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PSLE Foundation Mathematics

<p>A curve $C$ is given by the equation</p>
<p style="text-align: center">\[e^{\frac{x^2}{2}}y=\ln x^2,\]</p>
<p>where&nbsp;$x\ne0$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$x\ln x^2+e^{\frac{x^2}{2}}\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point $P$ where&nbsp;$x=1$. Given that the tangent at $P$ meets $C$ again at the point $Q$, find the coordinates of $Q$, giving your answer correct to 5 significant figures.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Determine the acute angle between the tangents to $C$ at the points $P$ and $Q$.<span class="inline-marks">[3]</span></div></li></ol>

Implicit Differentiation

Derivatives of Natural Logarithmic Functions

Derivatives of Exponential Functions

Tangents and Normals to Implicit Curves

Chain Rule

Implicit Curve Tangents and Angle of Intersection

Applications of Differentiation

Techniques of Differentiation

<p>It is given that&nbsp;$y = x^{xy}$, where&nbsp;$x &gt; 0$, $y &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y^2 (1 + \ln x)}{1 - \ln y}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the coordinates of the point on the curve $y = x^{xy}$ whose tangent is parallel to the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = x^{xy}$, where $x > 0$, $y > 0$.

Logarithmic Differentiation

Tangents and Normals Parallel to the Axes

2025 ACJC P1 Q3

<p>The equation of a curve is $2xy + (1 + y)^2 = x$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the equations of the two tangents which are parallel to the $y$-axis.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">The normal to the curve at the point $A(-4, 5)$ meets the curve again at the point $B$. Find the coordinates of point $B$.<span class="inline-marks">[4]</span></div></li></ol>

The equation of a curve is $2xy + (1 + y)^2 = x$.

2024 MI P1 Q4

<p>A curve $C$ has equation $3x^2 + 2xy - 4y^2 = 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3x + y}{4y - x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the acute angle between the tangent to $C$ at the point $(1, 1)$ and the $x$-axis.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Show that there is no point on $C$ where the tangent to the curve is parallel to the $x$-axis.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $3x^2 + 2xy - 4y^2 = 1$.

2024 VJC Promo Q5

<p>A curve has equation $x^{\frac{1}{2}} + y^{\frac{1}{2}} = 3$, for&nbsp;$x &gt; 0$ and $y &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = -\left(\frac{y}{x}\right)^{\frac{1}{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the equation of the normal to the curve at the point where&nbsp;$x = 1$ and determine the acute angle that the normal makes with the $x$-axis.<span class="inline-marks">[5]</span></div></li></ol>

A curve has equation $x^{\frac{1}{2}} + y^{\frac{1}{2}} = 3$, for $x > 0$ and $y > 0$.

N2021 P1 Q3 [Modified]

<p>The equation of a curve $C$ is $x^2 - 2xy + 2y^2 = k$, where&nbsp;$k$ is a constant.</p>
<p>Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.</p>
<p>Given that&nbsp;$C$ has two points for which the tangents are parallel to the line $y = x$, find the range of values of $k$.</p>

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

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

Given that $C$ has two points for which the tangents are parallel to the line $y = x$, find the range of values of $k$.

<p>The curve $C$ has the equation ${{x}^{2}}+{{y}^{2}}-2xy+3x=4$.</p>
<ol class="i-list"><li data-list-item-id="item-1771990757617-um4ttqujz"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.	<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1771990758224-l00qtoe99"><div class="list-item-paragraph">Find the equation of the tangent to $C$ which is parallel to the $y$-axis. <span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has the equation ${{x}^{2}}+{{y}^{2}}-2xy+3x=4$.

2023 JPJC Promo Q3

<p>Given the gradient function,$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{f}(x)}{\mathrm{g}\left( x \right)}$,<br>When tangent is parallel to $x$-axis $\to \mathrm{f}\left( x \right)=0$ <br>When tangent is parallel to $y$-axis $\to \mathrm{g}\left( x \right)=0$<br></p>

If $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}$, then

the tangent is parallel to the $x$-axis when $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$, so solve $\mathrm{f}(x)=0$ while checking $\mathrm{g}(x)\ne0$;

the tangent is parallel to the $y$-axis when the gradient is unbounded, so solve $\mathrm{g}(x)=0$ while checking the curve and numerator conditions.

