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H2 Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

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<p>Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations</p>
<p style="text-align: center">$x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),$</p>
<p>for&nbsp;$-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$, where&nbsp;$x$ and $y$ are measured in metres. The $x$-axis models the horizontal ground.</p>
<p>The maximum point on the curve is at $A$, and the line $AB$ is vertical. Points $C$ and $D$ are where the tangent to the curve is parallel to the $y$-axis. Point $E$ is the intersection of the lines $AB$ and $CD$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><ol class="i-list"><li data-list-item-id="imported-item-block-3-item-1-sub-0" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-0"><div class="list-item-paragraph">Find the values of $\theta$ at points $C$ and $D$, giving your answers correct to $2$ significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1-sub-1" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-1"><div class="list-item-paragraph">Hence find the greatest width, $CD$, of the loop.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>The $x$-coordinate of point $A$ is $28$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-5-item-0-sub-0" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-0"><div class="list-item-paragraph">Find the value of $\theta$ at point $A$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-1" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-1"><div class="list-item-paragraph">Find the $2$ possible values of $\theta$ at point $B$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-2" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-2"><div class="list-item-paragraph">Hence find the distance $AB$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>Engineers know that, for safety reasons, the distance $AE$ should be within $10\%$ of the distance $\frac{1}{2}CD$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="imported-item-block-7-item-0" data-list-item-path="imported-item-block-7-item-0"><div class="list-item-paragraph">Determine whether the design of this section of track satisfies this condition.<span class="inline-marks">[2]</span></div></li></ol>

Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations

$x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),$

for $-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$, where $x$ and $y$ are measured in metres. The $x$-axis models the horizontal ground.

The maximum point on the curve is at $A$, and the line $AB$ is vertical. Points $C$ and $D$ are where the tangent to the curve is parallel to the $y$-axis. Point $E$ is the intersection of the lines $AB$ and $CD$.

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

The $x$-coordinate of point $A$ is $28$.

Engineers know that, for safety reasons, the distance $AE$ should be within $10\%$ of the distance $\frac{1}{2}CD$.

Determine whether the design of this section of track satisfies this condition.

Tangents and Normals to Parametric Curves

Derivatives of Trigonometric Functions

Differentiation of Parametric Equations

Tangents and Normals Parallel to the Axes

Product Rule

Parametric Equations

N2025 P1 Q11

Applications of Differentiation

Techniques of Differentiation

Graphing Techniques

<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 $x > 0$, $y > 0$.

Implicit Differentiation

Logarithmic Differentiation

Tangents and Normals to Implicit Curves

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 parametric equations given by $x = 2t^2 + t$, $y = t^3 - 3t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the tangent on the curve which is parallel to the $y$-axis.</div></li><li><div class="list-item-paragraph">Find the coordinates of the point when the tangent at $t = -\frac{1}{2}$ meets the curve again.</div></li></ol>

A curve has parametric equations given by $x = 2t^2 + t$, $y = t^3 - 3t$.

<p>The curve $y = \mathrm{f}(x)$ is such that&nbsp;$\mathrm{f}''(x) = 2e^x + e^{-2x}$.</p>
<p>The $y$-axis is a normal to the curve at the point $Q$, where&nbsp;$y = 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$Q$ is a stationary point.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Hence, find an expression for&nbsp;$\mathrm{f}(x)$.</span><span class="inline-marks">[6]</span></div></li></ol>

The curve $y=\mathrm{f}(x)$ is such that $\mathrm{f}''(x)=2e^x+e^{-2x}$.

The $y$-axis is a normal to the curve at the point $Q$, where $y=1$.

Hence, find an expression for $\mathrm{f}(x)$.

