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PSLE

PSLE Foundation Mathematics

<p>A curve $C$ has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]</p>
<p>The points $P$ and $Q$ on $C$ have parameters $p$ and $2p$ respectively, where&nbsp;$p&gt;0$. The tangents to $C$ at $P$ and $Q$ meet at $R$.</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph">Show that the tangent to $C$ at the point with parameter $t$ has equation \[3tx-2y=t^3+3t+4.\]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph">Find the coordinates of $R$ in terms of $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph"><p>Show that the area of triangle $PQR$ is $\dfrac{5}{3}p^5$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-d" data-list-item-path="q1-d"><div class="list-item-paragraph">Given that the area of triangle $PQR$ is $\dfrac{160}{3}$, find the value of $p$.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]

The points $P$ and $Q$ on $C$ have parameters $p$ and $2p$ respectively, where $p>0$. The tangents to $C$ at $P$ and $Q$ meet at $R$.

Show that the tangent to $C$ at the point with parameter $t$ has equation \[3tx-2y=t^3+3t+4.\]

Find the coordinates of $R$ in terms of $p$.

Given that the area of triangle $PQR$ is $\dfrac{160}{3}$, find the value of $p$.

Tangents and Normals to Parametric Curves

Differentiation of Parametric Equations

Area enclosed by two parametric tangents

Techniques of Differentiation

Applications of Differentiation

<p>A curve $C$ has parametric equations \[x=3t-\frac{3}{t},\ y=t+\frac{1}{t},\quad \text{where}\hspace{0.5em} t&lt;0.\]</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-0" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-0"><div class="list-item-paragraph"><p>Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{t^2-1}{3(t^2+1)}$.</p></div><div class="list-item-after-content">Hence, show the equation of the tangent to $C$ at the point $P$ with parameter $p$ is</div><div class="list-item-after-content"><p>\[3\left(p^2+1\right)y=\left(p^2-1\right)x+12p.\]</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-1" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-1"><div class="list-item-paragraph">Show that the cartesian equation of $C$ may be written as&nbsp;$9y^2-x^2=36$, stating the restriction on the values of $y$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-2" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-2"><div class="list-item-paragraph">Sketch the graph of $C$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-3" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-3"><div class="list-item-paragraph"><p>The region bounded by $C$, the $y$-axis, the $x$-axis and the line $x=4.5$ is rotated through $2\pi$ radians about the $y$-axis. Find the volume of revolution of the solid formed, giving your answer correct to $2$ decimal places.</p><span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has parametric equations \[x=3t-\frac{3}{t},\ y=t+\frac{1}{t},\quad \text{where } t<0.\]

Show that the cartesian equation of $C$ may be written as $9y^2-x^2=36$, stating the restriction on the values of $y$.

Sketch the graph of $C$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes.

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2025 RVHS Promo Q11

Graphing Techniques

Applications of Integration

<p>A curve $C$ is defined parametrically by $x=6t,\ y=t^2$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-0" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-0"><div class="list-item-paragraph">Show that the equation of the normal at the point $P(6p, p^2)$ is $p^3+(18-y)p-3x=0$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-1" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-1"><div class="list-item-paragraph">The normal to the curve at the point $P$ meets the curve again at the point $Q(-54,81)$. Find the possible values of $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-2" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-2"><div class="list-item-paragraph">Sketch the graph of $C$, indicating clearly the coordinates of $Q$ and the possible points of $P$. Sketch also the normals at each of these points $P$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ is defined parametrically by $x=6t,\ y=t^2$.

Show that the equation of the normal at the point $P(6p, p^2)$ is $p^3+(18-y)p-3x=0$.

The normal to the curve at the point $P$ meets the curve again at the point $Q(-54,81)$. Find the possible values of $p$.

Sketch the graph of $C$, indicating clearly the coordinates of $Q$ and the possible points of $P$. Sketch also the normals at each of these points $P$.

2025 EJC Promo Q6

<p>A curve $C$ has parametric equations \[x = \sin 2\theta, \quad y = \cos^2 \theta \quad \text{for}\hspace{0.5em} 0 \le \theta \le \pi.\]</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta$, expressing your answer in a single trigonometric function.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find the rate of decrease of the gradient of $C$ at $\theta = \frac{\pi}{12}$ if&nbsp;$\theta$ is increasing at a constant rate of $\frac{\pi}{10}\text{ rad/s}$. Leave your answer in an exact form.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has parametric equations \[x = \sin 2\theta, \quad y = \cos^2 \theta \quad \text{for } 0 \le \theta \le \pi.\]

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta$, expressing your answer in a single trigonometric function.

Find the rate of decrease of the gradient of $C$ at $\theta = \frac{\pi}{12}$ if $\theta$ is increasing at a constant rate of $\frac{\pi}{10}\text{ rad/s}$. Leave your answer in an exact form.

