# Binomial Theorem (Myanmar Exam Board)

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Group (2015-2019)

1. (2015/Myanmar /q8b )
If the $2^{\text {nd }}$ and the $3^{\text {rd }}$ term in $(a+b)^{n}$ are in the same ratio as the $3^{\text {rd }}$ and $4^{\text {th }}$ $\begin{array}{ll}\text { in }(a+b)^{n+3}, \text { then find'n. } & \text { (5 marks) }\end{array}$

2. (2015/FC /q8b )
Given that $\left(p-\frac{1}{2} x\right)^{6}=r-96 x+s x^{2}+\cdots$, find $p, r, s . \quad$ (5 marks)

3. (2016/Myanmar /q8b )
The expansion of $(3+4 x)^{n}$, the coefficients of $x^{4}$ and $x^{5}$ are in the ratio of $5: 16$. Find the value of $n$

4. (2016/FC /q8b )
Write down and simplify the first four terms in the binomial expansion of $(1-2 x)^{7}$. Use it to find the value of $(0.98)^{7}$, correct to four decimal places.

5. (2017/Myanmar /q8b )
In the expansion of $(1+x)^{a}+(1+x)^{b}$, the coefficients of $x$ and $x^{2}$ are equal for all positive integers $a$ and $b$, prove that $3(a+b)=a^{2}+b^{2}$.
Q8(b) Solution

6. (2017/FC /q8b )
The first four terms in the binomial expression of $(a+b)^{n}$, in descending powers of $a$, are $w, x, y$ and $z$ respectively. Show that $(n-2) x y=3 n w z$. (5 marks)

7. (2018/Myanmar /q8b )
In the expansion of $(1-2 x)^{n}$, the sum of the coefficients of $x$ and $x^{2}$ is 16 Given that $n$ is positive, find the value of $n$ and the coefficient of $x^{3}$.
Click for Solution

8. (2018/FC /q8b )
Use the binomial theorem to find the value of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$.

9. (2019/Myanmar /q2a )
2. (a) Find and simplify the coefficient of $\mathrm{x}^{7}$ in the expansion of $\left(\mathrm{x}^{2}+\frac{2}{\mathrm{x}}\right)^{8}, \mathrm{x} \neq 0 .$ (3 marks) Click for Solution

10. (2019/Myanmar /q7b )
If the coefficients of $\mathrm{x}^{\mathrm{r}}$ and $\mathrm{x}^{\mathrm{r}+2}$ in the expansion of $(1+\mathrm{x})^{2 \mathrm{~m}}$ are equal, show that $r=n-1 .$ $(5$ marks) Click for Solution

11. (2019/FC /q2a )
Find and simplify the coefficient of $x^{6}$ in the expansion of $\left(x-\frac{3}{x}\right)^{14}, x \neq 0 .(3$ marks $)$ Click for Solution 2(a)

12. (2019/FC /q7b )
If the coefficients of $x^{r}$ and $x^{r-2}$ in the expansion of $(1+x)^{2 n}$ are equal, show that $r=n-1$ (5 marks) Click for Solution 7(b)

1.  $n=5$
2.  $p=2,r=64,s=60$
3.  $n=16$
4.  $1-14 x+84 x^{2}-280 x^{3}+\cdots, 0.8681$
5.  prove
6.  Prove
7.  $n=4$, $-32$
8.  $64 x^{6}-96 x^{4}+36 x^{2}-2$
9.  448
10.  Proof
11.  81081
12.  Prove

Group (2014)

1. Write down and simplify the first four terms in the expansion of $\left(1-\frac{1}{2} x\right)^{10}$. Find the coefficient of $x^{2}$ in the expansion of $(5+4 x)\left(1-\frac{1}{2} x\right)^{10}$. $\quad\mbox{ (5 marks)}$

2. Expand $\left(\frac{1}{2}-2 x\right)^{5}$ up to the term in $x^{3}$. If the coefficient of $x^{2}$ in the expansion of $\left(1+a x+3 x^{2}\right)\left(\frac{1}{2}-2 x\right)^{5}$ is $\frac{13}{2}$, find the coefficient of $x^{3}$. $\quad\mbox{ (5 marks)}$