Finding Equation of Curve from Second Derivatives

Stationary Points and Their Nature

2025 AHS PRELIMS P2 Q6

Differentiation

Integration

<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>A curve has the equation $y = \sqrt{ax^2 - 3x + c}$, where&nbsp;$a$ and $c$ are constants.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Given that the curve passes through $(0, \sqrt{6})$ and that the tangent to the curve at $x = \frac{3}{2}$ is parallel to the $x$-axis, find the values of $a$ and $c$.</div></li></ol>
<p>Hence find</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the coordinates of the points on the curve where the gradient of the curve is $\frac{1}{2}$, and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">the range of values of $x$ where the gradient of the curve is positive.</div></li></ol>

A curve has the equation $y = \sqrt{ax^2 - 3x + c}$, where $a$ and $c$ are constants.

Given that the curve passes through $(0, \sqrt{6})$ and that the tangent to the curve at $x = \frac{3}{2}$ is parallel to the $x$-axis, find the values of $a$ and $c$.

the coordinates of the points on the curve where the gradient of the curve is $\frac{1}{2}$, and

the range of values of $x$ where the gradient of the curve is positive.

Derivative as the Gradient of a Curve

Increasing and Decreasing Functions

<p>A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where&nbsp;$k$ is a constant. Given that there exists at least one point on the curve at which the tangent to the curve is perpendicular to the $y$-axis, find</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the range of values of $k$, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the $x$-coordinate(s), in terms of $k$, of such point(s) on the curve.</div></li></ol>

A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where $k$ is a constant. Given that there exists at least one point on the curve at which the tangent to the curve is perpendicular to the $y$-axis, find

the $x$-coordinate(s), in terms of $k$, of such point(s) on the curve.

Parameter Conditions for Stationary Points

<p>A curve has the equation $y = 2x^3 - 3x^2 - 36x + m$, where&nbsp;$m$ 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">Find, in terms of $m$, the coordinates of the point(s) on the curve at which the tangent to the curve is perpendicular to the $y$-axis.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence, or otherwise, find the possible value(s) of $m$, such that the $x$-axis is a tangent to the curve.</div></li></ol>

A curve has the equation $y = 2x^3 - 3x^2 - 36x + m$, where $m$ is a constant.

Find, in terms of $m$, the coordinates of the point(s) on the curve at which the tangent to the curve is perpendicular to the $y$-axis.

Hence, or otherwise, find the possible value(s) of $m$, such that the $x$-axis is a tangent to the curve.

<p>A curve has the equation $y = \frac{ax^2 + x}{2x - 1},$ where&nbsp;$a$ is a constant and $a\notin\{0,-2\}$. Find, respectively, the range of values of $a$ such that</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the gradient of the curve is always non-positive, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">there exists at least one point on the curve at which the tangent to the curve is parallel to the $x$-axis.</div></li></ol>

A curve has the equation $y = \frac{ax^2 + x}{2x - 1},$ where $a$ is a constant and $a\notin\{0,-2\}$. Find, respectively, the range of values of $a$ such that

the gradient of the curve is always non-positive, and

there exists at least one point on the curve at which the tangent to the curve is parallel to the $x$-axis.

<p>Find the coordinates of the point(s) on the curve $y = \sqrt{x} + \frac{1}{\sqrt{x}}$ where the tangent to the curve is perpendicular to the $y$-axis.</p>

Find the coordinates of the point(s) on the curve $y = \sqrt{x} + \frac{1}{\sqrt{x}}$ where the tangent to the curve is perpendicular to the $y$-axis.

<p>Find the coordinates of the point(s) on the curve $y = x^{2} + 6x - 7$ where the tangent to the curve is parallel to the $x$-axis.</p>

Find the coordinates of the point(s) on the curve $y = x^{2} + 6x - 7$ where the tangent to the curve is parallel to the $x$-axis.

<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>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.

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

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 $C$ 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.

Tangents and Normals at a Point

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.

2025 DHS P2 Q3 

<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>There are two points on the curve $y=\sqrt[3]{4x^3+9x^2-12x}$ at which the tangents are perpendicular to the line x=7.</p>
<p>Calculate the $x$–coordinates of these points.</p>

There are two points on the curve $y=\sqrt[3]{4x^3+9x^2-12x}$ at which the tangents are perpendicular to the line x=7.

Calculate the $x$–coordinates of these points.

<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 ${{x}^{2}}-xy+{{y}^{2}}=12$.

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