$\def\frac{\dfrac}$

$\def\D{\displaystyle}\def\iixi#1#2{\D\left(\begin{array}{c}#1\\#2\end{array}\right)}$

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 $\D x^2.$

[8]

2 (CIE 2012, s, paper 22, question 6)

(a) Find the coefficient of $\D x^3$ in the expansion of

(i) $\D (1 - 2x)^7,$ [2]

(ii) $\D (3 + 4x)(1 - 2x)^7.$ [3]

(b) Find the term independent of $\D x$ in the expansion of $\D\left(x+\frac{3}{x^2}\right)^6.$ [3]

3 (CIE 2012, w, paper 11, question 6)

(i) Find the first 3 terms, in descending powers of $\D x,$ in the expansion of $\D \left(x+\frac{2}{x^2}\right)^6.$ [3]

(ii) Hence find the term independent of $\D x$ in the expansion of $\D \left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6.$ [2]

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 $\D p.$ [4]

(ii) Use your value of $\D p$ to find the term independent of $\D x$ in the expansion of $\D (p + x)^6 \left(1-\frac{1}{x}\right)^2.$ [3]

5 (CIE 2012, w, paper 22, question 4)

(i) Find the coefficient of $\D x^5$ in the expansion of $\D (2 - x)^8.$ [2]

(ii) Find the coefficient of $\D x^5$ in the expansion of $\D (l + 2x)(2 - x)^8.$ [3]

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. [2]

(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 $\D n.$ [5]

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 $\D x.$ [3]

(ii) Hence find the coefficient of $\D x^3$ in the expansion of $\D(1+3x)(1-x)(2+x)^6.$ [4]

8 (CIE 2013, w, paper 21, question 6)

(a) (i) Find the coefficient of $\D x^3$ in the expansion of $(1-2x)^6.$ [2]

(ii) Find the coefficient of $\D x^3$ in the expansion of $\D \left(1+\frac{x}{2}\right)(1-2x)^6.$ [3]

(b) Expand $\D \left(2\sqrt{x}+\frac{1}{\sqrt{x}}\right)^4$ in a series of powers of $\D x$ with integer coefficients. [3]

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 $\D b$. [5]

(ii) Hence find the value of the remainder when the expression is divided by $\D 2x + 1.$ [2]

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 $\D r.$ [3]

(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 $\D b.$ [4]

11 (CIE 2014, s, paper 21, question 6)

(a) Find the coefficient of $x^5$ in the expansion of $\D (3-2x)^8.$ [2]

(b) (i) Write down the first three terms in the expansion of $\D (1+2x)^6$ in ascending powers of $\D x.$ [2]

(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 $\D a.$ [4]

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 $\D (1-4x)^5.$ [2]

(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 $\D b.$ [4]

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 $\D p.$ [3]

(ii) Using your value of $\D p,$ find the coefficient of $\D x^2$ in the expansion of $\D (3- x) (2 +px)^6.$ [3]

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 $\D r.$ [5]

(b) Find the value of the term that is independent of $\D x$ in the expansion of $\D \left(2x+\frac{1}{4x^3}\right)^{12}.$ [3]

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 $\D x.$ [3]

(ii) Find the term independent of $\D x$ in the expansion of $\D (2+x^2)^6\left(1-\frac{3}{x^2}\right)^2.$ [3]

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, $\D n.$ [6]

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 $\D q.$ [3]

(b) Find the term that is independent of $\D x$ in the expansion of $\D \left(x-\frac{2}{x^2}\right)^9.$ [3]

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 $\D x.$ [3]

(ii) Find the coefficient of $\D x^2$ in the expansion of $\D (1+ 2x) (2- 3x)^6.$ [2]

19 (CIE 2016, march, paper 22, question 5)

(i) Find, in ascending powers of $x$, the first 3 terms of the expansion of $(3+k x)^{7}$, where $k$ is a constant. Give each term in its simplest form. $[3]$

(ii) Given that, in the expansion of $(3+k x)^{7}$, the coefficient of $x^{2}$ is twice the coefficient of $x$, find the value of $k$.

20 (CIE $2016, \mathrm{~s}$, paper 12 , question 2)

(i) The first 3 terms in the expansion of $\left(2-\dfrac{1}{4 x}\right)^{5}$ are $a+\dfrac{b}{x}+\dfrac{c}{x^{2}}$. Find the value of each of the $[3]$ integers $a, b$ and $c$.

(ii) Hence find the term independent of $x$ in the expansion of $\left(2-\frac{1}{4 x}\right)^{5}(3+4 x)$. [2]

21 (CIE 2016, s, paper 21, question 8)

(a) (i) Use the Binomial Theorem to expand $(a+b)^{4}$, giving each term in its simplest form. $[2]$

(ii) I Ience find the term independent of $x$ in the expansion of $\left(2 x+\frac{1}{5 x}\right)^{4}$. [2]

(b) The coefficient of $x^{3}$ in the expansion of $\left(1+\frac{x}{2}\right)^{n}$ equals $\frac{5 h}{12}$. Find the value of the positive integer $n .$

22 (CIE 2016, w, paper 11, question 4)

(i) Find the first 3 terms in the expansion of $\left(2 x^{2}-\frac{1}{3 x}\right)^{5}$, in descending powers of $x$. [3]

(ii) Hence find the coefficient of $x^{7}$ in the expansion of $\left(3+\frac{1}{x^{3}}\right)\left(2 x^{2}-\frac{1}{3 x}\right)^{5}$.

