4.05c Partial fractions: extended to quadratic denominators

8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 3 } - x - 2 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form $$A + \frac { B } { x - 1 } + \frac { C x + D } { x ^ { 2 } + 1 }$$ where \(A , B , C\) and \(D\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 1\).

1. (a) Express

$$\frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { n ( n + 2 ) } { a ( 2 n + b ) ( 2 n + c ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. (a) Express \(\frac { 1 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   (b) Hence prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$ (c) Find the exact value of $$\sum _ { r = 9 } ^ { 25 } \frac { 5 } { 4 r ^ { 2 } - 1 }$$

2. (a) Write \(\frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }\) in partial fractions.
(b) Hence find $$\sum _ { r = 2 } ^ { n } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) } \quad n \geqslant 2$$ giving your answer in the form $$\frac { a n ^ { 2 } + b n + c } { 2 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(c) Hence determine the exact value of $$\sum _ { r = 15 } ^ { 20 } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }$$

1. (a) Express

$$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all positive integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where \(p\) and \(q\) are integers to be determined.
(c) Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$

1. (a) Express \(\frac { 1 } { ( r + 3 ) ( r + 4 ) }\) in partial fractions.
   (b) Hence, using the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 4 ) } = \frac { n } { a ( n + a ) }$$ where \(a\) is a constant to be found.
(c) Find the exact value of \(\sum _ { r = 15 } ^ { 30 } \frac { 1 } { ( r + 3 ) ( r + 4 ) }\) uestion 1 continued \includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_29_40_182_1914} \includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_33_37_201_1914}
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2. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 n } { 3 ( 2 n + 3 ) }\).

1. (a) Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
   (b) Hence show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.

5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),

1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
3. Find the coordinates of the point of contact each of these tangents makes with \(H\).

5 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

1 Express \(\frac { 2 x + 3 } { ( x + 3 ) \left( x ^ { 2 } + 9 \right) }\) in partial fractions.

2 It is given that \(\mathrm { f } ( x ) = \frac { x ( x - 1 ) } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) }\). Express \(\mathrm { f } ( x )\) in partial fractions and hence find the exact value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\).

3
- 2
0 \end{array} \right) .$$ Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\). Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\). 2 Show that the difference between the squares of consecutive integers is an odd integer. Find the sum to \(n\) terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } + \ldots$$ and deduce the sum to infinity of the series. 3 It is given that \(\phi ( n ) = 5 ^ { n } ( 4 n + 1 ) - 1\), for \(n = 1,2,3 , \ldots\). Prove, by mathematical induction, that \(\phi ( n )\) is divisible by 8 , for every positive integer \(n\).

7

1. Determine the values of \(A , B , C\) and \(D\) such that \(\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }\).
2. In this question you must show detailed reasoning. Hence determine the exact value of \(\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).

5 Express \(\frac { 12 x ^ { 3 } } { ( 2 x + 1 ) \left( 2 x ^ { 2 } + 1 \right) }\) using partial fractions.

3 In this question you must show detailed reasoning.

1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).

7 In this question you must show detailed reasoning.

1. Find the values of \(A , B\) and \(C\) for which \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\).
2. Hence express \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\) using partial fractions.
3. Using your answer to part (b), determine \(\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x\) expressing your answer in the form \(a + \ln b + c \pi\) where \(a\) is an integer, and \(b\) and \(c\) are both rational.

3

1. Using partial fractions and the method of differences, show that $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots + \frac { 1 } { \mathrm { n } ( \mathrm { n } + 2 ) } = \frac { 3 } { 4 } - \frac { \mathrm { an } + \mathrm { b } } { 2 ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) }$$ where \(a\) and \(b\) are integers to be determined.
2. Deduce the sum to infinity of the series. $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$

1. (a) Express as partial fractions

$$\frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) }$$ (b) Hence, show that $$\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) } d x = \ln ( a \sqrt { 2 } ) + b \pi$$ where \(a\) and \(b\) are constants to be determined.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
Given that

• the eccentricity of \(H\) is the reciprocal of the eccentricity of \(E\)
• the coordinates of the foci of \(H\) are the same as the coordinates of the foci of \(E\) determine
  1. the value of \(a\)
  2. the value of \(b\)

1

1. Given that $$\frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) } = A + B \left( \frac { 1 } { r } - \frac { 1 } { r + 1 } \right)$$ find the values of \(A\) and \(B\).
2. Hence find the value of $$\sum _ { r = 1 } ^ { 99 } \frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) }$$

8

1. 1. Given that \(z ^ { 6 } - 4 z ^ { 3 } + 8 = 0\), show that \(z ^ { 3 } = 2 \pm 2 \mathrm { i }\).
   2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. Show that, for any real values of \(k\) and \(\theta\), $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
3. Express \(z ^ { 6 } - 4 z ^ { 3 } + 8\) as the product of three quadratic factors with real coefficients.

8

1. Express \(\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\) where \(A\) and \(B\) are constants.
2. Hence find the exact value of \(\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.