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<p>HIMHEYS' Confectionery recently created three types of chocolate: Organic white, Organic milk and Organic dark, available in $250$ g bars. Cocoa butter, an essential ingredient in chocolate bars, makes up $25\%$, $20\%$ and $15\%$ of the mass of an Organic white, Organic milk and Organic dark chocolate bar respectively.</p>
<p>To prepare for the official launch of their chocolate bars at an upcoming Food Expo, HIMHEYS' decides to manufacture a total of $300$ bars for the event, with more than $70$ bars of each type.</p>
<p>The confectionary intends to use $14$ kg of cocoa butter in the production of the above batch of chocolate bars. If the number of milk chocolate bars is to be smaller than the number of white chocolate bars, determine how many organic chocolate bars of each type can be produced.<span class="inline-marks">[6]</span></p>

HIMHEYS' Confectionery recently created three types of chocolate: Organic white, Organic milk and Organic dark, available in $250$ g bars. Cocoa butter, an essential ingredient in chocolate bars, makes up $25\%$, $20\%$ and $15\%$ of the mass of an Organic white, Organic milk and Organic dark chocolate bar respectively.

To prepare for the official launch of their chocolate bars at an upcoming Food Expo, HIMHEYS' decides to manufacture a total of $300$ bars for the event, with more than $70$ bars of each type.

The confectionary intends to use $14$ kg of cocoa butter in the production of the above batch of chocolate bars. If the number of milk chocolate bars is to be smaller than the number of white chocolate bars, determine how many organic chocolate bars of each type can be produced.

2012 TJC P1 Q3

System of Linear Equations

<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 ${{S}_{n}}$.</p><p style="text-align: right;">[4]</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 ${{S}_{n}}$.

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

2023 SAJC Promo Q1

APGP

<p>When the polynomial ${{x}^{3}}+a{{x}^{2}}+bx+c$ is divided by $\left( x-1 \right)$, $\left( x-2 \right)$ and $\left( x-3 \right)$, the remainders are $8$, $12$ and $25$ respectively.</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;">[4]</p></li></ol>
<p>A curve has equation $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{3}}+a{{x}^{2}}+bx+c$, with the values of $a$, $b$ and $c$ found in part (i).</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Show that the gradient of the curve is always positive. Hence explain why the equation $\mathrm{f}\left( x \right)=0$ has only one real root and find this root.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the $x$- coordinates of the points where the tangent to the curve is parallel to the line $y=2x-3$.</span></div><p style="text-align: right;">[3]</p></li></ol>

When the polynomial ${{x}^{3}}+a{{x}^{2}}+bx+c$ is divided by $\left( x-1 \right)$, $\left( x-2 \right)$ and $\left( x-3 \right)$, the remainders are $8$, $12$ and $25$ respectively.

A curve has equation $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{3}}+a{{x}^{2}}+bx+c$, with the values of $a$, $b$ and $c$ found in part (i).

Show that the gradient of the curve is always positive. Hence explain why the equation $\mathrm{f}\left( x \right)=0$ has only one real root and find this root.

Find the $x$- coordinates of the points where the tangent to the curve is parallel to the line $y=2x-3$.

N2017 P1 Q5

<p>Terry wishes to buy a certain number of guppies, mollies, goldfish and neon tetras for his new aquarium. Three fish shops offer the following prices per fish:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759217610237-yph34dt11j.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMTc2MTAyMzcteXBoMzRkdDExai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzA5NDc2LCJleHAiOjE3ODU3MDk3NzZ9.Hp5sVMiSI7HZjxYaAuSy4WTyLOdB6pZiAmdJjwrAhT0" width="400" class="aligncenter" /></figure>
<p>Find the number of guppies, mollies, goldfish and neon tetras that Terry wishes to buy.</p><p style="text-align: right;">[4]</p>

Terry wishes to buy a certain number of guppies, mollies, goldfish and neon tetras for his new aquarium. Three fish shops offer the following prices per fish:

Find the number of guppies, mollies, goldfish and neon tetras that Terry wishes to buy.

