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<p>The coefficient of $x^3$ in the cubic polynomial, $\mathrm{f}(x)$, is $3$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the roots of the equation, $\mathrm{f}(x)=0$, are $-5$, $2p$ and $p$, find an expression for&nbsp;$\mathrm{f}(x)$ in terms of $p$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given also that the coefficient of $x$ is $-21$, find the possible value(s) of $p$.</span><span class="inline-marks">[3]</span></div></li></ol>

The coefficient of $x^3$ in the cubic polynomial, $\mathrm{f}(x)$, is $3$.

Given that the roots of the equation, $\mathrm{f}(x)=0$, are $-5$, $2p$ and $p$, find an expression for $\mathrm{f}(x)$ in terms of $p$.

Given also that the coefficient of $x$ is $-21$, find the possible value(s) of $p$.

Remainder and Factor Theorems

Polynomials and Long Division

Polynomials, Cubic Equations and Partial Fractions

<p>Factorise each of the following expressions completely.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-10x^2 - 19x - 6$</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$18 - 2(a + b)^2$</span><span class="inline-marks">[2]</span></div></li></ol>

Factorise each of the following expressions completely.

IP2 Class Test (May 2025) Q2

Quadratic Expressions, Equations, and Functions

<p>The polynomial $g(x) = mx^3 - 8x^2 - 9x + n$ has a factor of $3x - 2$ and leaves a remainder of $6$ when divided by $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $m$ and $n$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using the values of $m$ and $n$ from <strong>(a)</strong>, solve, in exact form, the equation $g(x) = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve the equation $9e^{2y} + 8e^y = 12 + 6e^{3y}$.</span><span class="inline-marks">[3]</span></div></li></ol>

The polynomial $g(x) = mx^3 - 8x^2 - 9x + n$ has a factor of $3x - 2$ and leaves a remainder of $6$ when divided by $x$.

Using the values of $m$ and $n$ from (a), solve, in exact form, the equation $g(x) = 0$.

Hence solve the equation $9e^{2y} + 8e^y = 12 + 6e^{3y}$.

2023 Ahmad Ibrahim Sec AM S3 EOY Q9

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1770806346981-qa5ypeh48l.png" style="max-width: min(270px, 100%); height: auto; display: block;" alt="Graph of two intersecting quadratic curves and a common tangent line" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the equation of the straight line in the form of $y = mx + c$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">the coordinates of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">the exact area of triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.

the equation of the straight line in the form of $y = mx + c$.

the coordinates of $A$, $B$ and $C$.

IP3 Class Test (May 2025) Bonus Question

Quadratic Equations and Inequalities

<p>Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.</span></div></li></ol>

Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,

Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.

Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.

Quadratic Functions - General

<p>Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.<span class="inline-marks">[5]</span></p>

Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.

2023 CSS AM S3 EOY Q3

<p>The equation of a polynomial is given by $p(x) = x^3 - 7x^2 + 18x - 24$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $x - 4$ is a factor of $p(x)$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the remainder when $p(x)$ is divided by $2x - 1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that $p(x) = 0$ only has one real root.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the answers from (a) and (c), solve $2^{3y} - 7(2^{2y}) + 18(2^y) = 24$.</span><span class="inline-marks">[2]</span></div></li></ol>

The equation of a polynomial is given by $p(x) = x^3 - 7x^2 + 18x - 24$.

Show that $x - 4$ is a factor of $p(x)$.

Find the remainder when $p(x)$ is divided by $2x - 1$.

Show that $p(x) = 0$ only has one real root.

Using the answers from (a) and (c), solve $2^{3y} - 7(2^{2y}) + 18(2^y) = 24$.

2023 CSS AM S3 EOY Q6

<p>Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.<span class="inline-marks">[4]</span></p>

Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.

IP3 Class Test (May 2025) Q1

<p>The function \f is defined by $f(x) = 2x^3 + (4 - 2a)x^2 - ax + 6a$, where&nbsp;$a$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$x + 2$ is a factor of $f(x)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the other quadratic factor of $f(x)$ in terms of $a$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If&nbsp;$f(x) = 0$ has only one real root, find the range of values of $a$.<span class="inline-marks">[2]</span></div></li></ol>

The function \f is defined by $f(x) = 2x^3 + (4 - 2a)x^2 - ax + 6a$, where $a$ is a constant.

Show that $x + 2$ is a factor of $f(x)$.

Find the other quadratic factor of $f(x)$ in terms of $a$.

If $f(x) = 0$ has only one real root, find the range of values of $a$.

IP3 Class Test (May 2025) Q5

<p>A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">As a result of the three members quitting, the amount each member had to pay was \$$5$ more than the initial amount.</div><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $x^2 + 25x - 900 = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Solve the equation $x^2 + 25x - 900 = 0$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the amount of money that each member had to pay initially.<span class="inline-marks">[2]</span></div></li></ol>

A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.

Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.

Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.

Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.

As a result of the three members quitting, the amount each member had to pay was \$$5$ more than the initial amount.\n\nForm an equation in $x$ and show that it reduces to $x^2 + 25x - 900 = 0$.

