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PSLE Foundation Mathematics

<p>The polynomial $\mathrm{f}(x)$ is such that&nbsp;$\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where&nbsp;$a$ and $b$ are integers. It is given that&nbsp;$\mathrm{f}(x)$ is divisible by $2x - 3$ and that&nbsp;$\mathrm{f}'(1) = 6$.</p>
<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Find the values of $a$ and $b$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Using your values of $a$ and $b$, solve the equation $\mathrm{f}(x) = 0$.<span class="inline-marks">[3]</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ is such that $\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where $a$ and $b$ are integers. It is given that $\mathrm{f}(x)$ is divisible by $2x - 3$ and that $\mathrm{f}'(1) = 6$.

Cubic Expressions and Equations

Power Rule and Derivatives of Algebraic Functions

Remainder and Factor Theorems

2024 FMSS PRELIMS P1 Q7

Differentiation

Polynomials, Cubic Equations and Partial Fractions

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

<p>Differentiate with respect to $x$ of the following expressions leaving your answers in terms of $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\frac{\sec (\ln x)}{x}$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\frac{e^{-2x}}{5^x}$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$\ln \sqrt{e^{1+\sec x}}$</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">$x^\pi$</div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">$\pi^x$</div></li><li data-list-item-id="item-f" data-list-item-path="item-f"><div class="list-item-paragraph">$2^{\ln x}$</div></li><li data-list-item-id="item-g" data-list-item-path="item-g"><div class="list-item-paragraph">$5^{\cos x}$</div></li><li data-list-item-id="item-h" data-list-item-path="item-h"><div class="list-item-paragraph">$x^x$</div></li><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$(\ln x)^x$</div></li></ol>

Differentiate with respect to $x$ of the following expressions leaving your answers in terms of $x$:

Derivatives of Trigonometric Functions

Derivatives of Natural Logarithmic Functions

Logarithmic Differentiation

Quotient Rule

Derivatives of Reciprocal Trigonometric Functions

Derivatives of Exponential Functions

Chain Rule

Techniques of Differentiation

Applications of Differentiation

<p>The diagram shows the curve $y = 2x^{2} + 3$, where&nbsp;$P(x, y)$ is a point moving along the curve for&nbsp;$0 &lt; x &lt; 6$. $Q$ is a point on the $x$-axis such that&nbsp;$PQ$ is parallel to the $y$-axis and $R(6, 0)$ is a fixed point on the $x$-axis.</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Express the area of triangle $PQR$, $T$ $\text{units}^{2}$, in terms of $x$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence find $\frac{\mathrm{d}T}{\mathrm{d}x}$.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Given that when&nbsp;$x = 2$, $T$ is increasing at the rate of $0.8$ $\text{units}^{2}/\text{s}$, find the corresponding rate of change of $x$ at this instant.</div></li></ol>

The diagram shows the curve $y = 2x^{2} + 3$, where $P(x, y)$ is a point moving along the curve for $0 < x < 6$. $Q$ is a point on the $x$-axis such that $PQ$ is parallel to the $y$-axis and $R(6, 0)$ is a fixed point on the $x$-axis.

Express the area of triangle $PQR$, $T$ $\text{units}^{2}$, in terms of $x$.

Hence find $\frac{\mathrm{d}T}{\mathrm{d}x}$.

Given that when $x = 2$, $T$ is increasing at the rate of $0.8$ $\text{units}^{2}/\text{s}$, find the corresponding rate of change of $x$ at this instant.

Connected Rates of Change

<p>The equation of a curve is $y = 5x^2 - 7x + k$, where&nbsp;$k$ is a constant.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Write down an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">The equation of the tangent to the curve at $x = p$ is $y + 2x = 3$. Find the values of $p$ and $k$.</div></li></ol>

The equation of a curve is $y = 5x^2 - 7x + k$, where $k$ is a constant.

Write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

The equation of the tangent to the curve at $x = p$ is $y + 2x = 3$. Find the values of $p$ and $k$.

Tangents and Normals at a Point

<p>Find the derivative of each of the following functions with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = (x - 1)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1) \cdots (x^{2048} + 1)$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = \frac{x(x^2)(x^3) \cdots (x^{100})}{x^{\frac{1}{2}} \left( x^{\frac{3}{2}} \right) \left( x^{\frac{5}{2}} \right) \cdots \left( x^{\frac{199}{2}} \right)}$</div></li></ol>

Find the derivative of each of the following functions with respect to $x$.

