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PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find the term independent of $x$ in the expansion of $\left(2x^2 + \frac{3}{x}\right)^9$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><ol class="i-list"><li data-list-item-id="subitem-1781511615480-i4ty918y5" data-list-item-path="item-2/subitem-1781511615480-i4ty918y5"><div class="list-item-paragraph">Find the first three terms of the expansion of $(2 - 3x)^5$ in ascending powers of $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1781511646569-ijhb25enu" data-list-item-path="item-2/subitem-1781511646569-ijhb25enu"><div class="list-item-paragraph">Hence, find the value of $n$ such that the coefficient of $x$ in the expansion of $ (2 - 3x)^5(1 - 3x)^n$ is $-720$<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Binomial Expansion for Positive Integer Powers

Coefficients in Products of Binomial Expansions

General Terms and Coefficients in Binomial Expansions

Finding Parameters in Binomial Expansions

2024 GESS PRELIMS P1 Q5

Binomial Theorem

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Write down the first three terms in the expansion, in descending powers of $x$, of $ \left(x^2 + \frac{m}{x} \right)^9$, where&nbsp;$m$ is an integer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Given that the coefficient of $x^3$ in the expansion of $ \left(x^2 + \frac{m}{x} \right)^9$ is $-126$,</div><ol class="i-list"><li data-list-item-id="subitem-1781507291750-bcufrqavo"><div class="list-item-paragraph">show that&nbsp;$m = -1$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1781507292113-kvd75ax8q"><div class="list-item-paragraph">hence, find the term independent of $x$ in the expression of $ \left(2 - \frac{1}{x^3} \right) \left(x^2 + \frac{m}{x} \right)^9$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2024 FMSS PRELIMS P2 Q7

<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>

Tangents and Normals at a Point

Differentiation by First Principles

Increasing and Decreasing Functions

Techniques of Differentiation

Applications of Differentiation

Differentiation

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write and simplify, in terms of $n$, the first four terms in the expansion of $(1 + 2x)^n$ in ascending powers of $x$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the ratio of the coefficient of $x^2$ to that of $x^3$ in the expansion of $(1 + 2x)^n$ is $3 : 20$, find the value of $n$ where $n$ is a positive integer.</span><span class="inline-marks">[3]</span></div></li></ol>

Write and simplify, in terms of $n$, the first four terms in the expansion of $(1 + 2x)^n$ in ascending powers of $x$.

Given that the ratio of the coefficient of $x^2$ to that of $x^3$ in the expansion of $(1 + 2x)^n$ is $3 : 20$, find the value of $n$ where $n$ is a positive integer.

2023 CSS AM S3 EOY Q10 

<p>Find, in ascending powers of $x$, the first four terms in the expansion of $(3+x)^{10}$.</p>

Find, in ascending powers of $x$, the first four terms in the expansion of $(3+x)^{10}$.

<p>For a positive integer $n$:</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$</p></div></div>
<p>There are $n+1$ terms. Powers of $a$ decrease as powers of $b$ increase; their sum is $n$.</p>
<p class="paragraph-header">Binomial coefficients</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">$\binom nr=\dfrac{n!}{r!(n-r)!}\qquad r=0,1,\ldots,n$</p></div></div>
<p>$0!=1\qquad\binom n0=\binom nn=1\qquad\binom n1=n\qquad\binom nr=\binom n{n-r}$</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Coefficient</strong></th><th style="text-align: center; --cell-justify: center"><strong>Evaluation</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\binom{10}{2}$</td><td style="text-align: center; --cell-justify: center">$\dfrac{10\times9}{2!}=45$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\binom73$</td><td style="text-align: center; --cell-justify: center">$\dfrac{7\times6\times5}{3!}=35$</td></tr></tbody></table></div></div>
<p>Use the calculator $n\mathrm Cr$ function for larger coefficients.</p>

$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$

There are $n+1$ terms. Powers of $a$ decrease as powers of $b$ increase; their sum is $n$.

