1.02y Partial fractions: decompose rational functions

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).

10 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\). State the equations of the asymptotes of \(C\). Show that \(y \leqslant \frac { 25 } { 12 }\) at all points of \(C\). Find the coordinates of any stationary points of \(C\). Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.

The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where \(a , b , c\) and \(d\) are constants. The curve cuts the \(y\)-axis at \(( 0 , - 2 )\) and has asymptotes \(x = 2\) and \(y = x + 1\).

1. Write down the value of \(d\).
2. Determine the values of \(a , b\) and \(c\).
3. Show that, at all points on \(C\), either \(y \leqslant 3 - 2 \sqrt { 6 }\) or \(y \geqslant 3 + 2 \sqrt { 6 }\).

Express \(\frac { 2 x ^ { 2 } - x + 5 } { x ^ { 2 } - 1 }\) in the form \(2 + \frac { A } { x - 1 } + \frac { B } { x + 1 }\), where \(A\) and \(B\) are integers to be found. The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - x + 5 } { x ^ { 2 } - 1 }\). Show that there are two distinct values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Sketch \(C\), stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points.

2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).

10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 2 x - 3 } { ( \lambda x + 1 ) ( x + 4 ) }$$ where \(\lambda\) is a constant.

1. Find the equations of the asymptotes of \(C\) for the case where \(\lambda = 0\).
2. Find the equations of the asymptotes of \(C\) for the case where \(\lambda\) is not equal to any of \(- 1,0 , \frac { 1 } { 4 } , \frac { 1 } { 3 }\).
3. Sketch \(C\) for the case where \(\lambda = - 1\). Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of \(C\) with the coordinate axes.

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).

1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

7 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$ Find the equations of the asymptotes of \(C\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 2\) at all points on \(C\). Sketch C.

4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).

9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-14_61_1566_513_328}
2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
3. Find the coordinates of the turning points of \(C\).
4. Sketch \(C\).

9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-14_61_1566_513_328}
2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
3. Find the coordinates of the turning points of \(C\).
4. Sketch \(C\).

9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-14_61_1566_513_328}
2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
3. Find the coordinates of the turning points of \(C\).
4. Sketch \(C\).

10 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } - 3 x } { x ^ { 2 } + 1 }\). Verify that the equation of \(C\) may be written in the form \(y = - \frac { 1 } { 2 } + \frac { ( 3 x - 1 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\) and also in the form \(y = \frac { 9 } { 2 } - \frac { ( x + 3 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\). Hence show that \(- \frac { 1 } { 2 } \leqslant y \leqslant \frac { 9 } { 2 }\). Without differentiating, write down the coordinates of the turning points of \(C\). State the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of the intersections with the coordinate axes and the asymptote.

2 Express $$\frac { 2 n + 3 } { n ( n + 1 ) }$$ in partial fractions and hence use the method of differences to find $$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$ in terms of \(N\). Deduce the value of $$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$

7 In a chemical process, the mass \(M\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = \frac { M } { t \left( 1 + t ^ { 2 } \right) }$$

1. Find \(\int \frac { t } { 1 + t ^ { 2 } } \mathrm {~d} t\).
2. Find constants \(A , B\) and \(C\) such that $$\frac { 1 } { t \left( 1 + t ^ { 2 } \right) } = \frac { A } { t } + \frac { B t + C } { 1 + t ^ { 2 } } .$$
3. Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { \sqrt { 1 + t ^ { 2 } } } ,$$ where \(K\) is a constant.
4. When \(t = 1 , M = 25\). Calculate \(K\). What is the mass of the chemical in the long term?

5

1. Write \(\frac { 4 } { 4 x + 1 } - \frac { 3 } { 3 x + 2 }\) in the form \(\frac { C } { ( 4 x + 1 ) ( 3 x + 2 ) }\), where \(C\) is a constant.
   (l mark)
2. Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { 10 } { ( 4 x + 1 ) ( 3 x + 2 ) } d x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.
   (6 marks)

5

1. Express \(\frac { 1 } { ( 1 + x ) ( 2 + x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 2 + x }\), where \(A\) and \(B\) are integers.
2. Use the substitution \(u = \frac { \mathrm { d } y } { \mathrm {~d} x }\) to solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { 1 } { ( 1 + x ) ( 2 + x ) } \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + x } { 1 + x }$$ given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
   [0pt] [11 marks]

6

1. Find the first three terms in the expansion of \(( 8 - 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).

12

1. Use the substitution \(u = \mathrm { e } ^ { x } - 2\) to show that $$\int \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = \int \frac { 7 u + 6 } { u ^ { 2 } ( u + 2 ) } \mathrm { d } u$$
2. Hence show that $$\int _ { \ln 4 } ^ { \ln 6 } \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. \section*{END OF QUESTION PAPER}

1

1. Differentiate the following.
   1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
   2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).

4

1. Express \(\frac { 1 } { ( x - 1 ) ( x + 2 ) }\) in partial fractions
   [0pt] [2]
2. In this question you must show detailed reasoning. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 } { ( x - 1 ) ( x + 2 ) } \mathrm { d } x\).
   Give your answer in its simplest form.

11. \(\frac { - 6 x ^ { 2 } + 24 x - 9 } { ( x - 2 ) ( 1 - 3 x ) } \equiv A + \frac { B } { x - 2 } + \frac { C } { 1 - 3 x }\) a. Find the values of the constants \(A , B\) and \(C\).
b. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
c. Prove that \(\mathrm { f } ( x )\) is an increasing function.

14. a. Express \(\frac { 1 } { ( 3 - x ) ( 1 - x ) }\) in partial fractions.
(2) A scientist is studying the mass of a substance in a laboratory.
The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation $$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$ Given that when \(t = 0 , x = 0\) b. solve the differential equation and show that the solution can be written as $$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$ c. Find the mass, \(x\) grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.
d. Find the limiting value of \(x\) as \(t\) increases.

1. The curve \(C\) with equation

$$y = \frac { p - 3 x } { ( 2 x - q ) ( x + 3 ) } \quad x \in \mathbb { R } , x \neq - 3 , x \neq 2$$ where \(p\) and \(q\) are constants, passes through the point \(\left( 3 , \frac { 1 } { 2 } \right)\) and has two vertical asymptotes
with equations \(x = 2\) and \(x = - 3\) with equations \(x = 2\) and \(x = - 3\)

1. 1. Explain why you can deduce that \(q = 4\)
   2. Show that \(p = 15\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-38_616_889_842_587} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows a sketch of part of the curve \(C\). The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
2. Show that the exact value of the area of \(R\) is \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are rational constants to be found.