Connected Rates of Change

2026 TMJC WA2 Q3

<p>The asteroid, a curve $C$ which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where&nbsp;$0 \le t \le \frac{1}{2}\pi$ and $a$ is a positive constant.</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 the equation of the tangent to $C$ at the point $P$ with parameter $p$.<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"><div class="list-item-paragraph">The tangent at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$. Show that the length $AB$ depends only on $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Find a cartesian equation of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph">The region bounded by $C$ and the $x$- and $y$-axes is rotated through $360^\circ$ about the $y$-axis. Find the exact value of the volume of revolution of the solid formed.<span class="inline-marks">[4]</span></div></li></ol>

The asteroid, a curve $C$ which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where $0 \le t \le \frac{1}{2}\pi$ and $a$ is a positive constant.

Find the equation of the tangent to $C$ at the point $P$ with parameter $p$.

The tangent at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$. Show that the length $AB$ depends only on $a$.

The region bounded by $C$ and the $x$- and $y$-axes is rotated through $360^\circ$ about the $y$-axis. Find the exact value of the volume of revolution of the solid formed.

Derivatives of Trigonometric Functions

Chain Rule

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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 Parallel to the Axes

Product Rule

Parametric Equations

N2025 P1 Q11

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">\[x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for}\hspace{0.5em} \lambda &gt; 0.\]</p>
<p>The point $P$ is a variable point on $C$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">With reference to the origin $O$, $OP$ forms the diagonal of the rectangle $OQPR$, where vertices $Q$ and $R$ lie on the $x$- and $y$-axis respectively. Using differentiation, find the value of $\lambda$ which maximises the area of rectangle $OQPR$. You need to show that your answer gives a maximum.<span class="inline-marks">[5]</span></div></li></ol>
<p>When the area of rectangle $OQPR$ is at its maximum, the rate of change of the $x$-coordinate of the point $P$ is $1$ unit per second.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence determine the rate of change of the $y$-coordinate of the point $P$ at this instant.<span class="inline-marks">[4]</span></div></li></ol>

\[x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for } \lambda > 0.\]

The point $P$ is a variable point on $C$.

When the area of rectangle $OQPR$ is at its maximum, the rate of change of the $x$-coordinate of the point $P$ is $1$ unit per second.

Quotient Rule

Problems Involving Maximisation and Minimisation

2024 RI Promo Q4

<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>A curve $C$ has parametric equations</p>
<p style="text-align: center">\[x = 1 - 2 \cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for}\hspace{0.5em} 0 \le \theta \le \frac{3\pi}{8}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of the point where&nbsp;$C$ cuts the $y$-axis, giving your answer in exact form.<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">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2} \cot 2\theta$. Find the coordinates of the points and the gradient of $C$ where&nbsp;$\theta = \frac{\pi}{4}$. What can be said about the tangent to $C$ as&nbsp;$\theta \to 0$?<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence sketch $C$, labelling the exact coordinates of axial intercepts and endpoints, and showing clearly the features of the curve at the points where&nbsp;$\theta = 0$ and $\theta = \frac{\pi}{4}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find a cartesian equation of $C$.<span class="inline-marks">[2]</span></div></li></ol>

\[x = 1 - 2 \cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for } 0 \le \theta \le \frac{3\pi}{8}.\]

Find the coordinates of the point where $C$ cuts the $y$-axis, giving your answer in exact form.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2} \cot 2\theta$. Find the coordinates of the points and the gradient of $C$ where $\theta = \frac{\pi}{4}$. What can be said about the tangent to $C$ as $\theta \to 0$?

Hence sketch $C$, labelling the exact coordinates of axial intercepts and endpoints, and showing clearly the features of the curve at the points where $\theta = 0$ and $\theta = \frac{\pi}{4}$.

Derivative as the Gradient of a Curve

2024 ASRJC Promo Q8

<p>A curve $C$ has parametric equations $x = t^2 + t$, $y = t^2 - t$, $t \in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the graph of $C$, indicating the coordinates of the points where the curve crosses either axis.<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 coordinates of the point on $C$ where the tangent to $C$ is parallel to the line $5y = 4x - 20$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point $(0, 2)$ and show algebraically that this tangent does not cut $C$ again.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has parametric equations $x = t^2 + t$, $y = t^2 - t$, $t \in \mathbb{R}$.

Sketch the graph of $C$, indicating the coordinates of the points where the curve crosses either axis.

Find the coordinates of the point on $C$ where the tangent to $C$ is parallel to the line $5y = 4x - 20$.

Find the equation of the tangent to $C$ at the point $(0, 2)$ and show algebraically that this tangent does not cut $C$ again.

