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PSLE

PSLE Foundation Mathematics

<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

Factorisation by Special Identities

2025 CGS PRELIMS P2 Q1

Polynomials, Cubic Equations and Partial Fractions

Exponential and Logarithmic Functions

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>

Solving Logarithm Equations

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.

2024 FMSS PRELIMS P2 Q1

<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 $.

2024 CGSS P1 Q4

<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">Solve the equation $\log_5 (3y + 1) - \frac{1}{\log_y 5} + \log_2 8 = 4$.</span><span class="inline-marks">[4]</span></div></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">By expressing the equation $\ln(6\mathrm{e}^x + 1) + x = 0$ as a quadratic equation in $\mathrm{e}^x$, solve the equation $\ln(6\mathrm{e}^x + 1) + x = 0$, giving values of $x$ in logarithmic form.</span><span class="inline-marks">[4]</span></div></li></ol>

Solve the equation $\log_5(3y+1)-\dfrac{1}{\log_y 5}+\log_2 8=4$.

By expressing the equation $\ln(6e^x+1)+x=0$ as a quadratic equation in $e^x$, solve the equation $\ln(6e^x+1)+x=0$, giving values of $x$ in logarithmic form.

(A Math) 2025 CCHY PRELIMS P2 Q6

<p>By using substitution or otherwise, find the values of $x$ for which $2^{2x-1} = 2^{x+3} - 24$, giving your answer where appropriate, to one decimal place.<span class="inline-marks">[5]</span></p>

By using substitution or otherwise, find the values of $x$ for which $2^{2x-1}=2^{x+3}-24$, giving your answer where appropriate, to one decimal place.

2025 AHS P1 Q2

<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">Given that&nbsp;$u = \left(\frac{3}{2}\right)^x$, express $9^x + 6^x = 6(4^x)$ as a quadratic equation in $u$.</span><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-content"><span class="list-item-text">Hence, or otherwise, solve $9^x + 6^x = 6(4^x)$.</span><span class="inline-marks">[3]</span></div></li></ol>

Given that $u = \left(\frac{3}{2}\right)^x$, express $9^x + 6^x = 6(4^x)$ as a quadratic equation in $u$.

Hence, or otherwise, solve $9^x + 6^x = 6(4^x)$.

2025 DSS PRELIMS P1 Q7

<ol class="abc-list"><li><div class="list-item-paragraph">Solve $6^x = \frac{10}{3} - 6^{-x}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of $y = \ln x$, showing any points of intersection with the axes.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">To solve $e^{1-2x} = x^3$, a straight line can be drawn on the same axes as the graph in part <strong>(b)</strong>:</div><ol class="i-list"><li><div class="list-item-paragraph">Determine the equation of the straight line to be drawn.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, state the number of solutions for&nbsp;$e^{1-2x} = x^3$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Sketch the graph of $y = \ln x$, showing any points of intersection with the axes.

Sketching Logarithmic Functions and Solving Equations Graphically

IP3 Class Test (May 2025) Q10 (2023 Jurong Sec AM Prelim P2 Q6)

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The curve with equation $y = ax^2 + bx + a$, where $a$ and $b$ are constants, lies completely above the $x$-axis.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the conditions which must apply to $a$ and $b$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Give an example of possible values for $a$ and $b$ which satisfy the conditions in part <strong>(i)</strong>.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Using a suitable substitution, explain why the equation $3^{2x-1} = 3^{x+2} - k$ has no solution if $k &gt; 60.75$.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve with equation $y = ax^2 + bx + a$, where $a$ and $b$ are constants, lies completely above the $x$-axis.

Using a suitable substitution, explain why the equation $3^{2x-1} = 3^{x+2} - k$ has no solution if $k > 60.75$.

Nature of Roots of Quadratic Equations

Quadratic Expressions Positive or Negative for All Real Values

2023 Ahmad Ibrahim Sec 3 AM EOY Q6

Quadratic Functions, Equations and Inequalities

<p>By using a suitable substitution, solve the equation $2^{3+2x} + 2^{5+x} = 2^x + 4$.</p>

By using a suitable substitution, solve the equation $2^{3+2x} + 2^{5+x} = 2^x + 4$.