2011 TJC C2 BT Q1

<p>Sherry has an initial sum of ${\$35 000}$ which she divided into three portions $A$, $B$ and $C$.</p>
<p>She is considering placing the portions of money into three separate banks that provide returns through savings account and investment account. The following table shows the returns on savings account and investment account provided by the three banks for a period of one year:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759139692443-c82zf3f2oaj.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxMzk2OTI0NDMtYzgyemYzZjJvYWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwOTQ3NiwiZXhwIjoxNzg1NzA5Nzc2fQ.VUKKMOzmAoplij3G_olsiw9PClnI8URBEY4TXvwixBU" width="400" class="aligncenter" /></figure>
<p>Sherry did some calculations and realized that:</p>
<p style="margin-left: 3rem">I. If she were to place the portions of money $A$, $B$ and $C$ into the savings accounts of POSD, OCBB and Non-Standard banks respectively, her total returns will be $\$151.15$ at the end of a year.</p>
<p style="margin-left: 3rem">II. If she were to invest the portions of money $A$, $B$ and $C$ through the investment accounts of POSD, OCBB and Non-Standard banks respectively, her total returns will be $\$426$ at the end of a year.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact amounts of the three portions of money $A$, $B$ and $C$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Sherry decides that she will place one of the portions of her money into the savings account of one bank and the remaining two portions into the investment accounts of the remaining two banks separately. Determine the maximum amount of money she can have after a period of one year.</span></div><p style="text-align: right;">[2]</p></li></ol>

Sherry has an initial sum of ${\$35 000}$ which she divided into three portions $A$, $B$ and $C$.

She is considering placing the portions of money into three separate banks that provide returns through savings account and investment account. The following table shows the returns on savings account and investment account provided by the three banks for a period of one year:

Sherry did some calculations and realized that:

I. If she were to place the portions of money $A$, $B$ and $C$ into the savings accounts of POSD, OCBB and Non-Standard banks respectively, her total returns will be $\$151.15$ at the end of a year.

II. If she were to invest the portions of money $A$, $B$ and $C$ through the investment accounts of POSD, OCBB and Non-Standard banks respectively, her total returns will be $\$426$ at the end of a year.

Find the exact amounts of the three portions of money $A$, $B$ and $C$.

Sherry decides that she will place one of the portions of her money into the savings account of one bank and the remaining two portions into the investment accounts of the remaining two banks separately. Determine the maximum amount of money she can have after a period of one year.

Linear Equations - General

ASRJC Tutorial 3 System of Linear Equations Q7

Linear Equations and Inequalities

<p>Suppose that in a fixed population of breakfast-food consumers, there are just three brands, $X$, $Y$, and $Z$ that share that market.</p>
<p>Suppose that each year,</p>
<p>${10\%}$ of the brand $X$ buyers shift to brand $Y$ and ${20\%}$ to brand $Z$ and</p>
<p>${20\%}$ of the brand $Y$ buyers shift to brand $X$ and ${20\%}$ to brand $Z$</p>
<p>${10\%}$ of the brand $Z$ buyers shift to brand $X$ and none to brand $Y$</p>
<p>Suppose that it is found that despite all this shifting, the market shares of these three brands are nevertheless unchanged from year to year. What are these market shares?</p><p style="text-align: right;">[4]</p>

Suppose that in a fixed population of breakfast-food consumers, there are just three brands, $X$, $Y$, and $Z$ that share that market.

${10\%}$ of the brand $X$ buyers shift to brand $Y$ and ${20\%}$ to brand $Z$ and

${20\%}$ of the brand $Y$ buyers shift to brand $X$ and ${20\%}$ to brand $Z$

${10\%}$ of the brand $Z$ buyers shift to brand $X$ and none to brand $Y$

Suppose that it is found that despite all this shifting, the market shares of these three brands are nevertheless unchanged from year to year. What are these market shares?

