$\def\frac{\dfrac}$
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. (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)
Answer(2015-2019)
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. $\quad\mbox{ (5 marks)}$
Answer(2014)
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)
Answer (2013)
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
Group (2012)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 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 $(a+b)^{n+3}$. $\mbox{ (5 marks)}$ |
Answer (2011)
| $\quad\;$ | $\,$ | |
|---|---|---|
| 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 $\left(x-\frac{1}{\sqrt{x}}\right)^{12}$. $\text{ (5 marks)}$ |
Answer (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 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. | $924 x^{3} ;495$ |
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