n}}=2{{u}_{n-1}}-15{{n}^{2}}+60n+A$, where&nbsp;$A$ is a constant, and $n\ge 2$. It is given that&nbsp;${{u}_{1}}=2$ and ${{u}_{2}}=4$.</p>
<ol class="i-list"><li data-list-item-id="item-1760419390205-gqnw8l5dd" data-list-item-path="item-1760419390205-gqnw8l5dd"><div class="list-item-paragraph">Find $A$ and ${{u}_{3}}$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is known that the $n$th term of this sequence is given by ${{u}_{n}}=p\left( {{2}^{n+1}} \right)-q{{n}^{2}}+r$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760419403225-zvrnnn44q" data-list-item-path="item-1760419403225-zvrnnn44q"><div class="list-item-paragraph">Find $p$, $q$ and $r$.<span class="inline-marks">[3]</span></div></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{n}}=2{{u}_{n-1}}-15{{n}^{2}}+60n+A$, where $A$ is a constant, and $n\ge 2$. It is given that ${{u}_{1}}=2$ and ${{u}_{2}}=4$.

It is known that the $n$th term of this sequence is given by ${{u}_{n}}=p\left( {{2}^{n+1}} \right)-q{{n}^{2}}+r$.

2023 JPJC P1 Q1

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where $P$ is a constant and $n\in {{\mathbb{Z}}^{+}}$.</p>
<p>The terms of the sequence are defined by their previous terms, for example, ${{u}_{2}}=3{{u}_{1}}+P$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that ${{u}_{1}}=1$ and ${{u}_{2}}=7$, find $P$ and ${{u}_{3}}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>lt is known that the ${{n}^{\mathrm{th}}}$ term of this sequence is given by</p>
<p style="text-align: center">${{u}_{n}}=a\left( {{3}^{n}} \right)+bn+c$,</p>
<p>where $a$, $b$ and $c$ are constants.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find $a$, $b$ and $c$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find $\sum\limits_{r=1}^{n}{{{u}_{r}}}$ in terms of $n$.</div><p style="text-align: right;">[4]</p></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where $P$ is a constant and $n\in {{\mathbb{Z}}^{+}}$.

The terms of the sequence are defined by their previous terms, for example, ${{u}_{2}}=3{{u}_{1}}+P$.

lt is known that the ${{n}^{\mathrm{th}}}$ term of this sequence is given by

${{u}_{n}}=a\left( {{3}^{n}} \right)+bn+c$,

Summations Involving APGP

2020 CJC Promo Q9

<p>The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}=a\left( {{3}^{n}} \right)+bn+c$, where&nbsp;$a$, $b$ and $c$ are constants and $n\ge 1$.</p>
<ol class="i-list"><li data-list-item-id="item-1759659297137-lllfommog" data-list-item-path="item-1759659297137-lllfommog"><div class="list-item-paragraph">Given that&nbsp;${{u}_{1}}=5$, ${{u}_{2}}=9$ and ${{u}_{3}}=33$, find $a$, $b$ and $c$ .<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759659297737-9443jsa5d" data-list-item-path="item-1759659297737-9443jsa5d"><div class="list-item-paragraph">Show that&nbsp;${{u}_{n+1}}=A~\left( {{3}^{n}} \right)+B$, where&nbsp;$A$ and $B$ are constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759659299567-vd449hgf3" data-list-item-path="item-1759659299567-vd449hgf3"><div class="list-item-paragraph">Using your answer in part <strong>(ii)</strong>, find $\sum\limits_{r=2}^{n}{{{u}_{r+1}}}$ in terms of $n$. (You need not simplify your answer.)<span class="inline-marks">[3]</span></div></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}=a\left( {{3}^{n}} \right)+bn+c$, where $a$, $b$ and $c$ are constants and $n\ge 1$.

Manipulating Summation Limits

$u_n = S_n - S_{n-1} $

2020 EJC P1 Q7

<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">A sequence ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$, … is given by</span></div><div class="list-item-paragraph" style="text-align: center">${{a}_{1}}=\frac{1}{2}$ and ${{a}_{n}}=\frac{{{n}^{2}}}{{{2}^{n}}}$ for $n\ge 1$.</div><div class="list-item-paragraph">Let ${{S}_{n}}$ be the sum of the first $n$ terms of the above sequence. Find the values of $p$ and $q$ such that ${{S}_{n}}=6-\frac{{{n}^{2}}+pn+q}{{{2}^{n}}}$ where $p$ and $q$ are constants.</div></li></ol>

A sequence ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$, … is given by

2023 HCI CT1 Q5 [Modified]

<p>The curve $C$ with equation $y=\frac{a{{x}^{2}}+bx+c}{3x+1}$ passes through the point with coordinates $\left( -1,-4 \right)$ and has a turning point at $\left( -3,-2 \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $a$, $b$ and $c$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence sketch the graph of curve $C$, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes.</span></div><p style="text-align: right;">[3]</p></li></ol>

The curve $C$ with equation $y=\frac{a{{x}^{2}}+bx+c}{3x+1}$ passes through the point with coordinates $\left( -1,-4 \right)$ and has a turning point at $\left( -3,-2 \right)$.

