A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<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 $p$.

Differentiation of Parametric Equations

Area enclosed by two parametric tangents

Techniques of Differentiation

<p>Find the derivative of each of the following w.r.t $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>$y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} $</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>$y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) $</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>$ y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }$</p></div></li></ol>

Find the derivative of each of the following w.r.t $x$:

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

Differentiation

<p>The diagram shows part of the graph $y = x^{\cos 2x}$, for&nbsp;$0 \le x \le 5$, which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by $x$ units and its vertical distance travelled is denoted by $y$ units.</p>
<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/1783573222728-q8fnjduy6xq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzMyMjI3MjgtcThmbmpkdXk2eHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNjgzNSwiZXhwIjoxNzg5NjI3MTM1fQ.bMkkYkpLHo7kjVcgg4BqdxkgSMVkek90YZKjoSdveok" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<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;$\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]$.<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">At the point on the graph where&nbsp;$x = \pi$, find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of $8$ units per hour.<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">Find the acute angle that the tangent to the graph where&nbsp;$x = \pi$ makes with the horizontal.<span class="inline-marks">[1]</span></div></li></ol>

The diagram shows part of the graph $y = x^{\cos 2x}$, for $0 \le x \le 5$, which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by $x$ units and its vertical distance travelled is denoted by $y$ units.

2024 ACJC P2 Q1

Applications of Differentiation

<p>Evaluate the following: </p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(4^{2x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\ln(2x)}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)$</div></li></ol>

<p>Given that</p>
<p style="text-align: center">\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]</p>
<p>where&nbsp;$A$ and $B$ are constants with $x &gt; -1$.</p>
<p>Show that&nbsp;$(x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0$.<span class="inline-marks">[5]</span></p>

\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]

where $A$ and $B$ are constants with $x > -1$.

Show that $(x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0$.

2018 AJC C1BT1 Q3

<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\tan x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(3^{\ln x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)$</div></li></ol>

<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>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\cos^{-1}\left(3x^2\right)$,<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">$\ln\left(\frac{\sqrt{2x+1}}{x^3}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

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

2024 JPJC 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>

Techniques of Differentiation - General

Binomial Theorem

<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 $ .

<p>Given the following relations between $x$ and $y$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{y}^{x}}=y+x$ </div></li><li><div class="list-item-paragraph">$y={{x}^{\text{sec}\hspace{0.5em}x}}$ </div></li><li><div class="list-item-paragraph">$y={{x}^{{{x}^{2}}}}$ </div></li></ol>

Given the following relations between $x$ and $y$, find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

<p>Find the $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the following equations in terms of $x$ and $y$:</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{x}^{2}}+{{y}^{2}}=25$</div></li><li><div class="list-item-paragraph">${{x}^{3}}+4{{y}^{2}}=x+2$</div></li><li><div class="list-item-paragraph">$\cos x+\tan y={{y}^{3}}$				</div></li></ol>

Find the $\frac{\mathrm{d}y}{\mathrm{d}x}$ of the following equations in terms of $x$ and $y$:

<p style="text-align: center">$\frac{\mathrm{d}}{\mathrm{d}x}\left( {{e}^{x}} \right)={{e}^{x}}$</p>
<p style="text-align: center">$\frac{\mathrm{d}}{\mathrm{d}x}\left( {{a}^{x}} \right)={{a}^{x}}\ln a$, where&nbsp;$a&gt;0$ 	</p>
<p style="text-align: center">$\frac{\mathrm{d}}{\mathrm{d}x}\left( \ln x \right)=\frac{1}{x}$</p>
<p style="text-align: center">$\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\log }_{a}}x \right)=\frac{1}{x\ln a}$, where&nbsp;$a&gt;0$</p>

