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<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 $n$.

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>

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

<p>The expansion of $(1+kx)^n$ in ascending powers of $x$ is $1+\frac{7}{3}x+\frac{7}{3}x^2+...$.<br></p>
<ol class="abc-list"><li data-list-item-id="item-1772949331739-4z7z0guj3"><div class="list-item-paragraph">Show that&nbsp;$k=\frac{1}{3}$ and calculate the value of $n$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1772949332663-ncgfsz59f"><div class="list-item-paragraph">Explain why all terms in the expansion of $ \left(px^2-\frac{1}{x^4} \right)^{12}$ contains only even powers of $x$.<span class="inline-marks">[4]</span></div></li></ol>

The expansion of $(1+kx)^n$ in ascending powers of $x$ is $1+\frac{7}{3}x+\frac{7}{3}x^2+...$.


2025 MGS 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 $3$.

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 $k$.

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