Binomial (IB Standard Level)

$\newcommand{\D}{\displaystyle}  \def\iixi#1#2{\D\left(\begin{array}{c} #1\\#2 \end{array}\right)}$
1.) Consider the expansion of $\D (x + 2)^{11}.$
(a) Write down the number of terms in this expansion.
(b) Find the term containing $\D x^2.$ (Total 5 marks)
2.) (a) Expand $\D (2 + x)^4$ and simplify your result.
(b) Hence, find the term in $\D x^2$ in $\D (2 + x)^4\left(1+\frac{1}{x^2}\right).$ (Total 6 marks)
3.) Find the term in $\D x^4$ in the expansion of $\D \left(3x^2-\frac{2}{x}\right)^5$. (Total 6 marks)
4.) The fifth term in the expansion of the binomial $\D (a + b)^n$ is given by $\D \iixi{10}{4}p^6(2q)^4.$
(a) Write down the value of $\D n$.
(b) Write down $\D a$ and $\D b,$ in terms of $\D p$ and/or $\D q.$
(c) Write down an expression for the sixth term in the expansion.  (Total 6 marks)
5.) Let $\D f(x) = x^3- 4x + 1.$
Expand $\D (x + h)^3.$
6.) Find the term in $\D x^3$ in the expansion of $\D \left(\frac{2}{3}x-3\right)^8.$ (Total 5 marks)
7.) (a) Expand $\D (x-2)^4$ and simplify your result.
(b) Find the term in $\D x^3$ in $\D (3x + 4)(x-2)^4.$  (Total 6 marks)
8.) Consider the expansion of the expression $\D (x^3-3x)^6.$
(a) Write down the number of terms in this expansion.
(b) Find the term in $\D x^{12}.$ (Total 6 marks)
9.) One of the terms of the expansion of $\D (x + 2y)^{10}$ is $\D ax^8 y^2.$ Find the value of a. (Total 6 marks)
10.) (a) Expand $\D \left(e+\frac{1}{e}\right)^4$ in terms of $\D e.$
(b) Express $\D \left(e+\frac{1}{e}\right)^4+\left(e-\frac{1}{e}\right)^4$ as the sum of three terms.  (Total 6 marks)
11.) Consider the expansion of $\D (x^2- 2)^5.$
(a) Write down the number of terms in this expansion.
(b) The first four terms of the expansion in descending powers of $\D x$ are \[x^{10} -10x^8 + 40x^6 + Ax^4 + ...\] Find the value of A. (Total 6 marks) 
12.) Given that $\D \left(3+\sqrt{7}\right)^3=p+\sqrt{q} $ where $\D p$ and $\D q$ are integers, find
(a) $\D p;$
(b) $\D q.$ (Total 6 marks)
13.) When the expression $\D (2 + ax)^{10}$ is expanded, the coefficient of the term in $\D x^3$ is 414720. Find the value of $\D a.$ (Total 6 marks)
14.) Find the term containing $\D x^3$ in the expansion of $\D (2- 3x)^8.$ (Total 6 marks)
15.) Find the term containing $\D x^{10}$ in the expansion of $\D (5 + 2x^2)^7.$ (Total 6 marks) 
16.) Complete the following expansion. $\D (2 + ax)^4 = 16 + 32ax +\cdots $ (Total 6 marks)
17.) Consider the expansion of $\D \left(3x^2-\frac{1}{x}\right)^9.$
(a) How many terms are there in this expansion?
(b) Find the constant term in this expansion. (Total 6 marks)
18.) Find the coefficient of $\D x^3$ in the expansion of $\D (2- x)^5.$ (Total 6 marks)
19.) Use the binomial theorem to complete this expansion. $\D (3x + 2y)^4 = 81x^4 + 216x^3 y + \cdots $ (Total 4 marks)
20.) Consider the binomial expansion . \[(1+x)^4=1+\iixi{4}{1}x+ \iixi{4}{2}x^2 +\iixi{4}{3}x^3+x^4.\]
(a) By substituting $\D x = 1$ into both sides, or otherwise, evaluate $\D \iixi{4}{1}+ \iixi{4}{2} +\iixi{4}{3}$
(b) Evaluate $\D \iixi{9}{1}+ \iixi{9}{2}+ \iixi{9}{3}+ \iixi{9}{4}+ \iixi{9}{5}+ \iixi{9}{6}+ \iixi{9}{7}+ \iixi{9}{8}.$ (Total 4 marks)
21.) Determine the constant term in the expansion of $\D \left(x-\frac{2}{x^2}\right)^9.$ (Total 4 marks)
22.) Find the coefficient of $\D a^5b^7$ in the expansion of $\D (a + b)^{12}.$ (Total 4 marks)
23.) Find the coefficient of $\D x^5$ in the expansion of $\D (3x - 2)^8.$ (Total 4 marks)
24.) Find the coefficient of $\D a^3b^4$ in the expansion of $(5a + b)^7.$ (Total 4 marks)


Answers


1. $12,28160x^2$
2. $16 32 24 8 1
25x^2$
3. $1080x^4$
4. $n=10,a=p,b=2q,10C5p^5(2q)^5$
5. $x^3+3x^2h+3xh^2+h^3$
6. $-4032x^3$
7. $ x^4-8x^3+24x^2-32x+16;40x^3$
8. $7;-540x^{12}$
9. $180$
10. $e^4+4e^2+6+4/e^2+1/e^4$
    $2e^4+12+2?e^4$
11. $6;A=-80$
12. $p=90,q=34$
13. $3$
14. $-48384x^3$
15. $16800x^{10}$
16. $...+24a^2x^2+8a^3x^3+a^4x^4$
17. $10;2268$
18. $-40$
19. $...+216x^2y^2+96xy^3+16y^4$
20. $14;510$
21. $-672$
22. $792$
23. $-108864$
24. $4375$

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