• A Level Maths

Demo

• A Level
• Binomial (Integer)

1. Expand $$(1+2x)^4$$

$1 + 8x + 24x^2 + 32x^3 + 16x^4$

2. Expand and simplify $$(3-x)^4 + (3+x)^4$$

$162 + 108x^2 + 2x^4$

3. Find, in ascending powers of $x$, the first three terms in the expansion of:

1. $(2+3x)^6$
2. $(3-4x)^5$

1. $64 + 576x + 2160x^2$
2. $243 - 1620x + 4320x^2$

4. Find, in ascending powers of $x$, the first three terms in the expansion of:

1. $(4x-\frac{1}{2})^7$
2. $(2-\frac{1}{4}x)^9$

1. $-\dfrac{1}{128} + \dfrac{7}{16}x - \dfrac{21}{2}x^2$
2. $512 - 576x + 288x^2$

5.

1. Find the first five terms, in ascending powers of $x$, of the expansion of $(1+2x)^{12}$.
2. Hence estimate $1.02^{12}$ to 5 decimal places.

1. $1 + 24x + 264x^2 + 1760x^3 + 7920x^4$
2. $1.26824$

6. In the binomial expansion of $(1+kx)^6$, where $k$ is a constant, the coefficient of $x^3$ is twice as large as the coefficient of $x^2$. Find $k$.

$\dfrac{3}{2}$

7.

1. Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(1+2x)^7$.
2. Hence find the coefficient of $x$ in the expansion of $(1+2x)^7(3+2x)^4$.

1. $1+14x+84x^2+280x^3$
2. $1350$

8.

1. Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(2+x)^9$.
2. Hence find the coefficient of $x^3$ in the expansion of $(1-\frac{1}{8}x)^2(2+x)^9$.

1. $512+2304x+4608x^2+5376x^3$
2. $4260$

9. Given $$(2+kx)^6 = a + bx + bx^2 + cx^3 + ...$$ where $k$ is a non zero constant, find the values of $a$, $b$ and $c$.

$a = 64$, $b = 153.6$, $c = 81.92$

10. Using the first three terms, in ascending powers of $x$, of the binomial expansion of $(2-3x)^{10}$, estimate the value of:

1. $1.97^{10}$
2. $3.94^{10}$

1. $880.768$
2. $901906.432$

11. Find the term which is independent of $x$ in the expansion of $$\left(4x^3-\dfrac{1}{2x}\right)^8$$

$7$

12. Given $$(1+\sqrt{2})^5 = a + b\sqrt{2}$$ find $a$ and $b$.

$a=41$ and $b=29$

13. In the binomial expansion of $\left(k-\dfrac{x}{2}\right)^6$, one of the terms is $960x^2$. Find the coefficient of the $x^3$ term.

$-160$

14. Given that $k$ is a non zero integer and $n$ is a positive integer, and that $$(1+kx)^n = 1 + 40x + 120k^2x^2$$ find $k$ and $n$.

$k=\dfrac{5}{2}$ and $n=16$

15. Given that $k$ is a positive constant and $a$ is a constant, and that $$(2-kx)^8 = 256 + ax + 1008x^2$$ find $k$ and $a$.

$k=\dfrac{3}{4}$ and $a = -768$

16. Find the coefficient of $x^2$ in the expansion of $$(2-3x)^2(1+4x)^7$$

$1017$

17.

1. Find the binomial expansion of $\left(1+\dfrac{1}{4}x\right)^{10}$ in ascending powers of $x$, up to and including the term in $x^3$.
2. Hence estimate $\left(\dfrac{41}{40}\right)^{10}$ to 3 significant figures.

1. $1 + \dfrac{5}{2}x+\dfrac{45}{16}x^2 + \dfrac{15}{8}x^3$
2. $1.28$

18. Find the first four terms, in ascending powers of $x$, of the expansion of $$(3+4x-4x^2)(1+2x)^6$$

$3 + 40x + 224x^2 + 672x^3$

19. Given $$(1+ax)^n = 1-30x+405x^2+bx^3$$ where $n$ is a positive integer and $a$ and $b$ are constants, find $a$, $b$ and $n$.

$n=10$, $a=-3$, $b=-3240$

20. Find the coefficient of $x^5$ in the expansion of $$(1-x)^5(1+x)^6$$

$10$

21. Use the binomial expansion of $(2x-4)^5$ to find the coefficient of:

1. $y^2$ in the expansion of $\left(\dfrac{y+16}{4}\right)^5$
2. $z^8$ in the expansion of $\left(\sqrt{2}z-2\right)^5\left(\sqrt{2}z+2\right)^5$

$-1024 + 2560x - 2560x^2 + 1280x^3 - 320x^4+32x^5$

1. $40$
2. $-320$

22. Given that $k$ is positive and $n$ is a positive integer, and that $$(1+kx)^n = 1 + \dfrac{7}{2}x + ax^2 + ax^3 + ...$$ where $a$ is a non zero constant, find $n$ and $k$.

$n=14$ and $k=\dfrac{1}{4}$

23. The constant term in the expansion of $$x^4\left(2x^2+\dfrac{m}{x}\right)^7$$ is $896$. Find $m$.

$2$

24. Use a suitable expansion to find the first 4 digits of $1003^{80}$.

$1270$