ASRJC Tutorial 3 Q6

<p>A company specializes in making toy cars. It makes three types of toy cars, Toy Car $A$, Toy Car $B$ and Toy Car $C$. The amount of materials required (in units) for each type of car is given below:</p>
<table><thead><tr><th></th><th>Plastic</th><th>Rubber</th><th>Metal</th></tr></thead><tbody><tr><td>Toy Car ${A}$</td><td>${2}$</td><td>${1}$</td><td>${1}$</td></tr><tr><td>Toy Car ${B}$</td><td>${3}$</td><td>${4}$</td><td>${5}$</td></tr><tr><td>Toy Car ${C}$</td><td>${1}$</td><td>${3}$</td><td>${4}$</td></tr><tr><td>Total amount of material available:</td><td>${200}$</td><td>${150}$</td><td>${170}$</td></tr></tbody></table>
<p>If the company makes a profit of $\$1$ , $\$2$  and $\$3$ from the sale of each Toy Car $A$, $B$ and $C$ respectively, find the maximum profit you can make given that all the materials are used.<span class="inline-marks">[4]</span></p>

A company specializes in making toy cars. It makes three types of toy cars, Toy Car $A$, Toy Car $B$ and Toy Car $C$. The amount of materials required (in units) for each type of car is given below:

If the company makes a profit of $\$1$ , $\$2$  and $\$3$ from the sale of each Toy Car $A$, $B$ and $C$ respectively, find the maximum profit you can make given that all the materials are used.

2008 CJC P1 Q1 [Modified]

<p>If two curves ${y = ax^3 + \sqrt{bx}}$ and ${y = \ln(px^2)}$ intersect at the points where ${x = 1}$, ${x = 2}$ and ${x = 4}$, find the values of ${a}$, ${b}$ and ${p}$, correct to $2$ decimal places.</p><p style="text-align: right;">[4]</p>

If two curves ${y = ax^3 + \sqrt{bx}}$ and ${y = \ln(px^2)}$ intersect at the points where ${x = 1}$, ${x = 2}$ and ${x = 4}$, find the values of ${a}$, ${b}$ and ${p}$, correct to $2$ decimal places.

2013 TJC P1 Q1

<p>Three space probes, Probe $A$, Probe $B$, and Probe $C$ with mass $600$ kg, $630$ kg, and ${900}$ kg respectively were sent into space to find the gravitational pull on the planets Venus, Mars, and Saturn. The sum of the weights of Probe $B$ on the three planets is $13860$ N.</p>
<p>The weight of Probe $C$ on Saturn is $2880$ N more than the weight of Probe ${A}$ on Venus. The average weight of Probe $A$ on Saturn, Probe $B$ on Venus, and Probe $C$ on Mars is $4870$ N.</p>
<p>Find the gravitational pull on each of the planets.</p><p style="text-align: right;">[3]</p>
<p>Hence find the weight of Probe $D$, which has mass $500$ kg, on planet Saturn.</p>
<p>[Weight (in N)$=$Mass (in kg)$\times $gravitational pull (ms$^{-2}$)]</p><p style="text-align: right;">[1]</p>

Three space probes, Probe $A$, Probe $B$, and Probe $C$ with mass $600$ kg, $630$ kg, and ${900}$ kg respectively were sent into space to find the gravitational pull on the planets Venus, Mars, and Saturn. The sum of the weights of Probe $B$ on the three planets is $13860$ N.

The weight of Probe $C$ on Saturn is $2880$ N more than the weight of Probe ${A}$ on Venus. The average weight of Probe $A$ on Saturn, Probe $B$ on Venus, and Probe $C$ on Mars is $4870$ N.

Find the gravitational pull on each of the planets.

Hence find the weight of Probe $D$, which has mass $500$ kg, on planet Saturn.

[Weight (in N)$=$Mass (in kg)$\times $gravitational pull (ms$^{-2}$)]

2010 IJC P2 Q1

<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 $100$.

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