Partial fractions then binomial expansion

8

1. Express \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

9

1. Express \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 12 } { ( x + 1 ) ( x - 3 ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

10 Let \(\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

8 Let \(f ( x ) = \frac { 16 - 17 x } { ( 2 + x ) ( 3 - x ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

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 \(x ^ { 3 }\).

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

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 \(x ^ { 2 }\).

8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).

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 \(x ^ { 2 }\).

9

1. Express \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

9

1. Express \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

8

1. Express \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

8

1. Express \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

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

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 \(x ^ { 3 }\).

9

1. Express \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

8

1. Express \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in partial fractions.
2. Hence expand \(\frac { 7 x ^ { 2 } + 8 } { ( 1 + x ) ^ { 2 } ( 2 - 3 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

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

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 \(x ^ { 2 }\).

8 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }\).

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 \(x ^ { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 14 - 3 x + 2 x ^ { 2 } } { ( 2 + x ) \left( 3 + x ^ { 2 } \right) }\).

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 \(x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{459b8403-a481-4ece-88c0-e7600a47c8e4-14_292_732_264_705} The diagram shows a trapezium \(A B C D\) in which \(A D = B C = r\) and \(A B = 2 r\). The acute angles \(B A D\) and \(A B C\) are both equal to \(x\) radians. Circular arcs of radius \(r\) with centres \(A\) and \(B\) meet at \(M\), the midpoint of \(A B\).

7 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + 8 x - 3 } { ( x - 2 ) \left( 2 x ^ { 2 } + 3 \right) }\).

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 \(x ^ { 2 }\).

10 Let \(\mathrm { f } ( x ) = \frac { 21 - 8 x - 2 x ^ { 2 } } { ( 1 + 2 x ) ( 3 - x ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).

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 \(x ^ { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 8 + 5 x + 12 x ^ { 2 } } { ( 1 - x ) ( 2 + 3 x ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

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

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 \(x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552} In the diagram, \(O A B C D\) is a solid figure in which \(O A = O B = 4\) units and \(O D = 3\) units. The edge \(O D\) is vertical, \(D C\) is parallel to \(O B\) and \(D C = 1\) unit. The base, \(O A B\), is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(B C\) is such that \(C N = 2 N B\).
   1. Express vectors \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
   2. Calculate the angle in degrees between the directions of \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\).
   3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\sqrt { \frac { 22 } { 5 } }\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3. (a) Express \(\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }\) in partial fractions.
(b) Hence, or otherwise, find the series expansion of $$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.

2. $$f ( x ) = \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ Given that, for \(x \neq \frac { 1 } { 2 } , \quad \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } = \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 1 - 2 x ) ^ { 2 } } , \quad\) where \(A\) and \(B\) are constants,

1. find the values of \(A\) and \(B\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
   (6)