IP2 Class Test (May 2025) Q11

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^3 = 3x^2 + 6x + 2$, giving your answers in exact form.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the equation $8x^3 - 12x^2 = 12x + 2$.</span><span class="inline-marks">[2]</span></div></li></ol>

Solve the equation $x^3 = 3x^2 + 6x + 2$, giving your answers in exact form.

Hence, solve the equation $8x^3 - 12x^2 = 12x + 2$.

IP3 Class Test (May 2025) Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $\frac{-6x + 11}{(x^2 - 2)(x - 1)}$ in partial fractions.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the equation $\frac{-12x + 22}{(x^2 - 2)(x - 1)} + \frac{10}{x - 1} = 9$.</span><span class="inline-marks">[3]</span></div></li></ol>

Express $\frac{-6x + 11}{(x^2 - 2)(x - 1)}$ in partial fractions.

Hence, solve the equation $\frac{-12x + 22}{(x^2 - 2)(x - 1)} + \frac{10}{x - 1} = 9$.

2023 CSS AM S3 EOY Q8

<p>Expand and simplify each of the following expressions.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( \frac{2x}{3}-\frac{1}{2} \right)}^{2}}$<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\left( x+6 \right)\left( 2y+3 \right)-\left( y+2 \right)\left( x-2 \right)$<span class="inline-marks">[2]</span></div></li></ol>

Expand and simplify each of the following expressions.

IP2 Class Test (May 2025) Q1

<p>Express $\frac{3{{x}^{3}}+10{{x}^{2}}+x+1}{{{x}^{3}}+3{{x}^{2}}}$ in partial fractions.<span class="inline-marks">[6]</span></p>

Express $\frac{3{{x}^{3}}+10{{x}^{2}}+x+1}{{{x}^{3}}+3{{x}^{2}}}$ in partial fractions.

IP3 Class Test (May 2025) Q7 (2023 Jurong Sec AM Prelim P1 Q2)

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.</span><span class="inline-marks">[2]</span></div></li></ol>

Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.

Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.

Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.

2023 CSS AM S3 EOY Q5

<p>The cubic polynomial $\text{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\text{f}(x)=0$ are $-1$, $3$ and $k$. Given that&nbsp;$\text{f}(x)$ has a remainder of $20$ when divided by $x-4$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ ;</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">the remainder when&nbsp;$\text{f}(x)$ is divided by $x+2$.<span class="inline-marks">[2]</span></div></li></ol>

The cubic polynomial $\text{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\text{f}(x)=0$ are $-1$, $3$ and $k$. Given that $\text{f}(x)$ has a remainder of $20$ when divided by $x-4$, find

the remainder when $f(x)$ is divided by $x + 2$.

IP3 Class Test (May 2025) Q6

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">by factorisation</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by completing the square</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">by the quadratic formula</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Solve the equation $x^2 - 6x - 13 = 3$ with the different methods:

Solve the equation $-4x^2 + 4x + 3 = 0$ with the different methods:

IP2 Class Test (May 2025) Q3

<p>Given that&nbsp;$3{{x}^{3}}+{{x}^{2}}-2x-3\equiv (Ax+B)(x-1)(x+2)+C(x-1)+D$, find the values of $A$, $B$, $C$ and $D$.<span class="inline-marks">[4]</span></p>

Given that $3{{x}^{3}}+{{x}^{2}}-2x-3\equiv (Ax+B)(x-1)(x+2)+C(x-1)+D$, find the values of $A$, $B$, $C$ and $D$.

IP3 Class Test (May 2025) Q2

<p>Sketch the graph of $y = (x - 1)(x + 5)$.</p>
<p>Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.<span class="inline-marks">[5]</span></p>
<p>Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.<span class="inline-marks">[1]</span></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769517826384-pn2smbhjeen.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Sketch the graph of $y = (x - 1)(x + 5)$.

Indicate clearly the coordinates of points where the graph crosses the $x$- and $y$-axes, the coordinates of the turning point and write down the equation of the line of symmetry of the curve.

Hence, or otherwise, solve the inequality $x^2 + 4x - 5 \le 0$.

IP2 Class Test (May 2025) Q12

<p>The diagram shows the graph of $y = x^2 - x - 5$ for&nbsp;$-3 \le x \le 4$.<br>Use the graph to answer the following questions. Present your solutions on the next page.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769516562543-yh7ojfj8sr.png" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph"><strong>Using the above graph,</strong></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find the value of $y$ when&nbsp;$x = 3$</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the coordinates of the turning point,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the equation of the line of symmetry,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">estimate the solutions of the equation $x^2 - x - 5 = 0$.</span><span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State the equation of the straight line to be drawn.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain how the straight line in <strong>(ii)(a)</strong> drawn on the graph can lead to an estimation of the solution(s) to the equation $x^2 - 3x - 2 = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $x^2 - 3x - 2 = 0$ graphically.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The diagram shows the graph of $y = x^2 - x - 5$ for $-3 \le x \le 4$.
Use the graph to answer the following questions. Present your solutions on the next page.

A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.

A Math (IGCSE) Question Bank