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<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 $n$.

Finding Parameters in Binomial Expansions

2025 NASS PRELIMS P1 Q4

Binomial Theorem

<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

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>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 $k$.

2023 ACSBR AM S3 EOY Q9 

<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.

<p>In the expansion of ${{(2+5x)}^{n}}$, the coefficients of ${{x}^{2}}$ and ${{x}^{3}}$ are in the ratio ${1:5}$, find the value of ${n}$.</p>

In the expansion of ${{(2+5x)}^{n}}$, the coefficients of ${{x}^{2}}$ and ${{x}^{3}}$ are in the ratio ${1:5}$, find the value of ${n}$.

<p>Given that the coefficient of the $x$ term in the binomial expansion of ${{\left( {{x}^{2}}+\frac{k}{x} \right)}^{8}}$ is $1792$, find the value of the constant $k$.</p>

Given that the coefficient of the $x$ term in the binomial expansion of ${{\left( {{x}^{2}}+\frac{k}{x} \right)}^{8}}$ is $1792$, find the value of the constant $k$.

<p>In the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$, the sum of coefficients of each term is $992$ more than the sum of the binomial coefficients, $\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)$, where&nbsp;$r=0,\,\,1,\,\,2,\,\,\ldots ,\,\,n$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $n$</div></li><li><div class="list-item-paragraph">Hence find the term with the greatest coefficient in the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$.</div></li></ol>

In the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$, the sum of coefficients of each term is $992$ more than the sum of the binomial coefficients, $\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)$, where $r=0,\,\,1,\,\,2,\,\,\ldots ,\,\,n$.

Sums of Coefficients in Polynomial Expansions

Greatest Coefficients in Binomial Expansions

<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

<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>Given that ${{\left( a+bx \right)}^{6}}=r-96x+60{{x}^{2}}+s{{x}^{3}}+\ldots $, find the possible values of $a$, of $b$, of $r$ and of $s$.</p>

Given that ${{\left( a+bx \right)}^{6}}=r-96x+60{{x}^{2}}+s{{x}^{3}}+\ldots $, find the possible values of $a$, of $b$, of $r$ and of $s$.

<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>The binomial expansion of ${{(1+px)}^{n}}$, where $n&gt;0$, in ascending powers of $x$ is$1-60x+66{{p}^{2}}{{x}^{2}}+\ldots $.  Find the values of $n$ and $p$.</p>

The binomial expansion of ${{(1+px)}^{n}}$, where $n>0$, in ascending powers of $x$ is$1-60x+66{{p}^{2}}{{x}^{2}}+\ldots $.  Find the values of $n$ and $p$.

<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 $81:1$.

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