# Binomial Theorem (IB SL)

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1 (IB/sl/2019/November/Paper1/g4)
[Maximum mark: 6]
Consider $\left(\begin{array}{l}11 \\ a\end{array}\right)=\frac{11 !}{a ! 9 !}$
 (a) Find the value of $a$.
 (b) Hence or otherwise find the coefficient of the term in $x^{9}$ in the expansion of $(x+3)^{11}$.

 2 (IB/sl/2019/May/paper2tz1/q6) [Maximum mark: 7 ] Consider the expansion of $\left(x^{2}+1.2\right)^{n}$ where $n \in \mathbb{Z}, n \geq 3$. Given that the coefficient of the term containing $x^{6}$ is greater than 200000 , find the smallest possible value of $n$.

 3 (IB/sl/2019/May/paper2tz2/q6) [Maximum mark: 7] In the expansion of the following expression, find the exact value of the constant term. $$x^{3}\left(\frac{1}{2 x}+x^{2}\right)^{15}$$

 4 (IB/sl/2018/November/Paper2/q6) [Maximum mark: 7] Consider the expansion of $\left(2 x^{4}+\frac{x^{2}}{k}\right)^{12}, k \neq 0 .$ The coefficient of the term in $x^{m}$ is five times the coefficient of the term in $x^{35}$. Find $k$.

 5 (IB/sl/2018/May/paper2tz2/q5) [Maximum mark: 6] Consider the expansion of $\left(2 x+\frac{k}{x}\right)^{9}$, where $k>0$. The coefficient of the term in $x^{3}$ is equal to the coefficient of the term in $x^{5}$. Find $k$.

 6 (IB/sl/2017/November/Paper2/q6) [Maximum mark: 6] In the expansion of $a x^{3}(2+a x)^{11}$, the coefficient of the term in $x^{5}$ is 11880 . Find the value of $a$,

 7 (IB/sl/2017/May/paper2tz1/q6) [Maximum mark: 7] Let $f(x)=\left(x^{2}+3\right)^{7}$. Find the term in $x^{5}$ in the expansion of the derivative, $f^{\prime}(x)$

8 (IB/s1/2016/November/Paper1/q3)
[Maximum mark: 7]
The values in the fourth row of Pascal's triangle are shown in the following table.
$$\begin{array}{|l|l|l|l|l|}\hline 1 & 4 & 6 & 4 & 1 \\\hline\end{array}$$
 (a) Write down the values in the fifth row of Pascal's triangle.
 (b) Hence or otherwise, find the term in $x^{3}$ in the expansion of $(2 x+3)^{5}$.

9 (IB/sl/2016/May/paper2tz1/q4)
[Maximum mark: 6]
 (a) Find the term in $x^{6}$ in the expansion of $(x+2)^{5}$.
 (b) Hence, find the term in $x^{\prime}$ in the expansion of $5 x(x+2)^{\circ}$. [2]

10 (IB/sl/2016/May/paper2tz2/q5)
[Maximum mark: 6]
Consider the expansion of $\left(x^{2}+\frac{2}{x}\right)^{10}$.
 (a) Write down the number of terms of this expansion.
 (b) Find the coeflicient of $x^{s}$.

 11 (IB/s1/2015/November/Paper1/q6) [Maximum mark: 7] In the expansion of $(3 x+1)^{n}$, the coefficient of the term in $x^{2}$ is $135 n$, where $n \in Z^{+}$. Find $n$.

12 (IB/s1/2015/May/paper2tz1/q2)
[Maximum mark: 5]
Consider the expansion of $(2 x+3)^{5}$.
 (a) Write down the number of terms in this expansion. $[1]$
 (b) Find the term in $x^{3}$. $[4]$

 13 (IB/sl/2015/May/paper2tz2/q4) [Maximum mark: 5] The third term in the expansion of $( x + k )^8$ is $63 x ^6 .$ Find the possible values of $k .$

[1](a) 2 (b) 495
[2] $27$
[3] $\frac{1365}{2048}$
[4] $\quad k=4$
[5] $k=\frac{6}{7}$
[6] $a=\frac{3}{4}$
[7] $17010 x^{5}$
[8](a) $1,5,10,10,5,1$ (b) $720 x^{3}$
[9](a) $\quad 672 x^{6}$ (b) $3360 x^{7}$
[10](a) 11 (b) 3360
[11] $n=31$
[12](a) $9$ (b) $108864 x^{3}$
[13] $k=\pm 1.5$