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<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

<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

<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 

<p>The coefficient of $x^3$ in the expansion of $(10+kx)\left(2-\frac{1}{2}x\right)^8$ is zero.</p>
<p>Find the value of the constant $k$.<span class="inline-marks">[5]</span></p>

The coefficient of $x^3$ in the expansion of $(10+kx)\left(2-\frac{1}{2}x\right)^8$ is zero.

<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>

<p>Given that the expansion of  ${{\left( 1-\frac{x}{2} \right)}^{n}}(3-x)=a-13x+5b{{x}^{2}}+\ldots $, find the values of $n$, $a$ and $b$.</p>

Given that the expansion of  ${{\left( 1-\frac{x}{2} \right)}^{n}}(3-x)=a-13x+5b{{x}^{2}}+\ldots $, find the values of $n$, $a$ and $b$.

<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>

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 $a$.



<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 of ${{\left( ax+\frac{1}{x} \right)}^{n}}$ in descending powers of $x$, where&nbsp;$a$ and $n$ are positive integers, the fourth term of the expansion is the constant term.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $n$ and hence, express the constant term in terms of $a$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Using your value of $n$ in <strong>(i)</strong>, determine if a term in $\frac{1}{{{x}^{3}}}$ in the expansion $\left( 1-x \right){{\left( ax+\frac{1}{x} \right)}^{n}}$ exists.<span class="inline-marks">[2]</span></div></li></ol>

In the expansion of ${{\left( ax+\frac{1}{x} \right)}^{n}}$ in descending powers of $x$, where $a$ and $n$ are positive integers, the fourth term of the expansion is the constant term.

2016 VS AM P1 Q4

<p>Given that the first two non-zero terms of the expansion of $\left( 1-px \right){{\left( 1-\frac{x}{3} \right)}^{n}}$are $1$ and $-\frac{7}{3}{{x}^{2}},$where&nbsp;$n$ is a positive integer, find the value of $p$ and of $n.$</p>

Given that the first two non-zero terms of the expansion of $\left( 1-px \right){{\left( 1-\frac{x}{3} \right)}^{n}}$are $1$ and $-\frac{7}{3}{{x}^{2}},$where $n$ is a positive integer, find the value of $p$ and of $n.$

<ol class="abc-list"><li><div class="list-item-paragraph">In the binomial expansion of ${{\left( x+\frac{k}{x} \right)}^{7}}$, where $k$ is a positive constant, the coefficients of ${{x}^{3}}$ and ${{x}^{5}}$ are the same.  Find the value of $k$.</div></li><li><div class="list-item-paragraph">Using the value of $k$ found in part <strong>(a)</strong>, find the coefficient of ${{x}^{5}}$ in the expansion of $(1-3{{x}^{2}}){{\left( x+\frac{k}{x} \right)}^{7}}$.

</div></li></ol>

<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 ${{\left( x+1 \right)}^{2}}{{\left( 2-\frac{x}{2} \right)}^{6}}.$

<ol class="i-list"><li data-list-item-id="item-1766131400754-5ckmlyf02" data-list-item-path="item-1766131400754-5ckmlyf02"><div class="list-item-paragraph">Given that the coefficient of ${{x}^{2}}$ in the expansion of ${{\left( 1-2x \right)}^{2}}{{\left( 1+px \right)}^{7}}$ is $32$, find the possible values of the constant $p$. </div></li><li data-list-item-id="item-1766131401256-r1z175h0i" data-list-item-path="item-1766131401256-r1z175h0i"><div class="list-item-paragraph">Hence find the coefficient of ${{x}^{2}}$ in the expansion of ${{\left( 1-x \right)}^{2}}{{\left( 1+\frac{px}{2} \right)}^{7}}.$</div></li></ol>

<ol class="abc-list"><li><div class="list-item-paragraph">Find the first $3$ terms in the expansion of ${{\left( {{x}^{2}}-2 \right)}^{10}}$ in ascending powers of $x$.</div></li><li><div class="list-item-paragraph">Hence find the coefficient of the ${{x}^{4}}$ term in the expansion of ${{\left( x+2 \right)}^{2}}{{\left( {{x}^{2}}-2 \right)}^{10}}$.</div></li></ol>

<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-paragraph">Show that the value of ${n}$ is ${12}$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Find the middle term.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the coefficient of ${x^6}$ in the expansion of ${\left(2 - x^3\right)\left(x^2 + \frac{1}{3x}\right)^{12}}$.<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 ${6:11}$.

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