Improper fractions requiring division

2 Express \(\frac { 6 x ^ { 2 } - 9 x - 16 } { 2 x ^ { 2 } - 5 x - 12 }\) in partial fractions.

5 Express \(\frac { 6 x ^ { 2 } - 2 x + 2 } { ( x - 1 ) ( 2 x + 1 ) }\) in partial fractions.
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4 Express \(\frac { 4 x ^ { 2 } - 13 x + 13 } { ( 2 x - 1 ) ( x - 3 ) }\) in partial fractions.

3. Express \(\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }\) in partial fractions.

7

1. Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12\) is divided by \(x ^ { 2 } + 4\).
2. Hence express \(\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } + 4 }\), where the values of the constants \(A , B , C\) and \(D\) are to be stated.
3. Use the result of part (ii) to find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x\).

10

1. Express \(\frac { x + 8 } { x ( x + 2 ) }\) in partial fractions.
2. By first using division, express \(\frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\) in the form \(P + \frac { Q } { x } + \frac { R } { x + 2 }\). A curve has parametric equations \(x = \frac { 2 t } { 1 - t } , y = 3 t + \frac { 4 } { t }\).
3. Show that the cartesian equation of the curve is \(y = \frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\).
4. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). Give your answer in the form \(L + M \ln 2 + N \ln 3\).

3 Express \(\frac { 2 x ^ { 3 } + x + 12 } { ( 2 x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

4 Express \(\frac { x ^ { 3 } } { ( x - 2 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

3

1. Express \(\frac { 3 + 13 x - 6 x ^ { 2 } } { 2 x - 3 }\) in the form \(A x + B + \frac { C } { 2 x - 3 }\).
2. Show that \(\int _ { 3 } ^ { 6 } \frac { 3 + 13 x - 6 x ^ { 2 } } { 2 x - 3 } \mathrm {~d} x = p + q \ln 3\), where \(p\) and \(q\) are rational numbers.
   [0pt] [4 marks]

9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (b) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(c) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
7. continued
8. $$f ( x ) = \frac { x ( 3 x - 7 ) } { ( 1 - x ) ( 1 - 3 x ) } , | x | < \frac { 1 } { 3 }$$ (a) Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = A + \frac { B } { 1 - x } + \frac { C } { 1 - 3 x }$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational.
(c) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
8. continued
8. continued

12. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
   (3)
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.