$y = (x - 1)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1) \cdots (x^{2048} + 1)$

$y = \frac{x(x^2)(x^3) \cdots (x^{100})}{x^{\frac{1}{2}} \left( x^{\frac{3}{2}} \right) \left( x^{\frac{5}{2}} \right) \cdots \left( x^{\frac{199}{2}} \right)}$

<p>Differentiate each of the following with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = \sqrt{x + \frac{1}{x}}$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = \frac{2}{\left(x^{2} + 1\right)^{3}}$</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">$y = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^{2}$</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">$y = \left(\frac{x^{4}}{2} + \frac{2}{x^{4}}\right)^{4}$</div></li></ol>

Differentiate each of the following with respect to $x$.

$y = \frac{2}{\left(x^{2} + 1\right)^{3}}$

$y = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^{2}$

$y = \left(\frac{x^{4}}{2} + \frac{2}{x^{4}}\right)^{4}$

<p>Find the derivative of each of the following with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = -5x^{-2} + \frac{1}{3}x^6 + 1$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = \frac{5}{2x^2} - \frac{3\sqrt{x}}{5} + 18\sqrt[9]{x}$</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">$y = \frac{-3x^5 + 6x^4 - 2x^2 + 4}{x^2}$</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">$y = \frac{x^3 + 2x^2 - x + 1}{\sqrt{x}}$</div></li></ol>

Find the derivative of each of the following with respect to $x$.

$y = \frac{5}{2x^2} - \frac{3\sqrt{x}}{5} + 18\sqrt[9]{x}$

$y = \frac{-3x^5 + 6x^4 - 2x^2 + 4}{x^2}$

$y = \frac{x^3 + 2x^2 - x + 1}{\sqrt{x}}$

<p>Find the derivative of each of the following with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = 3x^5 - x^7 + 8$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = -\frac{1}{3x} + \frac{6}{7x^2} + \frac{\sqrt[3]{x}}{8}$</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">$y = 2x^3 + \frac{3}{\sqrt{x}} - \frac{5}{x^5}$</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">$y = -\sqrt{3}x^3 + 3\sqrt[3]{3x}$</div></li></ol>

$y = -\frac{1}{3x} + \frac{6}{7x^2} + \frac{\sqrt[3]{x}}{8}$

$y = 2x^3 + \frac{3}{\sqrt{x}} - \frac{5}{x^5}$

<p>Find the derivative of each of the following with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = 2024$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = -\frac{5x^3}{x^9}$</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">$y = \sqrt[5]{x^4}$</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">$y = -\frac{3x^{11}}{2\sqrt{x^{11}}}$</div></li></ol>

<p>Find the derivative of each of the following with respect to $x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y = x^{10}$</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y = \frac{1}{x^{6}}$</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">$y = \sqrt[3]{x^{2}}$</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">$y = \frac{1}{\sqrt{x^{5}}}$</div></li></ol>

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768119184582-kjgszsinxbk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMTkxODQ1ODIta2pnc3pzaW54YmsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTE5NiwiZXhwIjoxNzkyNzg5NDk2fQ.364OUEQAnl-ZcdIV2QPMw45DK0rTyKJwycp3UWt6Hbc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>$PQR$ is an isosceles triangle with $PQ = PR = 50\text{cm}$ and $QR = 28\text{cm}$ as shown in the diagram above. A rectangle $ABCD$ is drawn inside the triangle with $AB$ on $QR$, and $D$ and $C$ on $PQ$ and $PR$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">If $AB = 2x\text{cm}$, show that the area $A \text{cm}^2$ of the rectangle is given by $A = \frac{48x(14-x)}{7}$.</div></li><li><div class="list-item-paragraph">Obtain an expression for $\frac{\\mathrm{d}A}{\\mathrm{d}x}$.</div></li><li><div class="list-item-paragraph">Given that $x$ can vary, find the value of $x$ for which the area of the rectangle is stationary.</div></li><li><div class="list-item-paragraph">Explain why this value of $x$ gives the largest area possible.</div></li></ol>

$PQR$ is an isosceles triangle with $PQ = PR = 50\text{cm}$ and $QR = 28\text{cm}$ as shown in the diagram above. A rectangle $ABCD$ is drawn inside the triangle with $AB$ on $QR$, and $D$ and $C$ on $PQ$ and $PR$ respectively.

Problems Involving Maximisation and Minimisation

2017 CHIJ AM SA1 P1

<p>A curve has equation $y = x^3 + 3x^2 - 9x + p$, where&nbsp;$p$ is a constant.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the $x$-coordinates of the two stationary points on the curve.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the two possible values of $p$ for which the curve has a stationary point on the $x$-axis.</span></div></li></ol>

A curve has equation $y = x^3 + 3x^2 - 9x + p$, where $p$ is a constant.

Write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Find the $x$-coordinates of the two stationary points on the curve.

Hence find the two possible values of $p$ for which the curve has a stationary point on the $x$-axis.

