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<p>It is given that&nbsp;$3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve $3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $3p^3-7p^2-22p+8=0$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.

Remainder and Factor Theorems

Polynomials, Cubic Equations and Partial Fractions

<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>It is given that the solution of the equation $x\sqrt{24} = x\sqrt{3} + \sqrt{6}$ is $\frac{a + \sqrt{b}}{7}$. <br>Find the values of the integers $a$ and $b$.</p>

It is given that the solution of the equation $x\sqrt{24} = x\sqrt{3} + \sqrt{6}$ is $\frac{a + \sqrt{b}}{7}$. 
Find the values of the integers $a$ and $b$.

Applications of Surds

Equation Involving Surds

Surds

<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>The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Find the perimeter of the square.</div></li></ol>

The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.

Properties of Circle

<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 $r$.

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Given that&nbsp;$ \mathrm{f}(x) = \frac{2x + 3}{\sqrt{4x - 3}}$, find $ \mathrm{f}'(x)$.</div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the set of values of $x$ for&nbsp;$y = \ln\left(\frac{3 + 8x}{8x - 5}\right)$ to be a decreasing function.</div></li></ol>

2024 CCHY P1 Q4

Applications of Differentiation

Techniques of Differentiation

<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that&nbsp;$3x^2 - 5x + 21$ is always positive for all real values of $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The curve $y = ax^n$, where&nbsp;$a$ and $n$ are constants, passes through $(2, 48)$, $(3, 108)$ and $(k, 192)$. Find the values of $a$, $n$ and $k$ where&nbsp;$k &gt; 0$.<span class="inline-marks">[3]</span></div></li></ol>

Nature of Roots of Quadratic Equations

2024 CCHY P1 Q3

Quadratic Equations and Inequalities

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>Find $\frac{\mathrm{d}}{\mathrm{d}x}\left[(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) \dots (x^{1024}+1)\right]$</p>

Find $\frac{\mathrm{d}}{\mathrm{d}x}\left[(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) \dots (x^{1024}+1)\right]$

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

<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>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></p>
<p></p>
<ol class="i-list"><li data-list-item-id="item-1781083736022-kfskn68ky"><div class="list-item-paragraph">Given that the line $y = 2x - k$ meets the curve $y = \frac{k}{4}x^2 - 2kx + 1$, find the range of values of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1781083737079-h01r9msxr"><div class="list-item-paragraph">A student claimed that when&nbsp;$k = -\frac{1}{2}$, the curve does not intersect the line. By showing your workings clearly, explain whether the statement is valid.<span class="inline-marks">[2]</span></div></li></ol>

Intersection of a Line and a Curve

2024 CCHY P1 Q1

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">If&nbsp;$a \cos^2 \theta + b \sin^2 \theta = c$, show that&nbsp;$\tan^2 \theta = \frac{a - c}{c - b}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-bi"><ol class="i-list"><li data-list-item-id="subitem-1781085512961-qeke46jc7"><div class="list-item-paragraph">Given that&nbsp;$\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x$, show that&nbsp;$6 \sin^2 x + \sin x - 2 = 0$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1781085513516-spykf9uxq"><div class="list-item-paragraph">Hence, solve the equation $\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x$ for&nbsp;$0 \le \theta \le 2\pi$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2024 CCHY P1 Q11

Trigonometric Equations and Identities

<p>A triangle has a base of $\left(\frac{1}{\sqrt{2}} - \frac{\sqrt{32}}{2} + \frac{16}{\sqrt{48}}\right)$ m and a height of $h$ m. Given that the area of the triangle is $\left(2\sqrt{2} - \sqrt{3}\right)$ m$^2$, find, without using calculator, the value of $h$ in the form $\frac{a\sqrt{6} + b}{5}$ where&nbsp;$a$ and $b$ are integers.<span class="inline-marks">[5]</span></p>

A triangle has a base of $\left(\frac{1}{\sqrt{2}} - \frac{\sqrt{32}}{2} + \frac{16}{\sqrt{48}}\right)$ m and a height of $h$ m. Given that the area of the triangle is $\left(2\sqrt{2} - \sqrt{3}\right)$ m$^2$, find, without using calculator, the value of $h$ in the form $\frac{a\sqrt{6} + b}{5}$ where $a$ and $b$ are integers.

IP Math Question Bank