3. Find the coefficient of $x^{2}$ in the expansion of $\left(2 x^{2}-x-3\right)^{6}$. $\quad\mbox{ (5 marks)}$

4. Find the coefficient of $x^{2}$ and $x^{3}$ in the expansion of $\left(x^{2}-x-2\right)^{7}$. $\quad\mbox{ (5 marks)}$

5. Evaluate the coefficients of $x^{5}$ and $x^{4}$ in the binomial expansion of $\left(\frac{x}{3}-3\right)^{7}$. $\quad\mbox{ (5 marks)}$ Hence evaluate the coefficient of $x^{5}$ in the expansion of $\left(\frac{x}{3}-3\right)^{7}(x+6)$. $\quad\mbox{ (5 marks)}$

6. In the expansion of $(1+2 x)^{11}$, the coefficient of $x^{3}$ is $k$ times the coefficient of $x^{2}$. Find $k$. $\quad\mbox{ (5 marks)}$

7. If the coefficient of $x^{4}$ in the expansion of $(3+2 x)^{6}$ is equal to the coefficient of $x^{4}$ in the expansion of $(2 k+3 x)^{6}$, find $k$. $\quad\mbox{ (5 marks)}$

8. The ratio of the coefficient of $x^{4}$ in the expansion of $(3-2 x)^{6}$ and the coefficient of $x^{4}$ in the expansion of $(k+2 x)^{7}$ is $1: 7.$ Find the value of $k$. $\quad\mbox{ (5 marks)}$

9. The first three terms in the binomial expasion of $(a+b)^{n}$, in ascending power of $b$, are denoted by $p, q$ and $r$ respectively. Show that $\frac{q^{2}}{p r}=\frac{2 n}{n-1}$. Given that $p=4, q=32$ and $r=96$, evaluate $n$. $\quad\mbox{ (5 marks)}$

10. In the binomial expansion of $(1+x)^{n}$ the first three terms are $1+3+4+\ldots$ Calculate the numerical values of $n$ and $x$, and the value of the fourth term of the expansion. $\quad\mbox{ (5 marks)}$

1. $1-5x+\frac{45}{4}x^2-15x^3+\cdots;\frac{145}{4}$
2. $\frac{1}{32}-\frac{5}{8} x+5 x^{2}-20 x^{3}+\cdots$; $-\frac{265}{8}$
3. $-1701$
4. $-224,784$
5. $\frac{7}{9}, \frac{-35}{3},-7$
6. $k=6$
7. $k=\pm \frac{2}{3} \quad$
8. $k=3$
9. $n=4$
10. $n=9,x=\frac{1}{3};\frac{28}{9}$

Group (2013)

1. Find the coefficient of $x^{3}$ in the expansion $\left(2+3 x+x^{2}\right)(1+x)^{6}$. (5 marks)

2. Find the coefficient of $x^{3}$ in the expansion of $(2 x-3)^{2}(1+3 x)^{8}$. (5 marks)

3. If the coefficients of $x$ and $x^{3}$ in the expansion of $(1+p x)^{8}$ are equal, find the value of $p$. (5 marks)

4. In the expansion of $\left(x^{2}+\frac{a}{x}\right)^{8}, a \neq 0$, the coefficient of $x^{7}$ is four times the coefficient of $x^{10}$. Find the value of ' $a^{2}$. (5 marks)

5. Given that the coefficient of $x^{3}$ in the expansion of $(1+a x)^{6}$ is equal to the coefficient of $x^{2}$ in the expansion of $\left(2-\frac{1}{3} x\right)^{10}$, find the value of $a$. (5 marks)

6. Find the value of ' $n$ ' if the coefficient of $x^{5}$ is six times the coefficient of $x^{4}$ in the expansion of $(3+2 x)^{n}$. (5 marks)

7. Given that $(1+a x)^{n}=1-12 x+63 x^{2}+\ldots$, find $a$ and $n$. (5 marks)

8. Find the middle term and the constant term in the expansion of $\left(2 x^{2}-\frac{1}{2 x}\right)^{12}$. (5 marks)

9. Write down the fourth term in the binomial expansion of $\left(p x+\frac{q}{x}\right)^{n}$. If this term is independent of $x$, find the value of $n$. With this value of $n$ calculate the values of $p$ and $q$ given that the fourth term is equal to 160 , both $p$ and $q$ are positive and $p-q=1$. (5 marks)

