Generalised Binomial Theorem and Partial Fractions

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are related by an equation of the form $$y = a x + \frac { b } { x + 2 }$$ where \(a\) and \(b\) are constants.

1. The variables \(X\) and \(Y\) are defined by \(X = x ( x + 2 ) , Y = y ( x + 2 )\). Show that \(Y = a X + b\).
2. The following approximate values of \(x\) and \(y\) have been found:

   \(x\)	1	2	3	4
   \(y\)	0.40	1.43	2.40	3.35

   1. Complete the table in Figure 1, showing values of \(X\) and \(Y\).
   2. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
   3. Estimate the values of \(a\) and \(b\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).

6

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. State the set of values of \(x\) for which the expansion is valid.

2

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - 1 }\) up to the term in \(x ^ { 3 }\).
   2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { 1 + 3 x }\) up to the term in \(x ^ { 3 }\).
2. Express \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) in partial fractions.
3. 1. Find the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) up to the term in \(x ^ { 3 }\).
   2. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) is valid.

9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.

1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]

Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).

1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]

Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

6 Let \(\mathrm { f } ( x ) = \frac { 9 x ^ { 2 } + 4 } { ( 2 x + 1 ) ( x - 2 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that, when \(x\) is sufficiently small for \(x ^ { 3 }\) and higher powers to be neglected, $$f ( x ) = 1 - x + 5 x ^ { 2 }$$

Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).

1. Express \(f(x)\) in partial fractions. [5]
2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

1 Find the first three terms in the binomial expansion of \(\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]

3. $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) } , \quad | x | < \frac { 2 } { 3 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$f ( x ) = \frac { A } { ( 3 x + 2 ) } + \frac { B } { ( 3 x + 2 ) ^ { 2 } } + \frac { C } { ( 1 - x ) }$$

1. find the values of \(B\) and \(C\) and show that \(A = 0\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
3. Find the percentage error made in using the series expansion in part (b) to estimate the value of \(\mathrm { f } ( 0.2 )\). Give your answer to 2 significant figures. \section*{LU}

7
Let \(f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\mathrm { f } ( x )\).

$$f(x) = 3 - \frac{x-1}{x-3} + \frac{x+11}{2x^2-5x-3}, \quad |x| < \frac{1}{2}.$$

1. Show that $$f(x) = \frac{4x-1}{2x+1}.$$ [4]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]

4. $$f ( x ) = \frac { 1 - 3 x } { \left( 1 + 3 x ^ { 2 } \right) ( 1 - x ) ^ { 2 } } , x \neq 1$$

1. Find the constants \(A , B , C\) and \(D\) such that $$\mathrm { f } ( x ) \equiv \frac { A x + B } { 1 + 3 x ^ { 2 } } + \frac { C } { 1 - x } + \frac { D } { ( 1 - x ) ^ { 2 } }$$
2. Find a series expansion for \(\mathrm { f } ( x )\) in ascending powers of \(x\) ,up to and including the term in \(x ^ { 4 }\) .
3. Find an equation of the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 0\) .