(a) Show that $\displaystyle \sqrt{4-x} = A\left(1 - \frac{x}{B}\right)^2$ where $\displaystyle A$ and $\displaystyle B$ are integers to be found.

(b) Hence expand $\displaystyle \sqrt{4-x}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$
Give each coefficient as an exact fraction in its lowest terms.

(c) Use your expansion with a suitable value of $\displaystyle x$ to obtain an estimate of $\displaystyle \frac{\sqrt{305}}{10}$
Give your answer correct to 5 decimal places.

Expand $\displaystyle \left(1 - \frac{x}{4}\right)^{-2}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$

Where appropriate, express each coefficient as an exact fraction in its lowest terms.

(a) Show that

The expansion of $\displaystyle \frac{a}{\sqrt{4+bx}}$ in ascending powers of $\displaystyle x$ can be written as

where $\displaystyle P$, $\displaystyle Q$, $\displaystyle R$ and $\displaystyle S$ are rational numbers.

(b) Show that $\displaystyle Q = -\frac{ab}{16}$ and $\displaystyle S = -\frac{5ab^3}{2048}$

and find $\displaystyle P$ and $\displaystyle R$ in terms of $\displaystyle a$ and $\displaystyle b$, as fractions in their lowest terms.

Given that $\displaystyle Q = \frac{128}{5}S$ and $\displaystyle R = \frac{9}{256}$

(c) show that $\displaystyle a = 3$ and $\displaystyle b = 1$

(d) Hence, using an appropriate value of $\displaystyle x$, find, to 3 decimal places, an approximate

value for $\displaystyle \frac{\sqrt{6}}{2}$

Given that

is the binomial expansion, in ascending powers of $\displaystyle x$, of $\displaystyle (1 + Ax)^n$

where $\displaystyle A$ and $\displaystyle n$ are rational numbers,

(a) find the value of $\displaystyle A$ and the value of $\displaystyle n$

(b) Hence find the value of the coefficient of $\displaystyle x^3$

Give your answer in the form $\displaystyle \frac{p}{q}$ where $\displaystyle p$ is a prime number and $\displaystyle q$ is an integer.

(a) Expand $\displaystyle (1 + 2x^2)^{-\frac{3}{4}}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^6$

Express each coefficient as an exact fraction in its lowest terms.

(b) Obtain a series expansion for $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^5$

Give each coefficient in terms of $\displaystyle k$ where appropriate.

The coefficient of the term in $\displaystyle x^5$ is fourteen times the coefficient of the term in $\displaystyle x^2$

(c) Find the value of $\displaystyle k$

(a) Expand $\displaystyle \frac{2}{\sqrt{1+3x}}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$

Express each coefficient as a fraction in its simplest terms where appropriate.

(b) State the range of values of $\displaystyle x$ for which the expansion is valid.

Given that $\displaystyle (2+3x)^{-1}$ can be expressed in the form $\displaystyle p(1+qx)^{-1}$, where $\displaystyle p$ and $\displaystyle q$ are constants,

(a) find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(b) Hence expand $\displaystyle (2+3x)^{-1}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

(c) Obtain a series expansion for $\displaystyle f(x)$, in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

(d) Hence use algebraic integration to obtain an estimate, to 4 decimal places, of

(a) Using the binomial expansion, or otherwise, find the complete expansion of

The quadratic equation

has roots $\displaystyle \alpha$ and $\displaystyle \beta$.

(b) Without solving the equation, find the value of

(c) Hence, form a quadratic equation with integer coefficients that has roots

(a) Write

in the form $\displaystyle a(1-bx)^{-3}$, where $\displaystyle a$ and $\displaystyle b$ are fractions in their lowest terms.

(b) Expand $\displaystyle \frac{3}{(3-x)^3}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Express each coefficient as a fraction in its lowest terms.

(c) (i) Use a suitable value of $\displaystyle x$ with your expansion in part (b), to obtain an approximation for $\displaystyle \frac{24}{125}$ to 5 decimal places.

(ii) Find the percentage error, to 2 decimal places, of your approximation from the actual value.

(a) Expand $\displaystyle (1+ax)^n$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Express each coefficient of $\displaystyle x$ in terms of $\displaystyle a$ and $\displaystyle n$, where $\displaystyle a$ and $\displaystyle n$ are constants and $\displaystyle n>2$.

The coefficient of $\displaystyle x$ is $\displaystyle 15$ and the coefficient of $\displaystyle x^2$ is equal to the coefficient of $\displaystyle x^3$.

(b) Find the value of $\displaystyle a$ and the value of $\displaystyle n$.

(c) Find the coefficient of $\displaystyle x^3$.

(a) Expand $\displaystyle \sqrt{1-x}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Give each coefficient as an exact fraction in its lowest terms.

(b) Using your expansion with a suitable value of $\displaystyle x$, obtain an approximation, to 6 decimal places, of $\displaystyle \sqrt{0.92}$.

(c) Hence find an approximation, to 5 decimal places, of $\displaystyle \sqrt{23}$.

(a) Expand

in ascending powers of $\displaystyle x$, up to and including the term in $\displaystyle x^4$. Give each term in its simplest form.

The coefficient of $\displaystyle x^r$ in the expansion is $\displaystyle c_r$. Given that

(b) find the value of $\displaystyle p$.

(a) Expand

in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^6$, expressing each coefficient as an exact fraction in its lowest terms.

(b) State the range of values of $\displaystyle x$ for which your expansion is valid.

The function $\displaystyle f$ is defined by

(c) Obtain a series expansion for $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^6$.

Give each coefficient in terms of $\displaystyle k$ where appropriate.

Given that the coefficient of $\displaystyle x^4$ in the series expansion of $\displaystyle f(x)$ is zero,

(d) find the value of $\displaystyle k$.

(e) Hence use algebraic integration to obtain an estimate, to 4 decimal places, of

(a) Expand simplifying each $\displaystyle -$ term in ascending as far as possible. powers of $\displaystyle x$ up to and including the term in $\displaystyle X^{3},$

(b) Write down the range of values of $\displaystyle x$ for which your expansion is valid.

(c) term Find in the series simplifying expansion each of term f(x) in as ascending far as possible. powers of $\displaystyle x$ up to and including the

The positive region y-axis is and bounded the line by with the curve equation with equation $\displaystyle y =$ f(x), the positive x-axis, the

(d) Using of giving your expansion your answer of f(x) to and decimal algebraic places. integration, find an estimate for the area

Given that

can be expressed in the form

where $\displaystyle p$ and $\displaystyle q$ are constants,

(a) find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(b) Hence expand $\displaystyle \sqrt{9-x}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

Using the expansion you found in part (b) with a suitable value of $\displaystyle x$,

(c) find an estimate to 5 decimal places for the value of