23 (CIE $2016, \mathrm{w}$, paper 13 , question 4)

(i) Find, in ascending powers of $x$, the first 3 terms in the expansion of $\left(2-\frac{x}{4}\right)^{6}$. [3]

(ii) Hence find the term independent of $x$ in the expansion of $\left(4+\frac{2}{x}+\frac{3}{x^{2}}\right)\left(2-\frac{x}{4}\right)$. [3]

24 (CIE 2017, march, paper 12, question 3) The first three terms in the expansion of $\left(a+\frac{x}{4}\right)^{5}$ are $32+b x+c x^{2}$. Find the value of each of the constants $a, b$ and $c$.

25 (CIE 2017, s, paper 12, question 4) The first 3 terms in the expansion of $\left(3-\frac{x}{6}\right)^{n}$ are $81+a x+b x^{2}$. Find the value of each of the constants $n, a$ and $b .$

26 (CIE 2017, s, paper 21, question 5)

(i) Given that $a$ is a constant, expand $(2+a x)^{4}$, in ascending powers of $x$, simplifying each term of your expansion. [2]

Given also that the coefficient of $x^{2}$ is equal to the coefficient of $x^{3}$,

(ii) show that $a=3$, $[1]$

(iii) use your expansion to show that the value of $1.97^{4}$ is $15.1$ to 1 decimal place.

27 (CIE 2017, s, paper 23, question 6) The first three terms of the binomial expansion of $(2-a x)^{n}$ are $64-16 h x+100 h x^{2}$. Find the value of each of the integers $n, a$ and $b .$

28 (CIE 2017, w, paper 12, question 3)

(i) Find, in ascending powers of $x$, the first 3 terms in the expansion of $\left(2-\frac{x^{2}}{4}\right)^{5}$ [3]

(ii) Hence find the term independent of $x$ in the expansion of $\left(2-\frac{x^{2}}{4}\right)^{5}\left(\frac{1}{x}-\frac{3}{x^{2}}\right)^{2}$. [3]

29 (CIE 2017, w, paper 13, question 7)

(i) Find, in ascending powers of $x$, the first 3 terms in the expansion of $\left(2-\frac{x^{2}}{4}\right)^{6}$. Give each term in its simplest form. [3]

(ii) IIcnce find the cocfficient of $x^{2}$ in the cxpansion of $\left(2-\frac{x^{2}}{4}\right)^{6}\left(\frac{1}{x}+x\right)^{2}$. $[4]$

30 (CIE 2017, w, paper 21, question 9)

(i) Expand $(1+x)^{4}$, simplifying all cocfficicnts.

(ii) Expand $(6-x)^{4}$, simplifying all cocfficients. [2]

(iii) Hence express $(6-x)^{4}-(1+x)^{4}=175$ in the form $a x^{3}+b x^{2}+c x+d=0$, where $a, b, c$ and $d$ are integers. [2]

(iv) Show that $x=2$ is a solution of the equation in part (iii) and show that this equation has no other real roots. $\quad[5]$

31 (CIE 2018, march, paper 12, question 5) The first 3 terms in the expansion of $(2+a x)^{n}$ are equal to $1024-1280 x+b x^{2}$, where $n, a$ and $b$ are constants.

(i) Find the value of each of $n, a$ and $b$. $[5]$

(ii) Hence find the term independent of $x$ in the expansion of $(2+a x)^{n}\left(x-\frac{1}{x}\right)^{2}$. [3]

32 (CIE 2018, s, paper 11, question 9)

(i) Find the first 3 terns in the expansion of $\left(2x-\dfrac{1}{16x}\right)^8$ in descending powers of $x$. [3]

(ii) Hence find the coefficient of $x^{4}$ in the expansion of $\left( 2 x-\frac{1}{16 x}\right)^{8}\left(\frac{1}{x^{2}}+1\right)^{2}$. [3]

33 (CIE 2018, s, paper 12, question 5)

(i) The first three terms in the expansion of $\left(3-\frac{1}{9 x}\right)^{9}$ can he written as $a+\frac{b}{x}+\frac{c}{x^2}$, Find the value of each of the constants $a,b $ and $c$. [3]

(ii) Use your values of $a,b$ and $c$ to find the term independent of $x$ in the expansion of

$\left(3-\frac{1}{9 x}\right)^{5}(2+9 x)^{2}.$ [3]

Answers

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) $\D 1008$

19. (i) $2187+5103 k x+5103 k^{2} x^{2}$

(ii) $k=2$

20. (i) $32-20 / x+5 / x^{2}$

(ii) 16

21. (ai) $a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}$

(ii) $24 / 25$ (b) $\mathrm{n}=6$

22. (i) $32 x^{10}-80 x^{7} / 3+80 x^{4} / 9$

(ii) $-48$

23. (i) $64-48 x+15 x^{2}$

(ii) 205

24. $a=2, b=20, c=5$

25. $a=-18, b=3 / 2$

26. (i) $16+32 a x+24 a^{2} x^{2}+8 a^{3} x^{3}$

$a^{4} x^{4}$

27. $n=6, a=5, b=60$

28. (i) $32-20 x^{2}+5 x^{4}$

(ii) 25

29. (i) $64-48 x^{2}+15 x^{4}$

(ii) $-17$

30. (i) $1+4 x+6 x^2+4 x^3+x^4$

(ii) $1296-864x+216 x^{2} -24 x^{3}+x^{4}$

(iii) $28 x^{3} -210 x^{2}+868 x -1120=0$

31. (i) $n=10, a=-1 / 4, b=720$

(ii) $-1328$

32. $256 x^{8}-64 x^{6}+7 x^{4}$

135

33. (i) $a=242, b=-45, c=10 / 3$

(ii) $-378$

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