<p>Solve the equation $2^{x+1} - 24\sqrt{2^x} + 64 = 0$.</p>

Solve the equation $2^{x+1} - 24\sqrt{2^x} + 64 = 0$.

<p>By using an appropriate substitution, solve the equation $0.2(2^x) + 0.8(2^{-x}) = 1$.</p>

By using an appropriate substitution, solve the equation $0.2(2^x) + 0.8(2^{-x}) = 1$.

<p>By using an appropriate substitution, solve the following equation $9^{x+1} - 28(3^x) + 3 = 0$.</p>

By using an appropriate substitution, solve the following equation $9^{x+1} - 28(3^x) + 3 = 0$.

<p>Use an appropriate substitution, or otherwise, solve $7^{2x+1} + 20(7^x) = 3$.</p>

Use an appropriate substitution, or otherwise, solve $7^{2x+1} + 20(7^x) = 3$.

<p>The polynomial ${\mathrm{g}(x) = mx^3 - 8x^2 - 9x + n}$ has a factor of ${3x - 2}$ and leaves a remainder of ${6}$ when divided by ${x}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of ${m}$ and ${n}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using the values of ${m}$ and ${n}$ from (a), solve, in exact form, the equation ${\mathrm{g}(x) = 0}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence solve the equation ${9e^{2y} + 8e^y = 12 + 6e^{3y}}$.<span class="inline-marks">[3]</span></div></li></ol>

The polynomial ${\mathrm{g}(x) = mx^3 - 8x^2 - 9x + n}$ has a factor of ${3x - 2}$ and leaves a remainder of ${6}$ when divided by ${x}$.

Cubic Expressions and Equations

Remainder and Factor Theorems

2023 AISS AM S3 EOY Q9

<ol class="abc-list"><li data-list-item-id="item-1760869906502-hbbg8kjmp" data-list-item-path="item-1760869906502-hbbg8kjmp"><div class="list-item-paragraph">The curve with equation ${y = ax^2 + bx + a}$, where&nbsp;${a}$ and ${b}$ are constants, lies completely above the ${x}$-axis.</div><ol class="i-list"><li data-list-item-id="subitem-1760869918022-3q9gusr0r" data-list-item-path="item-1760869906502-hbbg8kjmp/subitem-1760869918022-3q9gusr0r"><div class="list-item-paragraph">Write down the conditions which must apply to ${a}$ and ${b}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760869919042-j7hfklk1h" data-list-item-path="item-1760869906502-hbbg8kjmp/subitem-1760869919042-j7hfklk1h"><div class="list-item-paragraph">Give an example of possible values for&nbsp;${a}$ and ${b}$ which satisfy the conditions in part<strong> (i)</strong>.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760869907657-mbuguk1o5" data-list-item-path="item-1760869907657-mbuguk1o5"><div class="list-item-paragraph">Using a suitable substitution, explain why the equation ${3^{2x-1} = 3^{x+2} - k}$ has no solution if&nbsp;${k &gt; 60.75}$.<span class="inline-marks">[3]</span></div></li></ol>

2023 AISS AM S3 EOY Q6

<p>Solve the equation ${{5}^{x+2}}-{{25}^{x+0.5}}=2({{5}^{x+1}})$, giving your answer correct to $2$ decimal places.</p>

Solve the equation ${{5}^{x+2}}-{{25}^{x+0.5}}=2({{5}^{x+1}})$, giving your answer correct to $2$ decimal places.

<p>Solve the equation ${{2}^{2x}}-{{2}^{x+3}}+7=0$.</p>

Solve the equation ${{2}^{2x}}-{{2}^{x+3}}+7=0$.

<p>By using a suitable substitution, solve the equation ${{2}^{3+2x}}+{{2}^{5+x}}={{2}^{x}}+4$.</p>

By using a suitable substitution, solve the equation ${{2}^{3+2x}}+{{2}^{5+x}}={{2}^{x}}+4$.

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