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<p>Factorise each of the following expressions completely.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$2x^3 - 50x$<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$6y^2 + 9y - 42$<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$2ax - 5bx + 15by - 6ay$<span class="inline-marks">[2]</span></div></li></ol>

Factorise each of the following expressions completely.

Factorisation of quadratic expressions

XMS 2EM WA2 2021 Q6

Algebra

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Factorise $w^{2} - w - 6$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence write as a single fraction in its simplest form $\frac{5}{w^{2} - w - 6} + \frac{1}{2w - 6}$.</span></div></li></ol>

Hence write as a single fraction in its simplest form $\frac{5}{w^{2} - w - 6} + \frac{1}{2w - 6}$.

RSS 2EM EOY P2 2023 Q2b

<p>Given that&nbsp;$\mathbf{A} = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & k \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix}$,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find $\mathbf{A}^2 + 2\mathbf{B}$ in terms of $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{A}^2 + 2\mathbf{B} = 5\mathbf{C}$, find the value of $k$.</span></div></li></ol>

Given that $\mathbf{A} = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & k \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix}$,

find $\mathbf{A}^2 + 2\mathbf{B}$ in terms of $k$.

If $\mathbf{A}^2 + 2\mathbf{B} = 5\mathbf{C}$, find the value of $k$.

Application of Matrices

Matrices

<p>Aqil's father is $x^{2}$ years old while Aqil is $x$ years old now. In $5x$ years' time, Aqil's father is twice as old as him.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Form an equation in $x$ and show that it reduces to $x^{2}-7x=0$ .</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^{2}-7x=0$ .</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the age of Aqil's father now.</span><span class="inline-marks">[1]</span></div></li></ol>

Aqil's father is $x^{2}$ years old while Aqil is $x$ years old now. In $5x$ years' time, Aqil's father is twice as old as him.

Form an equation in $x$ and show that it reduces to $x^{2}-7x=0$ .

Hence, find the age of Aqil's father now.

Solving quadratic equations and its applications

XMS 2EM WA2 2021 Q5

<p>The sum of $\frac{3}{5x+4}$, $\frac{7}{4-5x}$ and $\frac{px+q}{16-25x^2}$, where&nbsp;$p$ and $q$ are constants, is $\frac{Ax+B}{25x^2-16}$.</p>
<p>Find an expression, in terms of $p$ and / or $q$, for&nbsp;$A$ and $B$.</p>

The sum of $\frac{3}{5x+4}$, $\frac{7}{4-5x}$ and $\frac{px+q}{16-25x^2}$, where $p$ and $q$ are constants, is $\frac{Ax+B}{25x^2-16}$.

Find an expression, in terms of $p$ and / or $q$, for $A$ and $B$.

Algebraic Fractions

Addition and Subtraction of Algebraic Fractions

Algebraic Identities

Algebraic Expressions, Equations, and Formulae

<p>Factorise the following expression completely</p>
<p style="text-align: center">$64-49a^2-28ab-4b^2$</p>

Factorise the following expression completely

Simplifying Algebraic Fractions involving Factorisation

Factorization of Algebraic Expressions

<p>A drone is hovering at a fixed altitude directly above the edge of a skyscraper. The drone’s camera tracks two delivery robots, Robot A and Robot B, which are moving along a straight road extending from the base of the skyscraper.</p>
<p>The angle of depression from the drone to Robot A is $28.5°$. The angle of depression from the drone to Robot B is $14.2°$. Robot A is known to be $320$ m away from the base of the skyscraper.<br></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the altitude at which the drone is hovering.</div></li><li><div class="list-item-paragraph">Find the distance between Robot A and Robot B. </div></li></ol>

A drone is hovering at a fixed altitude directly above the edge of a skyscraper. The drone’s camera tracks two delivery robots, Robot A and Robot B, which are moving along a straight road extending from the base of the skyscraper.

The angle of depression from the drone to Robot A is $28.5°$. The angle of depression from the drone to Robot B is $14.2°$. Robot A is known to be $320$ m away from the base of the skyscraper.


Applications of Trigonometric Ratios

Trigonometric Ratios

<p>Factorise the following expression completely</p>
<p style="text-align: center">$a^2-b^2+3b-3a$</p>

Further Applications of Algebraic Identities

<p>Simplify the following expressions.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$(2x + 3y)^2$,</span></div></li><li><div class="list-item-paragraph">$(x + 2) - (2x - 4)(x - 3)$</div></li></ol>

Expansion of quadratic expressions

OPSS 2EM MYE 2018 Q6

OPSS 2EM MYE 2018 Q13

<p>It is given that&nbsp;$\mathbf{M} = \begin{pmatrix} 2p & q \\ 0 & \frac{1}{3}r \end{pmatrix}$ and $\mathbf{N} = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathbf{N}^2$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$4\mathbf{M} = \mathbf{N}^2$, find the values of $p$, $q$ and $r$.</span></div></li></ol>

It is given that $\mathbf{M} = \begin{pmatrix} 2p & q \\ 0 & \frac{1}{3}r \end{pmatrix}$ and $\mathbf{N} = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix}$.

Given that $4\mathbf{M} = \mathbf{N}^2$, find the values of $p$, $q$ and $r$.

<p>If&nbsp;$\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.</p>

If $\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.

Equality of Indices

Index Notation

Indices

Laws of Indices

Exponential and Logarithmic Functions

<p>In triangle $AOB$, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $C$ lies on $AB$ such that&nbsp;$AC:AB = 1:3$.</p>
<p>Express $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, giving your answer in the simplest form.</p>
<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/1776508414841-i8f3zjzuriq.png" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In triangle $AOB$, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $C$ lies on $AB$ such that $AC:AB = 1:3$.

