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PSLE Foundation Mathematics

<p>The curve $C$ has equation</p>
<p>\[y=ax^2+bx+c,\]</p>
<p>where&nbsp;$a,b,c$ are constants. It passes through $P\left(1,\frac92\right)$.</p>
<p>The tangents to $C$ at the points where&nbsp;$x=1$ and $x=3$ have angles of inclination $\frac{\pi}{4}$ and $\frac{3\pi}{4}$, respectively. Angles of inclination are measured anticlockwise from the positive $x$-axis.</p>
<ol class="i-list"><li data-list-item-id="ac8a0ec9-c518-4df5-ae16-bc947f2a20bd" data-list-item-path="ac8a0ec9-c518-4df5-ae16-bc947f2a20bd"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="fdcca7ff-8d9a-4bce-afd6-954da5555f97" data-list-item-path="fdcca7ff-8d9a-4bce-afd6-954da5555f97"><div class="list-item-paragraph">The two tangents intersect at $T$. Find the exact area of the triangle bounded by the two tangents and the line $y=\frac92$.<span class="inline-marks">[3]</span></div></li></ol>

where $a,b,c$ are constants. It passes through $P\left(1,\frac92\right)$.

The tangents to $C$ at the points where $x=1$ and $x=3$ have angles of inclination $\frac{\pi}{4}$ and $\frac{3\pi}{4}$, respectively. Angles of inclination are measured anticlockwise from the positive $x$-axis.

Form three linear equations in $a,b,c$, and hence find their values.

The two tangents intersect at $T$. Find the exact area of the triangle bounded by the two tangents and the line $y=\frac92$.

Tangents and Normals at a Point

A parabola determined by tangent inclinations

System of Linear Equations

Applications of Differentiation

<p>The curve $C$ has equation</p>
<p>\[y=x^3+ax^2+bx+c,\]</p>
<p>where&nbsp;$a,b,c$ are constants.</p>
<p>The tangent to $C$ at the point where&nbsp;$x=1$ has equation $y=2x+1$. The normal to $C$ at the point where&nbsp;$x=-1$ is parallel to the line $x+10y=4$.</p>
<ol class="i-list"><li data-list-item-id="cb773b6e-3834-477c-be26-b4675e5a49e1" data-list-item-path="cb773b6e-3834-477c-be26-b4675e5a49e1"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="6331ed95-e7fb-4e5e-a814-3ab1a8c4631b" data-list-item-path="6331ed95-e7fb-4e5e-a814-3ab1a8c4631b"><div class="list-item-paragraph">Show that the gradient of $C$ is always positive. Hence explain why $C$ intersects the $x$-axis exactly once, and find the $x$-coordinate of this intersection correct to 3 significant figures.<span class="inline-marks">[3]</span></div></li></ol>

The tangent to $C$ at the point where $x=1$ has equation $y=2x+1$. The normal to $C$ at the point where $x=-1$ is parallel to the line $x+10y=4$.

Show that the gradient of $C$ is always positive. Hence explain why $C$ intersects the $x$-axis exactly once, and find the $x$-coordinate of this intersection correct to 3 significant figures.

