1.02k Simplify rational expressions: factorising, cancelling, algebraic division

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$

1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
   1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
   2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
3. 1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
   2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.

9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$

1. Write down the equations of the three asymptotes of \(C\).
2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
   (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$

1. 1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
   2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).

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   General Certificate of Education
   June 2005
   Advanced Subsidiary Examination MATHEMATICS
   MFP1
   Unit Further Pure 1 ASSESSMENT and
   QUALIFICATIONS
   ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
   Fill in the boxes at the top of this page.
   Fasten this insert securely to your answer book.

5

1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
2. Show that \(y\) cannot take values between - 3 and 1 .

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\) and f is a one-one function.

1. Describe a single transformation that transforms \(C\) to the curve with equation \(y = - \mathrm { f } ( - x )\) . The curve \(C\) is reflected in the line with equation \(y = - x\) to give the curve \(V\) . The equation of \(V\) is \(y = \mathrm { g } ( x )\) .
2. Explain why \(\mathrm { g } ^ { - 1 } ( x ) = - \mathrm { f } ( - x )\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-3_793_979_819_633} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 3 ( x - 1 ) } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ The curve has asymptotes with equations \(x = p\) and \(y = q\) and \(C\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\) .
3. Write down the value of \(p\) and the value of \(q\) .
4. Write down the coordinates of the point \(A\) and the coordinates of the point \(B\) . Given the definition of \(\mathrm { g } ( x )\) in part(b),
5. find the function g .
6. Solve \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = x\)

1 In this question you must show detailed reasoning. Solve the following equations.

1. \(\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0\)
2. \(\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0\)
3. \(3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }\)

16 Curve \(C\) has equation \(y = \frac { a x } { x + b }\) where \(a\) and \(b\) are constants.
The equations of the asymptotes to \(C\) are \(x = - 2\) and \(y = 3\) \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609} 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The gradient of \(C\) at the origin is \(\frac { 3 } { 2 }\) With reference to the graph, explain why there is exactly one root of the equation $$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$ 16
3. Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14

1. Write down the equation of the asymptote.
   14
2. Find the set of possible values of \(p\) □
   14
3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\) 14 (d) (i) Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14 (d) (ii) Hence, find the \(y\)-coordinate of point \(M\)

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\) The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\) [0pt] [5 marks]
      12
      1. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
         [0pt] [4 marks] 12
   2. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
      [0pt] [2 marks]
      12
   3. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
      [0pt] [1 mark]

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
   Fully justify your answer.
   9
2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
   \section*{Question 9 continues on the next page} 9
4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
   1. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)

7 The function f is defined by $$\mathrm { f } ( x ) = \frac { a x - 5 } { 2 x + b } \quad x \in \mathbb { R } , x \neq \frac { 9 } { 2 }$$ where \(a\) and \(b\) are integers.
The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = \frac { 9 } { 2 }\) and \(y = 3\) 7

1. Find the value of \(a\) and the value of \(b\) 7
2. Solve the inequality $$\mathrm { f } ( x ) \leq x + 2$$ Fully justify your answer.

5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
2. Sketch the curve, justifying all significant features.

12 The diagram shows the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) for \(x > - 1\). \includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}

1. Find the coordinates of the points where the curve crosses the axes.
2. Express \(\frac { x ^ { 2 } - 3 } { x + 1 }\) in the form \(A x + B + \frac { C } { x + 1 }\), where \(A , B\) and \(C\) are constants, and hence show that the exact area enclosed by the \(x\)-axis, the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) and the lines \(x = 2\) and \(x = 4\) is \(4 + \ln \frac { 9 } { 25 }\).

7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.

The function f is defined by \(\mathrm{f}(x) = \frac{2x + 1}{2x - 1}\) for \(x < \frac{1}{2}\).

1. 1. State the value of f\((-1)\). [1]
   2. \includegraphics{figure_5} The diagram shows the graph of \(y = \mathrm{f}(x)\). Sketch the graph of \(y = \mathrm{f}^{-1}(x)\) on this diagram. Show any relevant mirror line. [2]
   3. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of the function \(\mathrm{f}^{-1}\). [4]

The function g is defined by \(\mathrm{g}(x) = 3x + 2\) for \(x \in \mathbb{R}\).

1. Solve the equation \(\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)\). [3]

Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).

1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
3. Write down the equations of all the asymptotes of \(C\). [3]
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]

OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\). [4]
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]

The curve \(C\) has equation $$y = \frac{x^2}{x + \lambda},$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). [3] In separate diagrams, sketch \(C\) for the cases where

1. \(\lambda > 0\),
2. \(\lambda < 0\).

[4]

\(f(x) = \frac{(x^2 - 3)^2}{x^3}, x \neq 0\).

1. Show that \(f(x) \equiv x - 6x^{-1} + 9x^{-3}\). [2]
2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]

The function \(f\) is given by $$f : x \mapsto \frac{x}{x^2 - 1} - \frac{1}{x + 1}, \quad x > 1.$$

1. Show that \(f(x) = \frac{1}{(x-1)(x+1)}\). [3]
2. Find the range of \(f\). [2]

The function \(g\) is given by $$g : x \mapsto \frac{2}{x}, \quad x > 0.$$

1. Solve \(gf(x) = 70\). [4]

$$f(x) = \frac{2}{x-1} - \frac{6}{(x-1)(2x+1)}, \quad x > 1.$$

1. Prove that \(f(x) = \frac{4}{2x+1}\). [4]
2. Find the range of \(f\). [2]
3. Find \(f^{-1}(x)\). [3]
4. Find the range of \(f^{-1}(x)\). [1]

Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]

Express \(\frac{y + 3}{(y + 1)(y + 2)} - \frac{y + 1}{(y + 2)(y + 3)}\) as a single fraction in its simplest form. [5]

The function \(f\) is defined by \(f: x \mapsto \frac{3x-1}{x-3}, x \in \mathbb{R}, x \neq 3\).

1. Prove that \(f^{-1}(x) = f(x)\) for all \(x \in \mathbb{R}, x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(ff(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function \(g\), defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(fg(-2)\). [3]
2. Sketch the graph of the inverse function \(g^{-1}\) and state its domain. [3]

The function \(h\) is defined by \(h: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function \(h\) and state its range. [3]

Express \(\frac{x}{(x+1)(x+3)} + \frac{x+12}{x^2-9}\) as a single fraction in its simplest form. [6]