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<p>A curve $y=\dfrac{1+3\ln x}{x^2}$ has a minimum 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>

A curve $y=\dfrac{1+3\ln x}{x^2}$ has a minimum point at $P$.

Applications of Differentiation — General

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

2025 JVSS PRELIMS P1 Q9

Differentiation

Exponential and Logarithmic Functions

<ol class="abc-list"><li data-list-item-id="cgs-p2-q1-q-0-content-items-0" data-list-item-path="cgs-p2-q1-q-0-content-items-0"><div class="list-item-paragraph">Show that the equation $3e^{2x} + 7e^x = 6$ has only one solution and find its value correct to 2 significant figures.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cgs-p2-q1-q-0-content-items-1" data-list-item-path="cgs-p2-q1-q-0-content-items-1"><div class="list-item-paragraph">Factorise $(3x + 1)^3 - 8$ completely.<span class="inline-marks">[2]</span></div></li></ol>

Show that the equation $3e^{2x} + 7e^x = 6$ has only one solution and find its value correct to 2 significant figures.

Solving Exponential Equations Using Substitution

Cubic Expressions and Equations

2025 CGS PRELIMS P2 Q1

Polynomials, Cubic Equations and Partial Fractions

A drink cools according to T=22+58e-0.18t, where T is in degrees Celsius and t is in minutes. Find when T=35.

A drink cools according to $T=22+58e^{-0.18t}$, where $T$ is in degrees Celsius and $t$ is in minutes. Find when $T=35$.

Cooling model

Logarithmic equation with a domain check

Natural logarithm equation

Exponential equation by substitution

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Solve the equation $15(16^x) - 4^x = 6$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Solve the equation $\lg(2y) + 2\ln(y) = 2$.<span class="inline-marks">[4]</span></div></li></ol>

2024 GESS PRELIMS P2 Q5

<p>Show that the equation $2(\mathrm{e}^x - 3) = \mathrm{e}^{\frac{1}{2}x}$ has only one solution and find this value correct to $3$ significant figures.<span class="inline-marks">[4]</span></p>

Show that the equation $2(\mathrm{e}^x - 3) = \mathrm{e}^{\frac{1}{2}x}$ has only one solution and find this value correct to $3$ significant figures.

Quadratic Equations and Inequalities — General

Conversion between Exponential and Logarithm Form

2024 FMSS PRELIMS P2 Q1

Quadratic Functions, Equations and Inequalities

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that the solution of the equation $2^{3x+4} \times 5^{2x-1} = 16^x \times 5^{3x}$ is $\lg \frac{16}{5}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Express $2\log_2 x - \log_2(x - 4) = 3$ as a quadratic equation in $x$ and explain why there are no real solutions.<span class="inline-marks">[5]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q8

<p>Solve the equation $5^{x + 1} = 5^{2x - 1} - 30 $.<span class="inline-marks">[5]</span></p>

Solve the equation $5^{x + 1} = 5^{2x - 1} - 30 $.

Equality of Indices

2024 CGSS P1 Q4

<p>Solve the equation $2 \log_{3}{p} + \frac{1}{\log_{27}{3}} - 1 = \log_{3}{(8p + 1)} $.<span class="inline-marks">[5]</span></p>

Solve the equation $2 \log_{3}{p} + \frac{1}{\log_{27}{3}} - 1 = \log_{3}{(8p + 1)} $.

2024 CGSS P1 Q3

<p>Without using a calculator, evaluate $\frac{\log_5 9 + 2\log_5 6 - 4\log_5 3}{\log_{25} 4}$.<span class="inline-marks">[4]</span></p>

Without using a calculator, evaluate $\frac{\log_5 9 + 2\log_5 6 - 4\log_5 3}{\log_{25} 4}$.

Laws of Logarithms

2024 CCHY P1 Q7

<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that&nbsp;$3x^2 - 5x + 21$ is always positive for all real values of $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The curve $y = ax^n$, where&nbsp;$a$ and $n$ are constants, passes through $(2, 48)$, $(3, 108)$ and $(k, 192)$. Find the values of $a$, $n$ and $k$ where&nbsp;$k &gt; 0$.<span class="inline-marks">[3]</span></div></li></ol>

Nature of Roots of Quadratic Equations

2024 CCHY P1 Q3

<p>Let&nbsp;$f(x) = A\mathrm{e}^{kx} + 3$. Part of the graph of $f$ is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="262" style="width:300px;height:262px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The $y$-intercept is at $ (0, 13) $.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$A = 10$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$f(15) = 3.49$ (correct to $3$ significant figures), find the value of $k$.<span class="inline-marks">[3]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Using your value of $k$, find $f'(x)$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence, explain why $f$ is a decreasing function.</div></li><li><div class="list-item-paragraph">Write down the equation of the horizontal asymptote of the graph $f$.</div></li></ol></li></ol>
<p>Let&nbsp;$g(x) = -x^2 + 12x - 24$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the area enclosed by the graphs of $f$ and $g$.<span class="inline-marks">[6]</span></div></li></ol>

Let $f(x) = A\mathrm{e}^{kx} + 3$. Part of the graph of $f$ is shown below.

Definite Integrals and Area under Graph — General

Integration

Derivatives of Exponential, Logarithmic and Trigonometric Functions

<p>Solve the equation $9^{x-1} = \left( \frac{1}{3} \right)^{2x}$.<span class="inline-marks">[4]</span></p>

Solve the equation $9^{x-1} = \left( \frac{1}{3} \right)^{2x}$.

<p>If&nbsp;$\log_{a}{2} = x$ and $\log_{a}{5} = y$, find in terms of $x$ and $y$, expressions for<span class="inline-marks">[4]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\log_{2}{5}$;</div></li><li><div class="list-item-paragraph">$\log_{a}{20}$.</div></li></ol>

If $\log_{a}{2} = x$ and $\log_{a}{5} = y$, find in terms of $x$ and $y$, expressions for

<p>Let&nbsp;$\log_{10}{P} = x$, $\log_{10}{Q} = y$ and $\log_{10}{R} = z$. Express $\log_{10} \left( { \frac{P}{QR^3}} \right)^2$ in terms of $x, y$ and $z$.<span class="inline-marks">[4]</span></p>

Let $\log_{10}{P} = x$, $\log_{10}{Q} = y$ and $\log_{10}{R} = z$. Express $\log_{10} \left( { \frac{P}{QR^3}} \right)^2$ in terms of $x, y$ and $z$.

<p>Solve the equation $\log_{9}{81} + \log_{9}{\frac{1}{9}} + \log_{9}{3} = \log_{9}{x}$.<span class="inline-marks">[4]</span></p>

Solve the equation $\log_{9}{81} + \log_{9}{\frac{1}{9}} + \log_{9}{3} = \log_{9}{x}$.

<p>Solve the equation $ \log_{27}{x} = 1 - \log_{27}{(x - 0.4)}$.<span class="inline-marks">[6]</span></p>

Solve the equation $ \log_{27}{x} = 1 - \log_{27}{(x - 0.4)}$.

<p>Given that&nbsp;$\log_5{x} = y$, express each of the following in terms of $y$.<span class="inline-marks">[6]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">$ \log_5{x^2}$</div></li><li><div class="list-item-paragraph">$ \log_5{ \left( \frac{1}{x} \right) }$</div></li><li><div class="list-item-paragraph">$ \log_{25}{x}$</div></li></ol>

Given that $\log_5{x} = y$, express each of the following in terms of $y$.

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