Increasing and Decreasing Functions

Quadratic Expressions Positive or Negative for All Real Values

A cubic determined by a tangent and a normal

Inequalities

<p>The functions $\mathrm{p}$ and $\mathrm{q}$ are continuous for&nbsp;$1\leq x\leq5$. The function $\mathrm{p}$ is strictly increasing, while $\mathrm{q}$ is strictly decreasing and $\mathrm{q}(x)&gt;0$. Selected values are shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table class="table--original-style" style="table-layout: fixed; width: 300px;"><colgroup><col style="width: 16.6667%;"><col style="width: 16.6667%;"><col style="width: 16.6667%;"><col style="width: 16.6667%;"><col style="width: 16.6667%;"><col style="width: 16.6667%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$5$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\mathrm{p}(x)$</td><td style="text-align: center; --cell-justify: center">$-10$</td><td style="text-align: center; --cell-justify: center">$-8$</td><td style="text-align: center; --cell-justify: center">$-6$</td><td style="text-align: center; --cell-justify: center">$-4$</td><td style="text-align: center; --cell-justify: center">$5$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\mathrm{q}(x)$</td><td style="text-align: center; --cell-justify: center">$5$</td><td style="text-align: center; --cell-justify: center">$4$</td><td style="text-align: center; --cell-justify: center">$3$</td><td style="text-align: center; --cell-justify: center">$2$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr></tbody></table></div></div>
<p>Using the information in the table,</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph"><p>Find $\mathrm{p}\mathrm{q}(2)$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph"><p>Solve $\mathrm{p}^{-1}\mathrm{q}(x)=5$.</p><span class="inline-marks">[2]</span></div></li></ol>
<p>The function $\mathrm{r}$ is defined by $\mathrm{r}(x)=\frac{\mathrm{p}(x)}{\mathrm{q}(x)}$.</p>
<p>It is further given that&nbsp;$\mathrm{p}'(3)=3$ and $\mathrm{q}'(3)=-1$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph">Explain why $\mathrm{r}^{-1}$ does not exist on the given domain.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-d" data-list-item-path="q1-d"><div class="list-item-paragraph">Find the equation of the normal to $y=\mathrm{r}(x)$ at $x=3$.<span class="inline-marks">[3]</span></div></li></ol>

The functions $\mathrm{p}$ and $\mathrm{q}$ are continuous for $1\leq x\leq5$. The function $\mathrm{p}$ is strictly increasing, while $\mathrm{q}$ is strictly decreasing and $\mathrm{q}(x)>0$. Selected values are shown below.

The function $\mathrm{r}$ is defined by $\mathrm{r}(x)=\frac{\mathrm{p}(x)}{\mathrm{q}(x)}$.

It is further given that $\mathrm{p}'(3)=3$ and $\mathrm{q}'(3)=-1$.

Explain why $\mathrm{r}^{-1}$ does not exist on the given domain.

Find the equation of the normal to $y=\mathrm{r}(x)$ at $x=3$.

Composite Functions

One-to-One Functions and Inverse Existence

Quotient Rule

Inverse Functions

Normal from tabulated function values

Functions

Techniques of Differentiation

<p>Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.</p>
<p>A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.</p>
<ol class="abc-list"><li data-list-item-id="jvss-p1-q9-q-1-content-items-0" data-list-item-path="jvss-p1-q9-q-1-content-items-0"><div class="list-item-paragraph"><p>Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-1" data-list-item-path="jvss-p1-q9-q-1-content-items-1"><div class="list-item-paragraph"><p>Show that the $x$-coordinate of $P$ is $\mathrm{e}^{\frac{1}{6}}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-2" data-list-item-path="jvss-p1-q9-q-1-content-items-2"><div class="list-item-paragraph"><p>Find the equation of normal to the curve at the point $x=\mathrm{e}$. Leave your answer in terms of $\mathrm{e}$.</p><span class="inline-marks">[5]</span></div></li></ol>

Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.

A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.

Derivatives of Natural Logarithmic Functions

Stationary Points and Their Nature

2025 JVSS PRELIMS P1 Q9

Differentiation

<p>The curve $y=5-\mathrm{e}^{x}$ has a tangent at $A$, where the $x$-coordinate of $A$ is $\ln2$. The tangent meets the $y$-axis at $B$, and the curve meets the $x$-axis at $C$.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q16-parts-item-1" data-list-item-path="amarea-q16-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at $A$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q16-parts-item-2" data-list-item-path="amarea-q16-parts-item-2"><div class="list-item-paragraph">Find the coordinates of $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q16-parts-item-3" data-list-item-path="amarea-q16-parts-item-3"><div class="list-item-paragraph">Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.<span class="inline-marks">[5]</span></div></li></ol>

The curve $y=5-\mathrm{e}^{x}$ has a tangent at $A$, where the $x$-coordinate of $A$ is $\ln2$. The tangent meets the $y$-axis at $B$, and the curve meets the $x$-axis at $C$.