10. The first four terms in the binomial expansion of $(a+b)^{n}$, in descending powers of $a$, are $w, x, y$ and $z$ respectively. Show that $(n-2) x y=3 n w z$. (5 marks)

11. If $a_{1}, a_{2}, a_{3}$ and $a_{4}$ are any four consecutive coefficients in the expansion of $(1+x)^{n}$, then show that $\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{3}}{a_{3}+a_{4}}=\frac{2 a_{2}}{a_{2}+a_{3}}$. (5 marks)

1. 91
2. 10680
3. $\pm \frac{1}{\sqrt{7}}$
4. $a=2 \quad$
5. $a=4$
6. $n=49$
7. $n=8, a=-\frac{3}{2}$
8. $924x^4;\frac{495}{16}$
9. $^nC_3p^{n-3}q^3x^{n-6};n=6;p=2;q=1$
10. Show
11. Show

$\quad\;\,$$\, 1.Given that the coefficient of x^{2} in the expansion of (1+k x)^{5}+(1-4 x)^{3} is 138 . Calculate the positive value of k. (5 marks) 2.Given that the coefficient of x^{3} in the expansion of (a+x)^{5}+(1-2 x)^{6} is -120, calculate the possible values of a. (5 marks) 3.If the coefficient of x^{3} in the expansion of (1-a x)^{6}-\left(2-\frac{x}{2}\right)^{8} is 64, find a. (5 marks) 4.In the expansion of (a+b x)(1+x)^{6}, the coefficients of x^{2} and x^{3} are 48 and 85 respectively. Find the values of a and b. (5 marks) 5.Find the coefficients of x^{0} and x^{3} respectively in the expansions of \left(x-\frac{1}{x^{2}}\right)^{9}. Are they equal? (5 marks) 6.In the expansion of (2+3 x)^{n}, the coefficients of x^{3} and x^{4} are in the ratio 8: 15. Find the value of n. (5 marks) 7.In the expansion of (1+2 x)^{n}, the coefficient of x^{6} is 4 times the coefficient of x^{4}. Find n. (5 marks) 8.In the expansion of \left(1+\frac{3}{4}\right)^{n}, the fifth term and the third term are in the ratio 9: 16 . Find n. (5 marks) 9.If, in the expansion of (1+x)^{m}(1-x)^{n}, the coefficient of x and x^{2} a.e 3 and -6 respectively, then find the value of m. (5 marks) 10.Write down the fourth and the fifth terms of (a+b x)^{9}. If these terms are equal, show that x=\frac{2 a}{3 b}. (5 marks) 11.Find, in ascending powers of x, the first three terms of (1+k x)^{5}(1-4 x). If the coefficient of x is 16, find the value of k, and the coefficient of x^{2}. (5 marks) #### Answer (2012) \quad\;\, 1.k=3 2.a=\pm 2 3.a=2 4.a=2, b=3 5.-84,36 ; \mathrm{no} 6.n=8 7.n=10 8.n=6 9.m=12 10.{}^9C_3a^6b^3x^3;{}^9C_4a^5b^4x^4 11.1+(-4+5 k) x+\left(-20 k+10 k^{2}\right) x^{2}+\ldots- ;k=4 ; 80 ## Group (2011) \quad\;\,$$\,$
1.Find in ascending powers of $x$, the first three terms in the expansion of $(1+2 x)^{5}$ and $(3-x)^{4}$. Hence find the coefficient of $x^{2}$ in the expansion of $(1+2 x)^{5}(3-x)^{4}$. $\mbox{ (5 marks)}$

2.Using binomial theorem, find the coefficient of $x^{2}$ in the expansion of $\left(3+x-2 x^{2}\right)^{5}$. $\mbox{ (5 marks)}$

3.If the coefficient of $x^{2}$ in the expansion of $(4+k x)(2-x)^{6}$ is 384 , find the value of $k$ and the coefficient of $x^{3}$ in that expansion. $\mbox{ (5 marks)}$

4.Find the term independent of $x$ in the expansion of $\left(5 x+\frac{5}{x}\right)^{6}\left(5 x-\frac{5}{x}\right)^{6}$. $\mbox{ (5 marks)}$

