AS-Level Maths | Binomial Expansion

Question 1

$(a + b)^n = a^n + \dbinom{n}{1}a^{n-1}b + \cdots + \dbinom{n}{r}a^{n-r}b^r + \cdots + b^n$

Find the first four terms in the expansion in ascending powers of $x$ of

(a) $(1 + x)^{10}$

(b) $(1 + 2x)^8$

(c) $\left(1 - \tfrac{1}{2}x\right)^7$

(d) $(2 + 5x)^6$

Question 2

Find the coefficient indicated in the following expansions.

(a) $(1 + x)^{20}$,   coefficient of $x^3$

(b) $(1 - x)^{14}$,   coefficient of $x^4$

(c) $(1 + 4x)^9$,   coefficient of $x^2$

(d) $(1 - 3y)^{14}$,   coefficient of $y^3$

Question 3

Expand each of the following in ascending powers of $x$ up to and including the term in $x^2$.

(a) $(1 + x^2)(1 - 3x)^{10}$

(b) $(1 - 2x)(1 + x)^8$

(c) $(1 + x + x^2)(1 - x)^6$

(d) $(1 + 3x - x^2)(1 + 2x)^7$

Question 4

Find the term independent of $y$ in each of the following expansions.

(a) $\left(y + \dfrac{1}{y}\right)^8$

(b) $\left(2y - \dfrac{1}{2y}\right)^{12}$

(c) $\left(\dfrac{1}{y} + y^2\right)^6$

(d) $\left(3y - \dfrac{1}{y^2}\right)^9$

Question 5

(a) Expand $(1 + x)^6$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each coefficient.

(b) By substituting a suitable value of $x$ into your answer to part (a), obtain an estimate for

    (i) $1.02^6$

    (ii) $0.99^6$

giving your answers to 4 decimal places.

Question 6

(a) Expand $(1 + 2y)^8$ in ascending powers of $y$ up to and including the term in $y^3$, simplifying each coefficient.

(b) By substituting a suitable value of $y$ into your answer to part (a), obtain an estimate for

    (i) $0.98^8$

    (ii) $1.01^8$

giving your answers to 4 decimal places.