For $x=\mathrm{f}(t)$ and $y=\mathrm{g}(t)$,

\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}=\frac{\mathrm{g}'(t)}{\mathrm{f}'(t)},\qquad \frac{\mathrm{d}x}{\mathrm{d}t}\ne0.\]

Implicit and Parametric Tangents

<p>A curve $C$ has equation $y^2 = x + \mathrm{e}^{2x} y$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$(2y - \mathrm{e}^{2x}) \frac{\mathrm{d}y}{\mathrm{d}x} = 1 + 2y\mathrm{e}^{2x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the equations of the tangents to $C$ at the points where&nbsp;$x = 0$.<span class="inline-marks">[4]</span></div></li></ol>

A curve $C$ has equation $y^2 = x + \mathrm{e}^{2x} y$.

Show that $(2y - \mathrm{e}^{2x}) \frac{\mathrm{d}y}{\mathrm{d}x} = 1 + 2y\mathrm{e}^{2x}$.

Find the equations of the tangents to $C$ at the points where $x = 0$.

Tangents and Normals at a Point

2025 TMJC P1 Q2

Graphing Techniques

<p>A curve has equation $(x - y)^2 = 2\mathrm{e}^{xy}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x - y - y\mathrm{e}^{xy}}{x - y + x\mathrm{e}^{xy}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Given that the curve cuts the positive $y$-axis at point $A$, find the equation of the normal to the curve at $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">The normal to the curve at $A$ meets the curve at another point $B$. Find the coordinates of $B$.<span class="inline-marks">[3]</span></div></li></ol>

A curve has equation $(x - y)^2 = 2\mathrm{e}^{xy}$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x - y - y\mathrm{e}^{xy}}{x - y + x\mathrm{e}^{xy}}$.

Given that the curve cuts the positive $y$-axis at point $A$, find the equation of the normal to the curve at $A$.

The normal to the curve at $A$ meets the curve at another point $B$. Find the coordinates of $B$.

2025 SAJC P1 Q8

<p>The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-paragraph">It is given that&nbsp;$C$ has a tangent which is parallel to the $y$-axis. Show that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that&nbsp;$k \leq \frac{1}{54}$.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-c-9"><div class="list-item-content"><span class="list-item-text">Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.</span><span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where $k$ is a constant.

It is given that $C$ has a tangent which is parallel to the $y$-axis.\n\nShow that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that $k \leq \frac{1}{54}$.

Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.

Parameter Conditions for Graphs and Equations

2025 NYJC P2 Q4

<p>The curve $E$ has equation</p>
<p style="text-align: center">\[(x-a)^2 + 4y^2 = 4, \text{ where}\hspace{0.5em} 0 &lt; a &lt; 2.\]</p>
<ol class="i-list"><li data-list-item-id="item-i-3" data-list-item-path="item-i-3"><div class="list-item-paragraph">Sketch $E$.<span class="inline-marks">[2]</span></div></li></ol>
<p>You are now given that&nbsp;$a=1$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii-3" data-list-item-path="item-ii-3"><div class="list-item-paragraph">Show that the equation of a normal to $E$ at $x=0$ is $y=2\sqrt{3}x-\frac{\sqrt{3}}{2}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii-3" data-list-item-path="item-iii-3"><div class="list-item-paragraph">The region $R$ is bounded by $E$, the normal in part <strong>(ii)</strong> and the $y$-axis. Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[5]</span></div></li></ol>

\[(x-a)^2 + 4y^2 = 4, \text{ where } 0 < a < 2.\]

Show that the equation of a normal to $E$ at $x=0$ is $y=2\sqrt{3}x-\frac{\sqrt{3}}{2}$.

Volume of Revolution about the y-axis

Graphs of Conic Sections

2025 NJC P2 Q2

Applications of Integration

<p>The curve $C$ has equation $x^3 + y^3 - 3xy = k$, where&nbsp;$k$ is a positive constant.<br>$C$ intersects the $x$-axis at the point $A$.</p>
<ol class="abc-list"><li data-list-item-id="item-6-a" data-list-item-path="item-6-a"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at $A$. Give your answer in the form $y = px + q$, where&nbsp;$p$ and $q$ are constants to be found in terms of $k$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-6-b" data-list-item-path="item-6-b"><div class="list-item-paragraph">There are two points on $C$ where the tangents are parallel to the $y$-axis. Given that&nbsp;$k = 3$, find the exact $y$-coordinates of these two points.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $x^3 + y^3 - 3xy = k$, where $k$ is a positive constant.
$C$ intersects the $x$-axis at the point $A$.

