# Binomial (CIE)

$\def\D{\displaystyle}\def\iixi#1#2{\D\left(\begin{array}{c}#1\\#2\end{array}\right)}$
1 (CIE 2012, s, paper 11, question 9)
Find the values of the positive constants $\D p$ and $\D q$ such that, in  binomial expansion of $\D ( p + qx)^{10},$ the coefficient of $\D x^5$ is 252 and the coefficient of $\D x^3$ is 6 times the coefficient of $\D x^2.$

Solution:
$T_{r+1}=\D {\iixi{10}{r} } p^{10-r}(qx)^r= \left[{\iixi{10}{r} } p^{10-r}q^r\right]x^r$
Since the coefficient of $x^5$ is 252, $r=5.$ Thus
$252={\iixi{10}{5}}p^{10-5}q^5={\iixi{10}{5} }p^5q^5$   Therefore $\D p^5q^5=\frac{252}{\iixi{10}{5}}$ and hence $pq=1.\qquad \cdots \qquad (1)$
Coefficient of  $x^3=\iixi{10}{3}p^{10-3}q^3={\iixi{10}{3}}p^7q^3$
Coefficient of $x^2={\iixi{10}{2} }p^{10-2}q^2={\iixi{10}{2}}p^8q^2$
Thus ${\iixi{10}{3}}p^7q^3=6\times {\iixi{10}{2}}p^8q^2$   $\cdots$ (2)

Multiply (1) and (2),
${\iixi{10}{3}}p^8q^4=6\times {\iixi{10}{2}}p^8q^2$ . Thus $q=\D\frac 32$ and $p=\D\frac 23.$

Solution:
$T_{r+1}=\D {\iixi{10}{r} } p^{10-r}(qx)^r= \left[{\iixi{10}{r} } p^{10-r}q^r\right]x^r$
Since the coefficient of $x^5$ is 252, $r=5.$ Thus
$252={\iixi{10}{5}}p^{10-5}q^5={\iixi{10}{5} }p^5q^5$   Therefore $\D p^5q^5=\frac{252}{\iixi{10}{5}}$ and hence $pq=1.\qquad \cdots \qquad (1)$
Coefficient of  $x^3=\iixi{10}{3}p^{10-3}q^3={\iixi{10}{3}}p^7q^3$
Coefficient of $x^2={\iixi{10}{2} }p^{10-2}q^2={\iixi{10}{2}}p^8q^2$
Thus ${\iixi{10}{3}}p^7q^3=6\times {\iixi{10}{2}}p^8q^2$   $\cdots$ (2)

Multiply (1) and (2),
${\iixi{10}{3}}p^8q^4=6\times {\iixi{10}{2}}p^8q^2$ . Thus $q=\D\frac 32$ and $p=\D\frac 23.$

2 (CIE 2012, s, paper 22, question 6)
(a) Find the coefficient of $\D x^3$ in the expansion of
(i) $\D (1 - 2x)^7,$ 
(ii) $\D (3 + 4x)(1 - 2x)^7.$ 
(b) Find the term independent of $\D x$ in the expansion of $\D\left(x+\frac{3}{x^2}\right)^6.$ 

3 (CIE 2012, w, paper 11, question 6)
(i) Find the first 3 terms, in descending powers of $\D x,$ in the expansion of $\D \left(x+\frac{2}{x^2}\right)^6.$ 
(ii) Hence find the term independent of $\D x$ in the expansion of $\D \left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6.$  

4 (CIE 2012, w, paper 13, question 6)
In the expansion of $\D (p + x)^6,$ where $\D p$ is a positive integer, the coefficient of $\D x^2$ is equal to 1.5 times the coefficient of $\D x^3.$
(i) Find the value of $\D p.$ 
(ii) Use your value of $\D p$ to find the term independent of $\D x$ in the expansion of $\D (p + x)^6 \left(1-\frac{1}{x}\right)^2.$ 

5 (CIE 2012, w, paper 22, question 4)
(i) Find the coefficient of $\D x^5$ in the expansion of $\D (2 - x)^8.$ 
(ii) Find the coefficient of $\D x^5$ in the expansion of $\D (l + 2x)(2 - x)^8.$ 

6 (CIE 2013, s, paper 12, question 9)
(i) Given that $\D n$ is a positive integer, find the first 3 terms in the expansion of $\D\left(1+\frac{1}{2}x\right)^n.$  in ascending powers of x. 
(ii) Given that the coefficient of $\D x^2$ in the expansion of $\D(1 - x) \left(1+\frac{1}{2}x\right)^n$ is $\D\frac{25}{4},$  find the value of $\D n.$ 

7 (CIE 2013, s, paper 21, question 7)
(i) Find the first four terms in the expansion of $\D (2+ x)^6$  in ascending powers of $\D x.$ 
(ii) Hence find the coefficient of $\D x^3$ in the expansion of  $\D(1+3x)(1-x)(2+x)^6.$ 

8 (CIE 2013, w, paper 21, question 6)
(a) (i) Find the coefficient of $\D x^3$ in the expansion of $(1-2x)^6.$ 
(ii) Find the coefficient of $\D x^3$ in the expansion of $\D \left(1+\frac{x}{2}\right)(1-2x)^6.$ 
(b) Expand $\D \left(2\sqrt{x}+\frac{1}{\sqrt{x}}\right)^4$ in a series of powers of $\D x$ with integer coefficients. 