Stationary Points and Their Nature

<p>A sphere with radius $r$ cm, has a surface area $A$ cm$^2$ and volume $V$ cm$^3$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression for&nbsp;$\dfrac{\mathrm{d}A}{\mathrm{d}r}$ and $\dfrac{\mathrm{d}V}{\mathrm{d}r}$.</div></li><li><div class="list-item-paragraph">Given that the area is increasing at a rate of $2$ cm$^2$/s, find the rate of change of $r$ when radius is $3$ cm.</div></li><li><div class="list-item-paragraph">Hence, find the rate of change of $V$ at this instant.</div></li></ol>

A sphere with radius $r$ cm, has a surface area $A$ cm$^2$ and volume $V$ cm$^3$.

<p>Find $\frac{\\mathrm{d}y}{\\mathrm{d}x}$ for $y=\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2$</p>

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ for $y=\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2$

<p>Find the derivative of each of the following,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 4x^2 - \frac{3}{\sqrt{x}} + \frac{5}{2x^2}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{3\sqrt{x} - 3x^2 - 6x}{x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{\sqrt{x} - x^7 + 5x}{\sqrt{x}}$</span></div></li></ol>

Find the derivative of each of the following,

$y = 4x^2 - \frac{3}{\sqrt{x}} + \frac{5}{2x^2}$

$y = \frac{\sqrt{x} - x^7 + 5x}{\sqrt{x}}$

<p>Find the derivative of each of the following,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{f}(x) = -5x^8 + 4\sqrt[3]{x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{f}(x) = \sqrt[3]{x} - 2x^5 + 3$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = 7x^3 + \frac{2}{\sqrt{x}} + \frac{3}{x^5}$</span></div></li></ol>

$y = 7x^3 + \frac{2}{\sqrt{x}} + \frac{3}{x^5}$

<p>For each function below, find $\mathrm f'(x)$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 515px;"><colgroup><col style="width: 100.0000%;"></colgroup><tbody><tr style="height: 33px;"><td rowspan="2" style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center"><p style="text-align: left;"><strong>Addition and Subtraction Rule</strong></p><p style="text-align: center;">$\frac{\mathrm{d}}{\mathrm{d}x}(kx^n + qx^m) = k\frac{\mathrm{d}}{\mathrm{d}x}(x^n) + q\frac{\mathrm{d}}{\mathrm{d}x}(x^m) = knx^{n-1} + qmx^{m-1}$, where $k$ and $q$ are constants.</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>

<ol class="abc-list"><li data-list-item-id="item-addsub-a" data-list-item-path="item-addsub-a"><div class="list-item-paragraph">$\mathrm{f}(x) = 6x^5 - x^6 - 10$</div></li><li data-list-item-id="item-addsub-b" data-list-item-path="item-addsub-b"><div class="list-item-paragraph">$\mathrm{f}(x) = 2 - 5x^3 - 4\sqrt{x}$</div></li><li data-list-item-id="item-addsub-c" data-list-item-path="item-addsub-c"><div class="list-item-paragraph">$\mathrm{f}(x) = 6x - 5x^2 + 3\sqrt{x}$</div></li></ol>

For each function below, find $\mathrm f'(x)$.

<p style="text-align: left;"><strong>Addition and Subtraction Rule</strong></p><p style="text-align: center;">$\frac{\mathrm{d}}{\mathrm{d}x}(kx^n + qx^m) = k\frac{\mathrm{d}}{\mathrm{d}x}(x^n) + q\frac{\mathrm{d}}{\mathrm{d}x}(x^m) = knx^{n-1} + qmx^{m-1}$, where $k$ and $q$ are constants.</p>

$\mathrm{f}(x) = 6x - 5x^2 + 3\sqrt{x}$

Addition and Subtraction Rule

<p>Find the derivative of each of the following,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 7x^{11}$</span></div></li><li><div class="list-item-paragraph">$y = 2\sqrt[3]{x}$</div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{f}(x) = -\frac{10\sqrt{x}}{x^3}$</span></div></li></ol>

$\mathrm{f}(x) = -\frac{10\sqrt{x}}{x^3}$

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 320px;"><colgroup><col style="width: 320px;"></colgroup><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p style="text-align: left;"><strong>Scalar Multiple</strong></p><p style="text-align: center;">$\frac{\mathrm{d}}{\mathrm{d}x}(kx^n) = k\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = knx^{n-1}$, where $k$ is a constant.</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>Find the derivative of each of the following,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 4\sqrt{x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{f}(x) = -\frac{2}{x^5}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{f}(x) = -\frac{3x^2}{\sqrt{x^5}}$</span></div></li></ol>

<p style="text-align: left;"><strong>Scalar Multiple</strong></p><p style="text-align: center;">$\frac{\mathrm{d}}{\mathrm{d}x}(kx^n) = k\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = knx^{n-1}$, where $k$ is a constant.</p>

$\mathrm{f}(x) = -\frac{3x^2}{\sqrt{x^5}}$

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