Find the equation of the tangent to $C$ at $A$. Give your answer in the form $y = px + q$, where $p$ and $q$ are constants to be found in terms of $k$.

There are two points on $C$ where the tangents are parallel to the $y$-axis. Given that $k = 3$, find the exact $y$-coordinates of these two points.

2025 EJC P2 Q1

<p>A curve $C$ has equation $x^3+y^3-5xy=0$ where&nbsp;$x&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="item-5-a" data-list-item-path="item-5-a"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-5-b" data-list-item-path="item-5-b"><div class="list-item-paragraph">Find the coordinates of the point on $C$ at which the normal is parallel to the $y$-axis.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-5-c" data-list-item-path="item-5-c"><div class="list-item-paragraph">Determine the nature of the stationary point.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $x^3+y^3-5xy=0$ where $x>0$.

Find the coordinates of the point on $C$ at which the normal is parallel to the $y$-axis.

Determine the nature of the stationary point.

Stationary Points and Their Nature

2025 DHS P2 Q3 

<p>The Folium of Descartes is a curve, defined by the equation $x^3+y^3=3axy$, where&nbsp;$a$ is a real constant. It is given that&nbsp;$a \neq 0$ for this question.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay - x^2}{y^2 - ax}$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>The point $ \left(\frac{3a}{2}, \frac{3a}{2} \right)$ lies on this curve. Show that the equation of the normal at this point on the curve is independent of the value of $a$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that the curve has a stationary point at $(pa, qa)$, where&nbsp;$p$ and $q$ are positive constants, find the exact values of $p$ and $q$. You need not determine the nature of this stationary point.<span class="inline-marks">[4]</span></div></li></ol>

The Folium of Descartes is a curve, defined by the equation $x^3+y^3=3axy$, where $a$ is a real constant. It is given that $a \neq 0$ for this question.

Given that the curve has a stationary point at $(pa, qa)$, where $p$ and $q$ are positive constants, find the exact values of $p$ and $q$. You need not determine the nature of this stationary point.

2025 ASRJC P1 Q4

<p>It is given that&nbsp;$y = x^{xy}$, where&nbsp;$x &gt; 0$, $y &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y^2 (1 + \ln x)}{1 - \ln y}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the coordinates of the point on the curve $y = x^{xy}$ whose tangent is parallel to the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

Show that \[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y^2 (1 + \ln x)}{1 - \ln y}\]

Hence find the coordinates of the point on the curve $y = x^{xy}$ whose tangent is parallel to the $y$-axis.

<p>A curve $C$ has equation ${{x}^{2}}-xy+{{y}^{2}}=12$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y-2x}{2y-x}$.</div></li><li><div class="list-item-paragraph">Find the equation of the tangents to the curve that are parallel to $y-$axis.</div></li></ol>

A curve $C$ has equation ${{x}^{2}}-xy+{{y}^{2}}=12$.

Worked Example 2

<p>Find the equations of the tangent and normal to the curve $2y{{x}^{3}}-xy+5{{y}^{2}}+9=0$ at the point $\left( -2,1 \right)$.</p>

Find the equations of the tangent and normal to the curve $2y{{x}^{3}}-xy+5{{y}^{2}}+9=0$ at the point $\left( -2,1 \right)$.

Worked Example 1

<p>A curve $C$ has equation $x^{2} - xy + y^{2} = 12$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y - 2x}{2y - x}$.</div></li><li><div class="list-item-paragraph">Find the equation of the tangents to the curve that are parallel to the $y$-axis.</div></li></ol>

A curve $C$ has equation $x^{2} - xy + y^{2} = 12$.

<p>Find the equation of the tangent and normal to the curve $2x^{2} - 3xy + y^{2} = 5$ at the point $(4, 3)$.</p>

Find the equation of the tangent and normal to the curve $2x^{2} - 3xy + y^{2} = 5$ at the point $(4, 3)$.

Math Question Bank