9 (CIE 2013, w, paper 23, question 6)
The expression $\D 2x^3 + ax^2 + bx + 21$ has a factor $\D x + 3$ and leaves a remainder of 65 when divided by $\D x - 2.$
(i) Find the value of $\D a$ and of $\D b$. 
(ii) Hence find the value of the remainder when the expression is divided by $\D 2x + 1.$ 

10 (CIE 2014, s, paper 13, question 5)
(i) The first three terms in the expansion of $\D (2 - 5x)^6 ,$ in ascending powers of $\D x,$ are $\D p + qx + rx^2 .$ Find the value of each of the integers $\D p, q$ and $\D r.$ 
(ii) In the expansion of $\D (2 - 5x)^6 (a + bx)^3 ,$ the constant term is equal to 512 and the coefficient of $\D x$ is zero. Find the value of each of the constants $\D a$ and $\D b.$ 

11 (CIE 2014, s, paper 21, question 6)
(a) Find the coefficient of $x^5$ in the expansion of $\D (3-2x)^8.$ 
(b) (i) Write down the first three terms in the expansion of $\D (1+2x)^6$ in ascending powers of $\D x.$ 
(ii) In the expansion of $\D (1+ax)(1+2x)^6,$  the coefficient of $\D x^2$ is 1.5 times the coefficient of $\D x.$ Find the value of the constant $\D a.$ 

12 (CIE 2014, s, paper 22, question 5)
(i) Find and simplify the first three terms of the expansion, in ascending powers of $\D x,$ of $\D (1-4x)^5.$ 
(ii) The first three terms in the expansion of $\D (1-4x)^5(1+ax+bx^2)$  are $\D 1- 23x+ 222x^2$.  Find the value of each of the constants $\D a$ and $\D b.$ 

13 (CIE 2014, w, paper 11, question 6)
(i) Given that the coefficient of $\D x^2$ in the expansion of $\D(2+ px)^6$ is 60, find the value of the positive constant $\D p.$ 
(ii) Using your value of $\D p,$ find the coefficient of $\D x^2$ in the expansion of $\D (3- x) (2 +px)^6.$ 

14 (CIE 2014, w, paper 13, question 9)
(a) Given that the first 3 terms in the expansion of $\D (5-qx)^p$ are $\D 625- 1500x +rx^2,$ find the value of each of the integers $\D p, q$ and $\D r.$ 
(b) Find the value of the term that is independent of $\D x$ in the expansion of $\D \left(2x+\frac{1}{4x^3}\right)^{12}.$  

15 (CIE 2015, s, paper 11, question 3)
(i) Find the first 4 terms in the expansion of $\D (2+x^2)^6$ in ascending powers of $\D x.$ 
(ii) Find the term independent of $\D x$ in the expansion of $\D (2+x^2)^6\left(1-\frac{3}{x^2}\right)^2.$  

16 (CIE 2015, s, paper 22, question 7)
In the expansion of $\D (1+2x)^n$ , the coefficient of $\D x^4$ is ten times the coefficient of $\D x^2.$ Find the value of the positive integer, $\D n.$ 

17 (CIE 2015, w, paper 13, question 8)
(a) Given that the first 4 terms in the expansion of $\D (2 +kx)^8$  are $\D 256+ 256x+ px^2+ qx^3,$ find the value of $\D k,$ of $\D p$ and of $\D q.$ 
(b) Find the term that is independent of $\D x$ in the expansion of $\D \left(x-\frac{2}{x^2}\right)^9.$ 

18 (CIE 2015, w, paper 21, question 2)
(i) Find, in the simplest form, the first 3 terms of the expansion of $\D (2 -3x)^6,$  in ascending powers of $\D x.$ 
(ii) Find the coefficient of $\D x^2$ in the expansion of $\D (1+ 2x) (2- 3x)^6.$ 

1. $\D p = 2/3; q = 3/2$
2. (a) $\D -280;-504$ (b) $\D 135$
3. (i) $\D x^6 + 12x^3 + 60 + \cdots$
(ii) $\D 72$
4. (i) $\D p = 2,$(ii) $\D -80$
5. (i) $\D -448$ (ii) $\D 1792$
6. (i) $\D 1+n(x/2)+\frac{n(n-1)}{2} (x/2)^2$
(ii) $\D n = 10$
7. (i) $\D 64 + 192x + 240x^2 + 160x^3$
(ii) $\D 64$
8. (a)(i)$\D -160$ (ii) $\D -130$
(b) $\D 16x^2 + 32x + 24 + \frac{8}{x}+\frac{1}{x^2}$
9. (i) $\D a = 5; b = 4$(ii)$\D 20$
10. (i) $\D 64 - 960x + 6000x^2$
(ii) $\D a = 2; b = 10$
11. (a) $\D -48384$
(b)(i) $\D 1 + 12x + 60x^2$
(ii) $\D -4$
12. (i) $\D 1 - 20x + 160x^2$
(ii) $\D a = -3; b = 2$
13. (i) $\D p = 1/2$ (ii) $\D 84$
14. (a) $\D p = 4; q = 3; r = 1350$ (b) $\D 1760$
15. (i) $\D 64 + 192x^2 + 240x^4 + 160x^6,$ (ii) $\D 1072$
16. $\D n = 8$
17. (a) $\D k = 1/4; p = 112; q = 28$
(b) $\D -672$
18. (i) $\D 64 - 576x + 2160x^2$
(ii) $\D 1008$