2024 JPJC Promo Q10

<p>Given that&nbsp;$x=\ln \left( \cot \frac{1}{2}\theta  \right)-\cos \theta $, $y=-\sin \theta $ , show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\tan \theta $ .</p>

Given that $x=\ln \left( \cot \frac{1}{2}\theta  \right)-\cos \theta $, $y=-\sin \theta $ , show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\tan \theta $ .

Derivatives of Natural Logarithmic Functions

Derivatives of Reciprocal Trigonometric Functions

<p>A curve $C$ has a parametric equations</p>
<p style="text-align: center">$x=5{{t}^{2}}$, $y=2t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $t$ at the point on $C$ where the tangent has gradient $0.8$.</div></li><li><div class="list-item-paragraph">The normal at the point $P\left( 5{{p}^{2}},2p \right)$ on $C$ meets the $x-$axis at point $Q$. Find the Cartesian equation of the locus of the mid-point of $PQ$ as $p$ varies.</div></li></ol>

Locus of a Midpoint

Worked Example 3

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x = 5t^{2}$, $y = 2t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $t$ at the point on $C$ where the tangent has gradient $0.8$.</div></li><li><div class="list-item-paragraph">The normal at the point $P(5p^{2}, 2p)$ on $C$ meets the $x$-axis at point $Q$. Find the Cartesian equation of the locus of the midpoint of $PQ$ as&nbsp;$p$ varies.</div></li></ol>

<p>A curve has parametric equation $x=2t-1$ , $y=\frac{1}{{{t}^{2}}+1}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Prove that the equation of the tangent at the point with parameter $t$ is $(t^{2} + 1)^{2}y + tx = 3t^{2} - t + 1$.</div></li><li><div class="list-item-paragraph">The tangent at the point where&nbsp;$t = 3$ meets the curve again at the point where&nbsp;$t = q$. Find the value of $q$ and hence find the coordinates of the point of intersection.</div></li></ol>

A curve has parametric equation $x=2t-1$ , $y=\frac{1}{{{t}^{2}}+1}$.

<p>A curve is defined by the parametric equations $x = 2t - 1$, $y = t^{2} + 2$. Show that the equation of the tangent to the curve at the point $P(2p - 1, p^{2} + 2)$ is $y = px - p^{2} + p + 2$. Hence, find the equations of the tangents that pass through the point $(-1, -2)$.</p>

A curve is defined by the parametric equations $x = 2t - 1$, $y = t^{2} + 2$. Show that the equation of the tangent to the curve at the point $P(2p - 1, p^{2} + 2)$ is $y = px - p^{2} + p + 2$. Hence, find the equations of the tangents that pass through the point $(-1, -2)$.

<p>A curve $C$ is given parametrically by $x = \frac{1}{1 - t^{2}}$, $y = \frac{t}{1 - t^{2}}$. Show that the equation of the tangent at the point $\left(-\frac{1}{3}, -\frac{2}{3}\right)$ is $4y - 5x = -1$.</p>

A curve $C$ is given parametrically by $x = \frac{1}{1 - t^{2}}$, $y = \frac{t}{1 - t^{2}}$. Show that the equation of the tangent at the point $\left(-\frac{1}{3}, -\frac{2}{3}\right)$ is $4y - 5x = -1$.

<p>The parametric equations of a curve are $x = \frac{t}{1 - t^{3}}$ and $y = \frac{1}{1 - t^{3}}$, where&nbsp;$t \ne 1$. Find the equation of the normal to the curve at the point where&nbsp;$t = 2$.</p>

The parametric equations of a curve are $x = \frac{t}{1 - t^{3}}$ and $y = \frac{1}{1 - t^{3}}$, where $t \ne 1$. Find the equation of the normal to the curve at the point where $t = 2$.

<p>A curve is defined parametrically by the equations $x = \frac{2}{t}$ and $y = 1 - 2t$, where&nbsp;$t$ is a real parameter. Find the equation of the tangent to the curve at $t = -2$ and determine if it cuts the curve again.</p>

A curve is defined parametrically by the equations $x = \frac{2}{t}$ and $y = 1 - 2t$, where $t$ is a real parameter. Find the equation of the tangent to the curve at $t = -2$ and determine if it cuts the curve again.

<p>Given that&nbsp;$x = e^{t}\cos t$ and $y = e^{t}\sin t$, show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \tan \left(t + \frac{\pi}{4}\right)$.</p>

Given that $x = e^{t}\cos t$ and $y = e^{t}\sin t$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \tan \left(t + \frac{\pi}{4}\right)$.

<p>A curve has parametric equations given by $x = \theta - \sin \theta$ and $y = 1 - \cos \theta$. Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \cot \frac{\theta}{2}$.</p>

A curve has parametric equations given by $x = \theta - \sin \theta$ and $y = 1 - \cos \theta$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \cot \frac{\theta}{2}$.

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