$\binom nr=\dfrac{n!}{r!(n-r)!}\qquad r=0,1,\ldots,n$

$0!=1\qquad\binom n0=\binom nn=1\qquad\binom n1=n\qquad\binom nr=\binom n{n-r}$

Use the calculator $n\mathrm Cr$ function for larger coefficients.

Binomial Coefficients and Pascal's Triangle

<ol class="i-list"><li><div class="list-item-paragraph">Write down and simplify, the first $3$ terms in the expansion of ${{\left( 2-p \right)}^{7}}$ in ascending powers of $p$. <span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the value of $n$ where&nbsp;$n$ is a positive integer, given that the coefficient of ${{x}^{2}}$ is $96$ in the expansion of ${{\left( 1+x \right)}^{n}}{{\left( 2-x+{{x}^{2}} \right)}^{7}}$.<span class="inline-marks">[4]</span></div></li></ol>

2016 TKGS AM P1 Q4

<ol class="i-list"><li><div class="list-item-paragraph">Obtain the first three terms in the expansion of ${{\left( 1\mathrm{ }+x+{{x}^{2}} \right)}^{6}}$ in ascending powers of $x$.</div></li><li><div class="list-item-paragraph">Hence, find the term independent of $x$ in the expansion of ${{\left( 1-\frac{1}{3x} \right)}^{2}}{{(1+x+{{x}^{2}})}^{6}}.$</div></li></ol>

<ol class="i-list"><li><div class="list-item-paragraph">Find the binomial expansion of ${{\left( 2x+\mathrm{ }5 \right)}^{4}}$, simplifying all the terms.</div></li><li><div class="list-item-paragraph">Hence show that&nbsp;${{\left( 2x+5 \right)}^{4}}-{{\left( 2x-5 \right)}^{4}}$ can be written as&nbsp;$320{{x}^{3}}+kx$, where the value of the constant $k$ is to be stated.</div></li><li><div class="list-item-paragraph">Verify that&nbsp;$x=2$ is a root of the equation, ${{\left( 2x+5 \right)}^{4}}-{{\left( 2x-5 \right)}^{4}}=3680x-800$ and find the other possible values of $x$.</div></li></ol>

<p>Expand and simplify ${{\left( 3-\frac{1}{3}x \right)}^{4}}$ in ascending powers of $x$. Given that the coefficients of $x$ and ${{x}^{2}}$ in the expansion of $\left( 1+ax+b{{x}^{2}} \right){{\left( 3-\frac{1}{3}x \right)}^{4}}$ are $126$ and $-187\frac{1}{2}$ respectively, calculate the value of $a$ and of $b$.</p>

Expand and simplify ${{\left( 3-\frac{1}{3}x \right)}^{4}}$ in ascending powers of $x$. Given that the coefficients of $x$ and ${{x}^{2}}$ in the expansion of $\left( 1+ax+b{{x}^{2}} \right){{\left( 3-\frac{1}{3}x \right)}^{4}}$ are $126$ and $-187\frac{1}{2}$ respectively, calculate the value of $a$ and of $b$.

<p>Find the first three terms of the expansion of ${{(1+ax)}^{9}}$ where $a$ is a constant. The coefficient of ${{x}^{2}}$ in the expansion of $(2-x){{(1+ax)}^{9}}$ is zero.  Find the value of $a$.

</p>

Find the first three terms of the expansion of ${{(1+ax)}^{9}}$ where $a$ is a constant. The coefficient of ${{x}^{2}}$ in the expansion of $(2-x){{(1+ax)}^{9}}$ is zero.  Find the value of $a$.



<ol class="i-list"><li><div class="list-item-paragraph">Find the first three terms in the expansion, in ascending powers of $x$, of ${{\left( 2+x \right)}^{8}}$.</div></li><li><div class="list-item-paragraph">Hence, find the term in ${{x}^{2}}$ in the expansion of ${{\left( 2+x-{{x}^{2}} \right)}^{8}}$.</div></li></ol>

<p>Write down and simplify, the first three terms of the expansion in ascending powers of $x$,</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{\left( 2-3x \right)}^{10}}$,</div></li><li><div class="list-item-paragraph">${{\left( \frac{x}{2}+1 \right)}^{10}}$,</div></li><li><div class="list-item-paragraph">Hence, find the coefficient of ${{x}^{2}}$ in the expansion of ${{\left( \frac{4-4x-3{{x}^{2}}}{2} \right)}^{10}}$.</div></li></ol>

