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PSLE

PSLE Foundation Mathematics

<p>A curve $C$ has equation $y=x^{-3}\ln x$.</p>
<ol class="abc-list"><li data-list-item-id="p1q7-q-b2-i1" data-list-item-path="p1q7-q-b2-i1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q7-q-b2-i2" data-list-item-path="p1q7-q-b2-i2"><div class="list-item-paragraph">Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has equation $y=x^{-3}\ln x$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.

Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.

Derivatives of Natural Logarithmic Functions

Product Rule

Area Bounded by Curves

Stationary Points and Their Nature

N2022 P1 Q7

Applications of Differentiation

Applications of Integration

Techniques of Differentiation

Techniques of Integration

<p>The function $\mathrm{f}$ is defined by</p>
<p>\[\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,\]</p>
<p>where&nbsp;$a,b,c$ are constants.</p>
<p>The curve $y=\mathrm{f}(x)$ passes through $(0,3)$ and has a stationary point at $\left(\frac\pi6,4\right)$.</p>
<ol class="i-list"><li data-list-item-id="5051fe58-1e08-48fd-acb0-ee1b28acfb85" data-list-item-path="5051fe58-1e08-48fd-acb0-ee1b28acfb85"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their exact values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="f3299ee3-8d23-4137-be13-da91485900f1" data-list-item-path="f3299ee3-8d23-4137-be13-da91485900f1"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)+c$, where&nbsp;$R&gt;0$ and $0&lt;\alpha&lt;\frac\pi2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="13c4684e-6601-4b6f-bd4a-f416e8b586bc" data-list-item-path="13c4684e-6601-4b6f-bd4a-f416e8b586bc"><div class="list-item-paragraph">Hence solve $\mathrm{f}(x)=3$, giving all answers in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,\]

The curve $y=\mathrm{f}(x)$ passes through $(0,3)$ and has a stationary point at $\left(\frac\pi6,4\right)$.

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

Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)+c$, where $R>0$ and $0<\alpha<\frac\pi2$.

Hence solve $\mathrm{f}(x)=3$, giving all answers in exact form.

A trigonometric curve determined by a stationary point

System of Linear Equations

<p>The curve $C$ has equation</p>
<p>\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]</p>
<p>The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.</p>
<ol class="i-list"><li data-list-item-id="b818afb7-41c3-4440-814f-cebde95ce4c2" data-list-item-path="b818afb7-41c3-4440-814f-cebde95ce4c2"><div class="list-item-paragraph">Using differentiation, find the exact coordinates of the point on $C$ where&nbsp;$x$ has its greatest value.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="35fabeb6-60ee-401c-9944-333d45b502e4" data-list-item-path="35fabeb6-60ee-401c-9944-333d45b502e4"><div class="list-item-paragraph">Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92" data-list-item-path="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92"><div class="list-item-paragraph">Find the area of the finite region enclosed by $C$ and the line $x=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="9a5b99e2-d202-43e6-9ba3-93603dc40d0d" data-list-item-path="9a5b99e2-d202-43e6-9ba3-93603dc40d0d"><div class="list-item-paragraph">Find the volume generated when this region is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]

The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.

Using differentiation, find the exact coordinates of the point on $C$ where $x$ has its greatest value.

Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.

Sketching Exponential Functions and Solving Equations Graphically

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

Area bounded by 2 curves

An enclosed region beside a sideways exponential curve

Graphing Techniques

<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 $P$.

Tangents and Normals at a Point

Quotient Rule

2025 JVSS PRELIMS P1 Q9

Differentiation

<p>A curve is given by the equation $y = \frac{3x + 10}{\sqrt{x + 1}}$.</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 $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence find the $x$-coordinate of the turning point of the curve.<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">Use calculus to find the value of the second derivative at the turning point and state the nature of this turning point.<span class="inline-marks">[3]</span></div></li></ol>

A curve is given by the equation $y = \frac{3x + 10}{\sqrt{x + 1}}$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence find the $x$-coordinate of the turning point of the curve.

Use calculus to find the value of the second derivative at the turning point and state the nature of this turning point.