Find the equation of the tangent at $A$.

Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.

Area Bounded by a Curve and Straight Lines

Exponential curve with a tangent and the coordinate axes

Integration

<p>The tangent to the curve $y=2\ln(x+1)$ at $A$, where&nbsp;$x=3$, meets the $y$-axis at $B$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q17-parts-item-1" data-list-item-path="amarea-q17-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at $A$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q17-parts-item-2" data-list-item-path="amarea-q17-parts-item-2"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve, the tangent and the $y$-axis.<span class="inline-marks">[5]</span></div></li></ol>

The tangent to the curve $y=2\ln(x+1)$ at $A$, where $x=3$, meets the $y$-axis at $B$.

Hence find the exact area enclosed by the curve, the tangent and the $y$-axis.

Area under Logarithmic Curve

Logarithmic curve and its tangent beside the y-axis

<p>The tangent to the curve $y=2\sin x$ at $A$, where the $x$-coordinate of $A$ is $\dfrac{\pi}{3}$, meets the $y$-axis at $B$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q22-parts-item-1" data-list-item-path="amarea-q22-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at $A$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q22-parts-item-2" data-list-item-path="amarea-q22-parts-item-2"><div class="list-item-paragraph">Find the coordinates of $B$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amarea-q22-parts-item-3" data-list-item-path="amarea-q22-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve, the tangent and the $y$-axis.<span class="inline-marks">[5]</span></div></li></ol>

The tangent to the curve $y=2\sin x$ at $A$, where the $x$-coordinate of $A$ is $\dfrac{\pi}{3}$, meets the $y$-axis at $B$.

Sine curve and its tangent beside the y-axis

<p>The tangent to the curve $y=3\ln(x-1)$ at $A$, where&nbsp;$x=4$, meets the $x$-axis at $B$. The curve meets the $x$-axis at $C$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q18-parts-item-1" data-list-item-path="amarea-q18-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at $A$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q18-parts-item-2" data-list-item-path="amarea-q18-parts-item-2"><div class="list-item-paragraph">Find the coordinates of $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q18-parts-item-3" data-list-item-path="amarea-q18-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve, the tangent and the $x$-axis.<span class="inline-marks">[5]</span></div></li></ol>

The tangent to the curve $y=3\ln(x-1)$ at $A$, where $x=4$, meets the $x$-axis at $B$. The curve meets the $x$-axis at $C$.

Hence find the exact area enclosed by the curve, the tangent and the $x$-axis.

Logarithmic curve with a tangent and the x-axis

<p>The tangent to the curve $y=x^3$ at $P(1,\ 1)$ meets the curve again at $Q$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q08-parts-item-1" data-list-item-path="amarea-q08-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at $P$ and the coordinates of $Q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q08-parts-item-2" data-list-item-path="amarea-q08-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the tangent.<span class="inline-marks">[4]</span></div></li></ol>

The tangent to the curve $y=x^3$ at $P(1,\ 1)$ meets the curve again at $Q$.

Find the equation of the tangent at $P$ and the coordinates of $Q$.

Hence find the area enclosed by the curve and the tangent.

Cubic curve and its tangent

<p>The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinate of $P$ is $3$. The tangent meets the $y$-axis at $Q$, and the curve meets the positive $x$-axis at $R$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q30-parts-item-1" data-list-item-path="amarea-q30-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $P$ and the equation of the tangent $PQ$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q30-parts-item-2" data-list-item-path="amarea-q30-parts-item-2"><div class="list-item-paragraph">Find the coordinates of $Q$ and $R$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q30-parts-item-3" data-list-item-path="amarea-q30-parts-item-3"><div class="list-item-paragraph">Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.<span class="inline-marks">[5]</span></div></li></ol>

The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinate of $P$ is $3$. The tangent meets the $y$-axis at $Q$, and the curve meets the positive $x$-axis at $R$.

Find the coordinates of $P$ and the equation of the tangent $PQ$.