Write down and simplify, the first three terms of the expansion in ascending powers of $x$,

<ol class="i-list"><li><div class="list-item-paragraph">Write down the first three terms in the expansion, in descending powers of $x$, of ${{\left( 2x-\frac{1}{x} \right)}^{7}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the value of $a$ if the coefficient of ${{x}^{5}}$ in the expansion of $\left( 1+a{{x}^{2}} \right){{\left( 2x-\frac{1}{x} \right)}^{7}}$ is $224$.<span class="inline-marks">[3]</span></div></li></ol>

2016 NGHS AM P1 Q4

<p>In the expansion ${{\left( 2{{x}^{2}}+\frac{3}{x} \right)}^{n}}$, in descending powers of $x$, the ratio of the coefficients of the third and first term is $81:1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $n$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down the first three terms of the expansion.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the term that is independent of $x$.<span class="inline-marks">[2]</span></div></li></ol>

In the expansion ${{\left( 2{{x}^{2}}+\frac{3}{x} \right)}^{n}}$, in descending powers of $x$, the ratio of the coefficients of the third and first term is $81:1$.

2016 HIHS AM P2 Q3

<p>Find the first three terms in the binomial expansion of ${{\left( 1-8x \right)}^{12}}$.  Hence, use the expansion obtained to evaluate ${{\left( 0.992 \right)}^{12}}$, correct to three decimal places.</p>

Find the first three terms in the binomial expansion of ${{\left( 1-8x \right)}^{12}}$.  Hence, use the expansion obtained to evaluate ${{\left( 0.992 \right)}^{12}}$, correct to three decimal places.

Approximations Using Power Series

<p>Expand ${{\left( 1-2x \right)}^{8}}$ in ascending powers of $x$, up to and including the term ${{x}^{3}}$. Hence, evaluate ${{\left( 0.96 \right)}^{8}}$, correct to $3$ decimal places.</p>

Expand ${{\left( 1-2x \right)}^{8}}$ in ascending powers of $x$, up to and including the term ${{x}^{3}}$. Hence, evaluate ${{\left( 0.96 \right)}^{8}}$, correct to $3$ decimal places.

<p>Obtain the first $4$ terms in the expansion, in ascending powers of $x$, of ${{\left( 1-4x \right)}^{5}}$. Hence, find the value of ${{\left( 0.96 \right)}^{5}}$ correct to $3$ decimal places.</p>

Obtain the first $4$ terms in the expansion, in ascending powers of $x$, of ${{\left( 1-4x \right)}^{5}}$. Hence, find the value of ${{\left( 0.96 \right)}^{5}}$ correct to $3$ decimal places.

<p>Obtain the first four terms of ${{\left( 1-\frac{x}{2} \right)}^{8}}$ in ascending powers of $x$.  Hence, find the coefficient of ${{x}^{3}}$ in $(2+3x){{\left( 1-\frac{x}{2} \right)}^{8}}$.</p>

Obtain the first four terms of ${{\left( 1-\frac{x}{2} \right)}^{8}}$ in ascending powers of $x$.  Hence, find the coefficient of ${{x}^{3}}$ in $(2+3x){{\left( 1-\frac{x}{2} \right)}^{8}}$.

<p>Find, in ascending powers of $x$, the first three terms in the expansion of ${{\left( 2-\frac{x}{2} \right)}^{6}}.$ Hence, find the coefficient of ${{x}^{2}}$ in the expansion of ${{\left( x+1 \right)}^{2}}{{\left( 2-\frac{x}{2} \right)}^{6}}.$</p>

Find, in ascending powers of $x$, the first three terms in the expansion of ${{\left( 2-\frac{x}{2} \right)}^{6}}.$ Hence, find the coefficient of ${{x}^{2}}$ in the expansion of ${{\left( x+1 \right)}^{2}}{{\left( 2-\frac{x}{2} \right)}^{6}}.$

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