5.In the expansion of $\left(x^{2}-\frac{2}{x}\right)^{n}$ in descending power of $x$, the $5^{\text {th }}$ term is independent of $x .$ Find the value of $n$ and the $4^{\text {th }}$ term. $\mbox{ (5 marks)}$

6.In the binomial expansion of $\left(1+\frac{1}{4}\right)^{n}$, the third term is twice the fourth term. Calculate the value of $n$. $\mbox{ (5 marks)}$

7.In the expansion of $\left(1+\frac{1}{3}\right)^{n}$, the third term and the fourth term are in the ratio $3: 2$. Find the value of $n$ and the middle term of that expansion. $\mbox{ (5 marks)}$

8.If the $2^{\text {nd }}$ and the $3^{r d}$ terms in $(a+b)^{n}$ are in the same ratio as the $3^{\text {rd }}$ and the $4^{\text {th }}$ terms in $(a+b)^{n+3}$, find the value of $n .$ Calculate also the middle term of $(a+b)^{n+3}$. $\mbox{ (5 marks)}$

$\quad\;$$\, 1.(1+2 x)^{5}=1+10 x+40 x^{2}+\cdots \quad(3-x)^{4}=81-108 x+54 x^{2}+\cdots ; 2214 2.-540 3.k=3 ; 80 4.-4 \times 5^{13}=-4882812500 5.x=6 ;-160 x^{3} 6.x=8 7.x=8 ; \frac{70}{81} 8.n=5 ; 70 a^{4} b^{4} ## Group (2010) \quad\;\,$$\,$
1.Find the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(1-x)^{5}(1+x)^{6}$. $\text{ (5 marks)}$

2.Given that the coefficient of $x^{2}$ in the expansion of $(2 x+1)^{3}(2+k x)^{5}$ is $-336$, calculate the value of $k$. $\text{ (5 marks)}$

3.Given that the coefficient of $x^{2}$ in the expansion of $(2-x)(1+k x)^{5}$ is 165 , calculate the possible values of $k$. $\text{ (5 marks)}$

4.If the coefficients of $x^{4}$ and $x^{5}$ in the expansion of $(3+k x)^{10}$ are equal, find the value of $k$. $\text{ (5 marks)}$

5.In the expansion of $\left(x^{2}+\frac{a}{x}\right)^{8}, a \neq 0$, the coefficient of $x^{7}$ is four times the coefficient of $x^{10}$. Find the value of $a$. $\text{ (5 marks)}$

6.Evaluate the coefficients of $x^{5}$ and $x^{4}$ in the expansion of $\left(\frac{x}{3}-3\right)^{7}$. Evaluate the coefficient of $x^{5}$ in the expansion of $\left(\frac{x}{3}-3\right)^{7}(x+3)$. $\text{ (5 marks)}$

7.If the coefficient of $x^{4}$ in the expansion of $(3+2 x)^{6}$ is equal to the coefficient of $x^{4}$ in the expansion of $(k+3 x)^{6}$, find $k$. Find the $4^{\text {th }}$ term in the expansion of the second bit omial. $\text{ (5 marks)}$

8.Find the sum of the coefficients of the fourth term and the sixth term in the expansion of $\left(\frac{2 x}{3}-\frac{1}{x}\right)^{8}$. $\text{ (5 marks)}$

9.Write down the third and fourth terms in the expansion of $(a+b x)^{n}$. If these terms are equal find the value of $a$ in terms of $n, b$ and $x$. $\text{ (5 marks)}$

10.Write down the third and fourth terms in the expansion of $(a-b x)^{n}$. If these terms are equal find the value of $a$ in terms of $n, b$ and $x$. $\text{ (5 marks)}$

11.Find the middle term and the term independent of $x$ in the expansion of $\left(x-\frac{1}{\sqrt{x}}\right)^{12}$. $\text{ (5 marks)}$

$\quad\;\,$$\,$
1.$-5 /-5$
2.$-3$
3.$\frac{-11}{4}, 3$
4.$\frac{5}{2} \quad$
5.2
6.$\frac{7}{9}, \frac{-35}{3},-\frac{28}{3}$
7.$\frac{\pm 4}{3} ; \pm 1280 x^{3}$
8.$-\frac{5824}{243}$
9.$a=\left(\frac{n-2}{3}\right) b x$
10.$a=-\left(\frac{n-2}{3}\right) b x$
11.$924 x^{3} ;495$