Express $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, giving your answer in the simplest form.

Applications of Vectors

Vectors

<p>If&nbsp;$\frac{5x+2y}{2x-4y}=\frac{3}{4}$, find the value of $\frac{x}{y}$.</p>

If $\frac{5x+2y}{2x-4y}=\frac{3}{4}$, find the value of $\frac{x}{y}$.

Solving Equations involving Algebraic Fractions

<p>A rice distributor supplies rice to $4$ different supermarkets in Singapore. The number of bags of rice supplied per delivery to each supermarket, the sizes and cost price are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 120px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th colspan="3" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Number of bags of rice per delivery</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Number of deliveries in one month</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Size of each bag</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2.5$ kg</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$ kg</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$ kg</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $A$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $B$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $C$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$20$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$11$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $D$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$35$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Price per bag</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$4$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$7$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$12$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{P} = \begin{pmatrix} 40 & 25 & 10 \end{pmatrix}$, evaluate $6\mathbf{P}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $6\mathbf{P}$.</span></div></li></ol></li><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{Q} = \begin{pmatrix} 40 & 25 & 10 \\ 80 & 60 & 30 \\ 30 & 20 & 25 \\ 40 & 35 & 30 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 4 \\ 7 \\ 12 \end{pmatrix}$, evaluate $\mathbf{QR}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $\mathbf{QR}$.</span></div></li></ol></li><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down two matrices such that the product under the matrix multiplication give the total number of each size of bag the distributor supply to the four supermarkets in one month.</span></div></li><li><div class="list-item-paragraph">Evaluate the product of the two matrices in <strong>(c)(i)</strong>.</div></li></ol></li></ol>
<p>Using the product from part <strong>(c)(ii)</strong> and multiplying by another matrix, evaluate the total amount of money the distributor received from the four supermarkets in one month.</p>

A rice distributor supplies rice to $4$ different supermarkets in Singapore. The number of bags of rice supplied per delivery to each supermarket, the sizes and cost price are shown in the table below.

Using the product from part **(c)(ii)** and multiplying by another matrix, evaluate the total amount of money the distributor received from the four supermarkets in one month.

<p>Solve the simultaneous equations.</p>
<p style="text-align: center">$x-2y=8$<br>$3^x(27^y)=9$</p>

Simultaneous Equations

<p>Given that&nbsp;$\begin{pmatrix} 3 & h+2 \\ 3 & k \end{pmatrix} + \begin{pmatrix} 2 & -5 \\ 3 & 3k+1 \end{pmatrix} = \begin{pmatrix} 5 & 17-h \\ 6 & 3 \end{pmatrix}$, find the values of $h$ and $k$.</p>

Given that $\begin{pmatrix} 3 & h+2 \\ 3 & k \end{pmatrix} + \begin{pmatrix} 2 & -5 \\ 3 & 3k+1 \end{pmatrix} = \begin{pmatrix} 5 & 17-h \\ 6 & 3 \end{pmatrix}$, find the values of $h$ and $k$.

<p>It is given that matrices $\mathbf{P} = \begin{pmatrix} 3y & 1 \\ 2 & 3 \end{pmatrix}$, $\mathbf{Q} = \begin{pmatrix} y & 4 \\ 8 & 2 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 5 & 1 \\ 2 & 8 \end{pmatrix}$.</p>
<p>If&nbsp;$3\mathbf{P} - \frac{1}{2}\mathbf{Q} = \mathbf{R}$, find the value of $y$.</p>

It is given that matrices $\mathbf{P} = \begin{pmatrix} 3y & 1 \\ 2 & 3 \end{pmatrix}$, $\mathbf{Q} = \begin{pmatrix} y & 4 \\ 8 & 2 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 5 & 1 \\ 2 & 8 \end{pmatrix}$.

If $3\mathbf{P} - \frac{1}{2}\mathbf{Q} = \mathbf{R}$, find the value of $y$.

<p>The $n \mathrm{th}$ term of a sequence is given by $ T_{n} = \frac{3n +2}{47 - 2n}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find $T_{3}$.</div></li><li><div class="list-item-paragraph">Given that&nbsp;$ T_{k} = 13$, find the value of $k$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Explain why $3n + 2$ is always positive.</div></li><li><div class="list-item-paragraph">Hence or otherwise, find the least value of $n$ for which $T_{n} &lt; 0$.</div></li></ol></li></ol>

The $n \mathrm{th}$ term of a sequence is given by $ T_{n} = \frac{3n +2}{47 - 2n}$

General Term of a Sequence

2022 TKSS EOY P2 Q3

Number Patterns

<p>The diagram below shows the graph of $y = x^2 - x - 12$.</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/1776140313968-zhxc4w63y9.png" style="max-width: min(250px, 100%); height: auto; display: block;" alt="Graph of a parabola with intercepts A, B, and C" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">The graph cuts the $x$-axis at $B$ and $C$. Find the coordinates of $B$ and $C$.</div><div class="list-item-paragraph" style="text-align: right">Answer $B$ (.................... , ....................) <span class="inline-marks">[1]</span></div><div class="list-item-paragraph" style="text-align: right">Answer $C$ (.................... , ....................)<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the gradient of line $AC$.</div><div class="list-item-paragraph" style="text-align: right">Answer ....................................................<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the equation of the line of symmetry.</div><div class="list-item-paragraph" style="text-align: right">Answer ....................................................<span class="inline-marks">[1]</span></div></li></ol>

The diagram below shows the graph of $y = x^2 - x - 12$.

E Math Question Bank