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<p>Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.<span class="inline-marks">[5]</span></p>

Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.

General Binomial Theorem

A negative integer power

Binomial Theorem

<p>Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.<span class="inline-marks">[7]</span></p>

Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.

Combining two validity conditions

<p>For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where&nbsp;\(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.<span class="inline-marks">[7]</span></p>

For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where \(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.

Recovering a rational exponent

<p>The first three terms in the expansion of \((1+ax)^n\) are \(1-3x+\frac{33}{8}x^2\), where&nbsp;\(a\) and \(n\) are constants. Find the value of \(a\) and of \(n\).<span class="inline-marks">[5]</span></p>

The first three terms in the expansion of \((1+ax)^n\) are \(1-3x+\frac{33}{8}x^2\), where \(a\) and \(n\) are constants. Find the value of \(a\) and of \(n\).

2025 NASS PRELIMS P1 Q4

<p>Without expanding completely, prove that&nbsp;\( \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0\) for every positive integer \(n\).</p>

Without expanding completely, prove that \( \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0\) for every positive integer \(n\).

<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>

2024 GESS PRELIMS P1 Q5

<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>One of the terms of the expansion of \( (x + 2y)^{10}\) is \(ax^8y^2\). Find the value of \(a\).<span class="inline-marks">[6]</span></p>

One of the terms of the expansion of \( (x + 2y)^{10}\) is \(ax^8y^2\). Find the value of \(a\).

<ol class="abc-list"><li><div class="list-item-paragraph">Expand \( (x - 2)^4\) and simplify your result.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the term in \(x^3\) in \( (3x +4)(x - 2)^4\).<span class="inline-marks">[3]</span></div></li></ol>

<p>Find the term in \(x^3\) in the expansion of \( \left( \frac{2}{3}x - 3 \right)^8\).<span class="inline-marks">[5]</span></p>

Find the term in \(x^3\) in the expansion of \( \left( \frac{2}{3}x - 3 \right)^8\).

<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

Techniques of Differentiation

<p>The fifth term in teh expansion of the binomial \( (a + b)^n\) is given by \( \binom{10}{4} p^6 (2q)^4\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the value of \(n\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down \(a\) and \(b\), in terms of \(p\) and/or \(q\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Write down an expression for the sixth term in the expansion.<span class="inline-marks">[3]</span></div></li></ol>

The fifth term in teh expansion of the binomial \( (a + b)^n\) is given by \( \binom{10}{4} p^6 (2q)^4\).

<p>Find the term in \(x^4\) in the expansion of \( \left( 3x^2 - \frac{2}{x} \right)^5\).<span class="inline-marks">[6]</span></p>

Find the term in \(x^4\) in the expansion of \( \left( 3x^2 - \frac{2}{x} \right)^5\).

<ol class="abc-list"><li><div class="list-item-paragraph">Expand \( (2 +x)^4\) and simplify your result.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, find the term in \(x^2\) in \( (2  + x)^4 \left(1 + \frac{1}{x^2} \right) \).<span class="inline-marks">[3]</span></div></li></ol>

<p>Consider the expansion of \( (x +2)^{11}\)</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the number of terms in this expansion.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the term containing \(x^2\).<span class="inline-marks">[4]</span></div></li></ol>

Consider the expansion of \( (x +2)^{11}\)

<p>The binomial expansion of \((1 + px)^n\) where&nbsp;\(n &gt; 0\), in ascending powers of \(x\) is</p>
<p style="text-align: center">\(1 - 12x + 28p^2x^2 + qx^3 + \dots\)</p>
<p>Find the value of \(p\), of \(n\) and of \(q\).<span class="inline-marks">[6]</span></p>

The binomial expansion of \((1+px)^n\) where \(n > 0\), in ascending powers of \(x\) is

Find the value of \(p\), of \(n\) and of \(q\).

2025 AHS PRELIMS P1 Q6

<ol class="abc-list"><li data-list-item-id="item-5-a" data-list-item-path="item-5-a"><div class="list-item-content"><span class="list-item-text">By considering the general term in the binomial expansion of \(\left(x^4-\frac{1}{kx^2}\right)^6\), where&nbsp;\(k\) is a positive constant, explain why there are only even powers of \(x\) in this expansion.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-5-b" data-list-item-path="item-5-b"><div class="list-item-content"><span class="list-item-text">Given that the term independent of \(x\) in the binomial expansion of \(\left(x^4-\frac{1}{kx^2}\right)^6\) is \(\frac{5}{27}\), show that&nbsp;\(k\) is \(3\).</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1772870286453-hmj11qotg" data-list-item-path="item-1772870286453-hmj11qotg"><div class="list-item-paragraph">Hence find the term independent of \(x\) in \( \left(2-3x^6 \right) \left(x^4-\frac{1}{kx^2} \right)^6\).<span class="inline-marks">[4]</span></div></li></ol>

By considering the general term in the binomial expansion of \(\left(x^4-\frac{1}{kx^2}\right)^6\), where \(k\) is a positive constant, explain why there are only even powers of \(x\) in this expansion.

Given that the term independent of \(x\) in the binomial expansion of \(\left(x^4-\frac{1}{kx^2}\right)^6\) is \(\frac{5}{27}\), show that \(k\) is 3.

Hence find the term independent of \(x\) in \((2-3x^6)\left(x^4-\frac{1}{kx^2}\right)^6\).

2025 SCGS PRELIMS P2 Q6

<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>In the expansion of \(\left(x^2 + \frac{1}{3x}\right)^n\) in descending powers of \(x\), the coefficients of the second and the third term are in the ratio of \(6 : 11\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the value of \(n\) is \(12\).</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the middle term.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the coefficient of \(x^6\) in the expansion of \((2 - x^3)\left(x^2 + \frac{1}{3x}\right)^{12}\).</span><span class="inline-marks">[3]</span></div></li></ol>

In the expansion of \(\left(x^2 + \frac{1}{3x}\right)^n\) in descending powers of \(x\), the coefficients of the second and the third term are in the ratio of \(6 : 11\).

Find the coefficient of \(x^6\) in the expansion of \((2 - x^3)\left(x^2 + \frac{1}{3x}\right)^{12}\).

2023 Ahmad Ibrahim Sec AM S3 EOY Q11 

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that the coefficient of \(\frac{1}{x^2}\) in the binomial expansion of \(\left(x^2 + \frac{m}{x^4}\right)^5\) is \(360\), find the possible values of \(m\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the first three terms in the expansion of \((1 - 3x)^n\) in ascending powers of \(x\) are \(1 - 15x + kx^2\), find the value of \(n\) and of \(k\).</span><span class="inline-marks">[3]</span></div></li></ol>

Given that the coefficient of \(\frac{1}{x^2}\) in the binomial expansion of \(\left(x^2 + \frac{m}{x^4}\right)^5\) is \(360\), find the possible values of \(m\).

Given that the first three terms in the expansion of \((1 - 3x)^n\) in ascending powers of \(x\) are \(1 - 15x + kx^2\), find the value of \(n\) and of \(k\).

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