Second and Higher Derivatives

Chain Rule

N2025 P1 Q4

<p>The equation of a curve is $y = \frac{x}{e^{2x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Given that&nbsp;$x$ is changing at a constant rate of $0.18$ units per second, find the rate of change of $y$ when&nbsp;$x = 1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Explain if the curve has more than one stationary point.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = \frac{x}{e^{2x}}$.

Connected Rates of Change

Derivatives of Exponential Functions

2024 GESS PRELIMS P2 Q8

<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>The curve $C$ has equation</p>
<p style="text-align: center">\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where}\hspace{0.5em} k &gt; 0, \ k \ne 2.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Without the use of a calculator, find the coordinates of any points of intersection of $C$ with the axes.<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">Write down the equations of the asymptotes of $C$ in terms of $k$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$k = 1$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Show that the $x$-coordinates of the turning points of $C$ satisfy the equation \[x^4 - 7x^2 + 12x - 2 = 0.\] Hence find these $x$-coordinates.<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">Sketch $C$, labelling all essential features.<span class="inline-marks">[3]</span></div></li></ol>

\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where } k > 0, \ k \ne 2.\]

Sketching Rational Functions

2024 EJC Promo Q12 

<p>A curve is such that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{9}{(2x-1)^2}$, where&nbsp;$x \neq \frac{1}{2}$. The point $(2, 12)$ lies on the curve.</p>
<ol class="i-list"><li data-list-item-id="item-1778561612436-sp6wzqj4x"><div class="list-item-paragraph">Find the equation of the curve.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1778561619148-fviss5i9e"><div class="list-item-paragraph">Find the value of $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ when&nbsp;$x = 0$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1778561616412-w3ixzyy43"><div class="list-item-paragraph">A student concludes from <strong>(ii)</strong> that since the value of $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} &gt; 0$ at $x = 0$, hence there is a minimum point at $x = 0$. State with reason and mathematical working whether the above conclusion is correct.<span class="inline-marks">[2]</span></div></li></ol>

A curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{9}{(2x-1)^2}$, where $x \neq \frac{1}{2}$. The point $(2, 12)$ lies on the curve.

Finding the Equation of the Curve

AMKSS 4AM 2022 Prelims P1 Q12

Integration

<ol class="abc-list" start="2"><li data-list-item-id="item-3b-list" data-list-item-path="item-3b-list"><ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-3b-list/item-i"><div class="list-item-content"><span class="list-item-text">The equation of a curve is $y = \frac{2x}{1 - x^2}$. Find an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-3b-list/item-ii"><div class="list-item-paragraph">Using the result in <strong>(i)</strong> above, explain, with working, why the curve $y = \frac{2x}{1 - x^2}$ has no turning points.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

HSS 4Express AM MYE 2021

<p>A curve has equation $y=x^n\sqrt{a^2-x^2}$, where&nbsp;$a$ and $n$ are integer constants such that&nbsp;$a \ge 2$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$, $a$ and $n$, expressing your answer as a single fraction in a fully factorised form.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the curve has the same gradient value at both&nbsp;$x=1$ and $x=-1$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find a condition that&nbsp;$n$ must satisfy.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the curve has a gradient of $0$ at $x=\sqrt{2}$, find the values of $a$ and $n$. Justify your answers clearly.</span><span class="inline-marks">[4]</span></div></li></ol>

A curve has equation $y=x^n\sqrt{a^2-x^2}$, where $a$ and $n$ are integer constants such that $a \ge 2$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$, $a$ and $n$, expressing your answer as a single fraction in a fully factorised form.

It is given that the curve has the same gradient value at both $x=1$ and $x=-1$.

Find a condition that $n$ must satisfy.

Given that the curve has a gradient of $0$ at $x=\sqrt{2}$, find the values of $a$ and $n$. Justify your answers clearly.

Derivative as the Gradient of a Curve

NJC IP4 Techniques of Differentiation Tutorial Q3

<p>The equation of a curve is $y=\frac{ \ln x}{x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773466434665-oon3h3sb1"><div class="list-item-paragraph">Find the coordinates of the stationary point of this curve. <br>Leave your answer in terms of e.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773466436825-f0ta69f8n"><div class="list-item-paragraph">Determine the nature of the stationary point.<span class="inline-marks">[3]</span></div></li></ol>

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

Find the coordinates of the stationary point of this curve. Leave your answer in terms of $e$.