Exponential and Logarithmic Derivatives

<p>The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units$^3$, with variables $x$ and $y$.</p>
<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/1769312378450-pfu3lpl2glp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMTIzNzg0NTAtcGZ1M2xwbDJnbHAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyNjgzNSwiZXhwIjoxNzg5NjI3MTM1fQ.DnNkfG2HKbpjS7gNQ8HqWSjKMtlnZn6K2wXql7dav3w" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The top surface of the prism, $ABCD$, is an isosceles trapezoid with $AB$ of length $5x$ units, $DC$ of length $3x$ units, $AD = BC$ and $\angle ABC = \angle BAD = 60{}^\circ $. The rectangular sides $ABFE$ and $BCGF$ are perpendicular to both the top surface $ABCD$ and the bottom surface $EFGH$, with $AE = BF = DH = CG = y$ units.</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 the total external surface area $A$ of the trapezoidal prism is given by $A = 8x^2\sqrt{3} + \frac{12k}{x}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Using differentiation, find the value of $x$ in terms of $k$ at which $A$ is a minimum.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">It is given instead that the volume of the prism is $1000$ units$^3$ and its external surface area is $800$ units$^2$. Find the two possible values of $x$.<span class="inline-marks">[2]</span></div></li></ol>

The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units$^3$, with variables $x$ and $y$.

The top surface of the prism, $ABCD$, is an isosceles trapezoid with $AB$ of length $5x$ units, $DC$ of length $3x$ units, $AD = BC$ and $\angle ABC = \angle BAD = 60{}^\circ $. The rectangular sides $ABFE$ and $BCGF$ are perpendicular to both the top surface $ABCD$ and the bottom surface $EFGH$, with $AE = BF = DH = CG = y$ units.

Show that the total external surface area $A$ of the trapezoidal prism is given by $A = 8x^2\sqrt{3} + \frac{12k}{x}$.

Using differentiation, find the value of $x$ in terms of $k$ at which $A$ is a minimum.

It is given instead that the volume of the prism is $1000$ units$^3$ and its external surface area is $800$ units$^2$. Find the two possible values of $x$.

2025 VJC P2 Q2

<p>It is given that&nbsp;$2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the substitution $y = \ln \left( \frac{u}{x} \right)$ to show that the differential equation can be reduced to $\frac{\mathrm{d}u}{\mathrm{d}x} = \mathrm{f}(u)$, where the function $\mathrm{f}(u)$ is to be found.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$y$ has a minimum value at $x = 3$, solve the differential equation $2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y$, to find the particular solution for&nbsp;$y$ in terms of $x$.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of this particular solution.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y$.

Use the substitution $y = \ln \left( \frac{u}{x} \right)$ to show that the differential equation can be reduced to $\frac{\mathrm{d}u}{\mathrm{d}x} = \mathrm{f}(u)$, where the function $\mathrm{f}(u)$ is to be found.

Given that $y$ has a minimum value at $x = 3$, solve the differential equation $2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y$, to find the particular solution for $y$ in terms of $x$.

Sketch the graph of this particular solution.

2025 VJC P1 Q11

Differential Equations

<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 $x = 0$.

2025 TMJC P1 Q2

Graphing Techniques

<p>It is given that&nbsp;$y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.</span><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">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for&nbsp;$y$, up to and including the term in $x^2$, giving the coefficients in exact form.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Deduce the Maclaurin series for&nbsp;$\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.

Show that $2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.

Deduce the Maclaurin series for $\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.

2025 TJC P2 Q1

Maclaurin Series and Power Series

<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>A closed cylinder is designed to have a fixed external surface area of $p$ cm$^2$ such that its volume is maximum. Find the radius of the cylinder in terms of $p$.<span class="inline-marks">[5]</span></p>

A closed cylinder is designed to have a fixed external surface area of $p$ cm$^2$ such that its volume is maximum. Find the radius of the cylinder in terms of $p$.

2025 SAJC P1 Q2

<p>A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where&nbsp;$a$ and $b$ are real constants such that&nbsp;$a &gt; 0$, $a \neq \frac{1}{2}b$ and $x \neq -2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$C$ has stationary points, use differentiation to find the relationship between $a$ and $b$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is now given that&nbsp;$a = 1$ and $b = 3$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Prove algebraically that&nbsp;$y$ cannot lie between $-1$ and $3$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points.</span><span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where $a$ and $b$ are real constants such that $a > 0$, $a \neq \frac{1}{2}b$ and $x \neq -2$.

Given that $C$ has stationary points, use differentiation to find the relationship between $a$ and $b$.

It is now given that $a = 1$ and $b = 3$.

Prove algebraically that $y$ cannot lie between $-1$ and $3$.

Sketch $C$, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points.

Math Question Bank