Rational curve with a tangent and the coordinate axes

<p>The equation of a curve is $y = ax^3 + b$, where&nbsp;$a$ and $b$ are constants. The equation of the normal to the curve at the point where&nbsp;$x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.<span class="inline-marks">[6]</span></p>

The equation of a curve is $y = ax^3 + b$, where $a$ and $b$ are constants. The equation of the normal to the curve at the point where $x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.

Parallel and Perpendicular Lines

Equation of a Straight Line

2024 FMSS PRELIMS P2 Q2

Coordinate Geometry

<p>It is given that&nbsp;$y = \ln \left(\frac{2x}{5 - x} \right)$, where&nbsp;$0 &lt; x &lt; 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">If&nbsp;$y$ is decreasing at a rate of $10$ units/s, find the rate of change of $x$ when&nbsp;$y = \ln 8$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">The equation of the normal to the curve at point $A$ is $8y = 8\ln 2 + 25 - 10x$. Find the $x$-coordinate of point $A$.<span class="inline-marks">[4]</span></div></li></ol>

It is given that $y = \ln \left(\frac{2x}{5 - x} \right)$, where $0 < x < 5$.

Connected Rates of Change

2024 CGSS P1 Q9

<p>A curve $D$ has equation</p>
<p style="text-align: center"> $y = ax + b \ln x + \frac{c}{x},\quad x &gt; 0,$</p>
<p>where&nbsp;$a$, $b$ and $c$ are constants. It is given that&nbsp;$D$ has a stationary point at $x = \frac{3}{2}$ and the tangent to $D$ at the point where&nbsp;$x = 1$ is $y = x - 5$. Find the values of $a$, $b$ and $c$.<span class="inline-marks">[4]</span></p>

 $y = ax + b \ln x + \frac{c}{x},\quad x > 0,$

where $a$, $b$ and $c$ are constants. It is given that $D$ has a stationary point at $x = \frac{3}{2}$ and the tangent to $D$ at the point where $x = 1$ is $y = x - 5$. Find the values of $a$, $b$ and $c$.

2024 RI Promo Q1

<p>Let&nbsp;$f(x) = x^3 - 4x +1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Expand $ (x + h)^3$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Use the formula $f'(x) = \lim\limits_{h \to 0} \frac{f(x +h) - f(x)}{h}$ to show that the derivative of $f(x)$ is $3x^2 - 4$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The tangent to the curve of $f$ at the point $ \mathrm{P} (1,-2)$ is parallel to the tangent at a point $\mathrm{Q}$.<br>Find the coordinates of $\mathrm{Q}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The graph of $f$ is decreasing for&nbsp;$p &lt; x&lt; q$. Find the value of $p$ and of $q$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down the range of values for the gradient of $f$.<span class="inline-marks">[2]</span></div></li></ol>

Binomial Expansion for Positive Integer Powers

Differentiation by First Principles

Binomial Theorem

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the equations of the tangent to the curve $y=x^2$ at the point $(t, t^2)$, expressing the coefficients in terms of $t$ where necessary.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equations of all lines that are tangents to both the curves $y=x^2$ and $y=-x^2+8x-16$.</span><span class="inline-marks">[4]</span></div></li></ol>

Find the equations of the tangent to the curve $y=x^2$ at the point $(t, t^2)$, expressing the coefficients in terms of $t$ where necessary.

Find the equations of all lines that are tangents to both the curves $y=x^2$ and $y=-x^2+8x-16$.

NJC IP4 Tangents and Normals Tutorial Q3

<p>The equation of a curve is $y=x^3$. The distinct points $P$ and $Q$ lying on the curve have&nbsp;$x$-coordinates $p$ and $q$ respectively, where&nbsp;$p$ and $q$ are real constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the tangent to the curve at $Q$ passes through point $P$, show that\[p^2 + pq + cq^2 = 0\] for some constant $c$ to be determined. Hence, find $p$ in terms of $q$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.</span><span class="inline-marks">[4]</span></div></li></ol>

The equation of a curve is $y=x^3$. The distinct points $P$ and $Q$ lying on the curve have $x$-coordinates $p$ and $q$ respectively, where $p$ and $q$ are real constants.

Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.

NJC IP4 Tangents and Normals Tutorial Q2

<p>The equation of a curve is $y = -\frac{5}{8}x + \frac{6}{x}$, where&nbsp;$x &gt; 0$. Given that the gradient of the normal at point $P$ is $1$, find</p>
<ol class="abc-list"><li data-list-item-id="item-a-12" data-list-item-path="item-a-12"><div class="list-item-content"><span class="list-item-text">the coordinates of $P$,</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1772957522866-x2jn7boae" data-list-item-path="item-1772957522866-x2jn7boae"><div class="list-item-paragraph">the equation of the normal at the curve at <strong>P</strong><span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y=-\frac{5}{8}x+\frac{6}{x}$, where $x>0$. Given that the gradient of the normal at point $P$ is 1, find

the equation of the normal at the curve at $P$.

2025 AHS PRELIMS P1 Q8

<p>The curve $D$ has equation $y = \ln(1 + 2x^2)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use a non-calculator method to find the equation of the tangent to $D$ at the point where&nbsp;$x = 2$, giving your answer in the form $ax + by = c + d \ln 3$ where&nbsp;$a$, $b$, $c$ and $d$ are integers.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graphs of $D$ and the tangent found in part (a) on a single diagram, stating the coordinates of the point(s) of intersection of the two graphs and the points where the tangent meets the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the finite region bounded by the curve $D$, the $x$-axis and the tangent line. Give your answer correct to $3$ significant figures.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A point $P(x, y)$ moves along the curve $D$ in such a way that its $x$ coordinate is increasing at a constant rate of $3$ units per second. Find the rate of change of its $y$ coordinate at the instant when&nbsp;$x=1$.<span class="inline-marks">[2]</span></p>

The curve $D$ has equation $y = \ln(1 + 2x^2)$.

Use a non-calculator method to find the equation of the tangent to $D$ at the point where $x = 2$, giving your answer in the form $ax + by = c + d \ln 3$ where $a$, $b$, $c$ and $d$ are integers.

Sketch the graphs of $D$ and the tangent found in part (a) on a single diagram, stating the coordinates of the point(s) of intersection of the two graphs and the points where the tangent meets the axes.

Find the area of the finite region bounded by the curve $D$, the $x$-axis and the tangent line. Give your answer correct to $3$ significant figures.

A point $P(x, y)$ moves along the curve $D$ in such a way that its $x$ coordinate is increasing at a constant rate of $3$ units per second. Find the rate of change of its $y$ coordinate at the instant when $x=1$.

Sketching Logarithmic Functions and Solving Equations Graphically

Area Bounded by Curves

2024 SAJC H1 Q4

Graphing Techniques

Applications of Integration

<p>The equation of a curve is $y = \ln\left( \frac{3x - 7}{x + 1} \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find, respectively, the equations of the tangent and the normal to the curve at the point where the curve crosses the $x$-axis.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence find the area of the triangle formed by the tangent and the normal to the curve at the above point and the $y$-axis.</div></li></ol>

The equation of a curve is $y = \ln\left( \frac{3x - 7}{x + 1} \right)$.

Find, respectively, the equations of the tangent and the normal to the curve at the point where the curve crosses the $x$-axis.

Hence find the area of the triangle formed by the tangent and the normal to the curve at the above point and the $y$-axis.

<p>The equation of a curve is $y = x \ln x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find, respectively, the equations of the tangent and the normal to the curve at the point where&nbsp;$x = e$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence find the area of the quadrilateral formed by the tangent and the normal to the curve at the point where&nbsp;$x = e$ and the axes.</div></li></ol>

The equation of a curve is $y = x \ln x$.

Find, respectively, the equations of the tangent and the normal to the curve at the point where $x = e$.

Hence find the area of the quadrilateral formed by the tangent and the normal to the curve at the point where $x = e$ and the axes.

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