Determine the nature of the stationary point.

2025 MSHS PRELIMS P2 Q2

<p>A curve is such that&nbsp;$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=3e^{2x}-e^{-x}$. The curve cuts the y-axis at $P (0,1)$ and has a gradient of $4$ at $P$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773464662797-ow3lqjvav"><div class="list-item-paragraph">Show that the curve has no stationary points.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1773464663273-5sy4ovusl"><div class="list-item-paragraph">Find the equation of the curve.<span class="inline-marks">[4]</span></div></li></ol>

A curve is such that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=3e^{2x}-e^{-x}$. The curve cuts the $y$-axis at $P(0,1)$ and has a gradient of 4 at $P$.

Show that the curve has no stationary points.

Finding Equation of Curve from Second Derivatives

2025 MSHS PRELIMS P1 Q9

<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 $\mathrm{f}(x)$.

Tangents and Normals Parallel to the Axes

2025 AHS PRELIMS P2 Q6

<p>A curve is defined by the equation $y = \ln \left( \frac{\mathrm{e}^x}{x^2 + 10} \right)$.</p>
<p>Show that the curve has no turning points for all real values of $x$.<span class="inline-marks">[5]</span></p>

A curve is defined by the equation $y=\ln\left(\dfrac{e^x}{x^2+10}\right)$.

Show that the curve has no turning points for all real values of $x$.

CGSS PRELIMS P1 Q6

<p>A function has the equation $f(x) = x - \frac{2}{3x - 1}$.</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 range of values of $x$ for which the function to be valid.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Explain why the graph of $f(x)$ does not have a stationary point.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence state whether $f(x)$ is an increasing or decreasing function.</div></li></ol>

A function has the equation $f(x) = x - \frac{2}{3x - 1}$.

Find the range of values of $x$ for which the function to be valid.

Explain why the graph of $f(x)$ does not have a stationary point.

Hence state whether $f(x)$ is an increasing or decreasing function.

Increasing and Decreasing Functions

<p>A function has the equation $f(x) = \frac{4 - 6x}{2x - 1}.$</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 range of values of $x$ for which the function to be valid.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Explain why the graph of $f(x)$ does not have a stationary point.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence state whether $f(x)$ is an increasing or decreasing function.</div></li></ol>

A function has the equation $f(x) = \frac{4 - 6x}{2x - 1}.$

<p>A function has the equation $f(x) = \frac{4}{\sqrt{2x - 3}}.$</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 range of values of $x$ for which the function to be valid.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Explain why the graph of $f(x)$ does not have a stationary point.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence state whether $f(x)$ is an increasing or decreasing function.</div></li></ol>

A function has the equation $f(x) = \frac{4}{\sqrt{2x - 3}}.$

<p>A curve has the equation $y = x^3 + 3x^2 - 9x + p$, where&nbsp;$p$ 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 $p$, the coordinates of the stationary point(s) on the curve.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Determine the nature of the stationary point(s).</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence find the possible value(s) of $p$, such that the curve has a stationary point on the $x$-axis.</div></li></ol>

A curve has the equation $y = x^3 + 3x^2 - 9x + p$, where $p$ is a constant.

Find, in terms of $p$, the coordinates of the stationary point(s) on the curve.

Determine the nature of the stationary point(s).

Hence find the possible value(s) of $p$, such that the curve has a stationary point on the $x$-axis.

<p>A curve has the equation $y = \frac{3}{\sqrt{x}} + x$ and has a turning point at point $P$.</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 coordinates of $P$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Determine the nature of the turning point.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the equations of tangents to the curve at $x = 1$ and $x = 9$, respectively.</div></li></ol>

A curve has the equation $y = \frac{3}{\sqrt{x}} + x$ and has a turning point at point $P$.

Determine the nature of the turning point.

Find the equations of tangents to the curve at $x = 1$ and $x = 9$, respectively.

Math Question Bank