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IP Math

<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 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.<br>Find</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">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></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/public/question-images/pasted/1770806346981-qa5ypeh48l.png" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

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.
Find

IP3 Class Test (May 2025) Bonus Question

Quadratic Equations and Inequalities

<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

<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.

To solve $e^{1-2x} = x^3$, a straight line can be drawn on the same axes as the graph in part (b):

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

Exponential and Logarithmic Functions

<p>It is given that&nbsp;$\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain clearly why $1 &lt; x &lt; 2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$ and show that it has only one solution.</span><span class="inline-marks">[4]</span></div></li></ol>

It is given that $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$.

Hence solve $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$ and show that it has only one solution.

IP3 Class Test (May 2025) Q8 (2023 ACSBR AM Prelim P2 Q8)

<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

<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/1770801443737-mxzzan9k6s.png" style="max-width: min(150px, 100%); height: auto; display: block;" alt="The diagram shows a rectangle $ABCD$ with side $AB = (2\sqrt{2} + \sqrt{3})$ cm." class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a rectangle $ABCD$ in which $AB = (2\sqrt{2} + \sqrt{3})$ cm. Given that the area of the rectangle is $(9 + \sqrt{6})$ cm$^2$, find, without using a calculator,</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the exact value of $BC$ in the form $(\sqrt{a} - \sqrt{b})$ cm, where&nbsp;$a$ and $b$ are integers.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the exact value of $(AC)^2$ in the form $(c - \sqrt{d})$ cm$^2$, where&nbsp;$c$ and $d$ are integers.</span><span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $AB = (2\sqrt{2} + \sqrt{3})$ cm. Given that the area of the rectangle is $(9 + \sqrt{6})$ cm$^2$, find, without using a calculator,

the exact value of $BC$ in the form $(\sqrt{a} - \sqrt{b})$ cm, where $a$ and $b$ are integers.

the exact value of $(AC)^2$ in the form $(c - \sqrt{d})$ cm$^2$, where $c$ and $d$ are integers.

IP3 Class Test (May 2025) Q3

Surds

<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-paragraph">The sum and product of $m$ and $n$ are $18$ and $77$ respectively. Find the value(s) of $m-n$.</div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$x^2-6x+1=0$. Without using a calculator, find the value of $x^4+\frac{1}{x^4}$.</span></div></li></ol>

The sum and product of $x$ and $y$ are $18$ and $77$ respectively. Without finding the values of $x$ and $y$, determine the value(s) of $x-y$.

It is given that $x^2-6x+1=0$. Without using a calculator, find the value of $x^4+\frac{1}{x^4}$.

Sum and Product of Roots

Algebraic Identities

<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

<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/1770805603072-58nwvw5dd8k.png" style="max-width: min(250px, 100%); height: auto; display: block;" alt="The diagram shows an isosceles triangle $PQR$ on a coordinate plane." class="aligncenter custom-width" /></figure></div>
<p>The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where&nbsp;$h$ and $k$ are positive constants and $h &gt; k$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$h + k = 12$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is now given that&nbsp;$h = 7$.<br>The point $S$ is such that&nbsp;$PQRS$ is a kite. Given that&nbsp;$S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the area of the kite.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where $h$ and $k$ are positive constants and $h > k$.

It is now given that $h = 7$.\nThe point $S$ is such that $PQRS$ is a kite. Given that $S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.

IP3 Class Test (May 2025) Q11 (2023 ACSBR AM Prelim P2 Q5)

Coordinate Geometry

<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)

<p>The cubic polynomial $\mathrm{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\mathrm{f}(x)=0$ are $-1$, $3$ and $k$. Given that&nbsp;$\mathrm{f}(x)$ 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;$\mathrm{f}(x)$ is divided by $x+2$.<span class="inline-marks">[2]</span></div></li></ol>

The cubic polynomial $\mathrm{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\mathrm{f}(x)=0$ are $-1$, $3$ and $k$. Given that $\mathrm{f}(x)$ 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>The temperature, $T^\circ\mathrm{C}$, of glass being heated in a kiln is given by the formula $T = 2030 - 2000e^{-kt}$, where&nbsp;$t$ is the time in hours since the glass was placed in the kiln. The temperature of the glass after $90$ minutes is $1000^\circ\mathrm{C}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the temperature of the glass before it was heated.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The glass can be moulded into different shapes when its temperature reaches at least $1300^\circ\mathrm{C}$. What is the earliest time that the glass can be removed from the kiln and moulded?<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The instruction manual of the kiln gives the range of the temperature of the kiln as&nbsp;$T &lt; y^\circ\mathrm{C}$. State the value of $y$.<span class="inline-marks">[1]</span></div></li></ol>

The temperature, $T^\circ\mathrm{C}$, of glass being heated in a kiln is given by the formula $T = 2030 - 2000e^{-kt}$, where $t$ is the time in hours since the glass was placed in the kiln. The temperature of the glass after $90$ minutes is $1000^\circ\mathrm{C}$.

Find the temperature of the glass before it was heated.

The glass can be moulded into different shapes when its temperature reaches at least $1300^\circ\mathrm{C}$. What is the earliest time that the glass can be removed from the kiln and moulded?

The instruction manual of the kiln gives the range of the temperature of the kiln as $T < y^\circ\mathrm{C}$. State the value of $y$.

IP3 Class Test (May 2025) Q9 (2023 ACSBR AM Prelim P1 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 Question Bank