Hence sketch the graph of curve $C$, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes.

2022 ACJC Promo Q3

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The first three terms of a sequence are given by ${{u}_{1}}=10$, ${{u}_{2}}=6$, ${{u}_{3}}=5$. Given that ${{u}_{n}}$ is a quadratic polynomial in $n$, find ${{u}_{n}}$ in terms of $n$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the set of values of $n$ for which ${{u}_{n}}$ is greater than $100$.</span></div><p style="text-align: right;">[2]</p></li></ol>

The first three terms of a sequence are given by ${{u}_{1}}=10$, ${{u}_{2}}=6$, ${{u}_{3}}=5$. Given that ${{u}_{n}}$ is a quadratic polynomial in $n$, find ${{u}_{n}}$ in terms of $n$.

Find the set of values of $n$ for which ${{u}_{n}}$ is greater than $100$.

Problems involving inequalities

N2009 P1 Q1

<p>A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$. The graph of $y=\mathrm{f}\left( x \right)$ passes through the points $\left( 1,5 \right)$ and $\left( -1,-3 \right)$. The graph has a turning point at $x=1$, and $\displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)}\mathrm{d}x=6$.</p>
<p>Find the values of $a$, $b$, $c$ and $d$.<span class="inline-marks">[5]</span></p>

A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$. The graph of $y=\mathrm{f}\left( x \right)$ passes through the points $\left( 1,5 \right)$ and $\left( -1,-3 \right)$. The graph has a turning point at $x=1$, and $\displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)}\mathrm{d}x=6$.

Find the values of $a$, $b$, $c$ and $d$.

Stationary Points and Points of Inflection

Definite Integral

N2021 P1 Q1

Applications of Integration

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $\left( -4,2 \right)$ and $\left( 0,-6 \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $a$, $b$ and $c$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, stating the equations of the asymptotes.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$$\left( k>0 \right)$ such that the equation ${{\left( x+4 \right)}^{2}}+{{\left( \frac{{{x}^{2}}+ax+b}{c-x} \right)}^{2}}={{k}^{2}}$ has no real roots.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">When $C$ undergoes a translation in the direction of $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$.</span></div><p style="text-align: right;">[2]</p></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $\left( -4,2 \right)$ and $\left( 0,-6 \right)$.

Sketch $C$, stating the equations of the asymptotes.

By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$$\left( k>0 \right)$ such that the equation ${{\left( x+4 \right)}^{2}}+{{\left( \frac{{{x}^{2}}+ax+b}{c-x} \right)}^{2}}={{k}^{2}}$ has no real roots.

When $C$ undergoes a translation in the direction of $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$.

Parameter Conditions for Graphs and Equations

Circle–Curve Intersection Conditions

2013 PJC P2 Q5

<p>The curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-1}$, where&nbsp;$a$, $b$ and $c$ are non-zero constants.</p>
<ol class="abc-list"><li data-list-item-id="item-1758882211867-m09ok1l7r" data-list-item-path="item-1758882211867-m09ok1l7r"><div class="list-item-paragraph">It is given that&nbsp;$C$ passes through the point $\left( 3,\ \frac{23}{2} \right)$and has a minimum point at $\left( 2,\ 10 \right)$.</div><ol class="i-list"><li data-list-item-id="subitem-1758882215842-fxgbj0cvy" data-list-item-path="item-1758882211867-m09ok1l7r/subitem-1758882215842-fxgbj0cvy"><div class="list-item-paragraph">Find the values of $a$, $b$ and $c$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1758882216157-c9zsf5pyl" data-list-item-path="item-1758882211867-m09ok1l7r/subitem-1758882216157-c9zsf5pyl"><div class="list-item-paragraph">Sketch $C$, giving the coordinates of any turning points, points of intersection with the axes and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1758882216479-zvx3ekndr" data-list-item-path="item-1758882211867-m09ok1l7r/subitem-1758882216479-zvx3ekndr"><div class="list-item-paragraph">Find the set of values of $\frac{m}{k}$, where&nbsp;$m$ and $k$ are positive constants, such that the curve with equation $\frac{{{x}^{2}}}{{{k}^{2}}}-\frac{{{\left( y-1 \right)}^{2}}}{{{m}^{2}}}=1$ does not intersect $C$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1758882250161-01jszwi3l" data-list-item-path="item-1758882250161-01jszwi3l"><div class="list-item-paragraph">In the case when&nbsp;$a=-1$ and $b=-1$, find the set of values of $c$, where&nbsp;$c\ne 2$, such that&nbsp;$C$ has no stationary point.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-1}$, where $a$, $b$ and $c$ are non-zero constants.

It is given that $C$ passes through the point $\left( 3,\ \frac{23}{2} \right)$and has a minimum point at $\left( 2,\ 10 \right)$.

In the case when $a=-1$ and $b=-1$, find the set of values of $c$, where $c\ne 2$, such that $C$ has no stationary point.

Math Question Bank