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

333 questions

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AQA FP1 2011 January Q7
15 marks Standard +0.8
7 A graph has equation $$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$
  1. Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
  2. Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation $$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
  3. Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
  4. Hence find the coordinates of the two stationary points on the curve.
    (No credit will be given for methods involving differentiation.)
AQA FP1 2007 June Q7
9 marks Moderate -0.3
7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$
  1. Write down the equations of the two asymptotes to the curve.
  2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
  3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$
AQA FP1 2009 June Q8
15 marks Standard +0.8
8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$
  1. Write down the equations of the three asymptotes to the curve.
  2. Show that the curve has no point of intersection with the line \(y = - 1\).
    1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
    2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
  4. Hence find the coordinates of the two stationary points on the curve.
AQA FP1 2012 June Q5
11 marks Standard +0.3
5 The curve \(C\) has equation \(y = \frac { x } { ( x + 1 ) ( x - 2 ) }\).
The line \(L\) has equation \(y = - \frac { 1 } { 2 }\).
  1. Write down the equations of the asymptotes of \(C\).
  2. The line \(L\) intersects the curve \(C\) at two points. Find the \(x\)-coordinates of these two points.
  3. Sketch \(C\) and \(L\) on the same axes.
    (You are given that the curve \(C\) has no stationary points.)
  4. Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$
AQA FP1 2013 June Q9
14 marks Challenging +1.2
9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
  1. Find the equations of the three asymptotes of the curve.
    1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
    2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
    3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve and its asymptotes.
AQA FP1 2015 June Q8
11 marks Challenging +1.2
8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$
  1. State the equation of the asymptote of \(C\).
  2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
  3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.) \includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}
OCR MEI Further Pure with Technology 2024 June Q1
17 marks Standard +0.8
1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.
    1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
      • \(a = - 0.1\)
      • \(a = 0.5\)
      • State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
      • By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
      • Write down the equation of the vertical asymptote of the curve (}).
      The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
    2. Show that \(A = 0\).
    3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
    4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
    5. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
      1. Determine, in terms of \(a\), the area of the triangle XYZ.
      2. Find the maximum area of the triangle XYZ.
Edexcel FP1 AS 2020 June Q2
5 marks Standard +0.8
  1. Use algebra to determine the values of \(x\) for which
$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$
OCR MEI C2 Q3
12 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in factorised form.
  2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\).
OCR H240/01 2018 September Q6
7 marks Standard +0.3
6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).
Edexcel C3 Q2
10 marks Moderate -0.3
2. (a) Differentiate with respect to \(x\)
  1. \(3 \sin ^ { 2 } x + \sec 2 x\),
  2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).
AQA C1 2007 January Q1
11 marks Moderate -0.8
1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.
    1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)
AQA C4 2006 January Q1
8 marks Moderate -0.8
1
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
    1. Find f(1).
    2. Show that \(\mathrm { f } ( - 2 ) = 0\).
    3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).
AQA C4 2007 January Q2
6 marks Moderate -0.8
2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant. Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
  3. Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).
AQA C4 2008 January Q2
10 marks Moderate -0.3
2
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8\).
    1. Use the Factor Theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
    2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
    3. Simplify the algebraic fraction \(\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }\).
  2. Express the algebraic fraction \(\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }\) in the form \(A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }\), where \(A , B\) and \(C\) are integers.
AQA C4 2009 January Q1
8 marks Moderate -0.3
1
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
    1. Find \(\mathrm { f } ( - 1 )\).
    2. Use the Factor Theorem to show that \(2 x + 1\) is a factor of \(\mathrm { f } ( x )\).
    3. Simplify the algebraic fraction \(\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }\).
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 4 x ^ { 3 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(2 x + 1\), the remainder is 2 . Find the value of \(d\).
AQA C4 2009 January Q3
13 marks Standard +0.3
3
    1. Express \(\frac { 2 x + 7 } { x + 2 }\) in the form \(A + \frac { B } { x + 2 }\), where \(A\) and \(B\) are integers. (2 marks)
    2. Hence find \(\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x\).
    1. Express \(\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }\) in the form \(\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }\), where \(P , Q\) and \(R\) are constants.
    2. Hence find \(\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x\).
AQA C4 2010 January Q1
8 marks Moderate -0.3
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 15 x ^ { 3 } + 19 x ^ { 2 } - 4\).
    1. Find \(\mathrm { f } ( - 1 )\).
    2. Show that \(( 5 x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.
AQA C4 2005 June Q3
6 marks Moderate -0.8
3
  1. Find the remainder when \(2 x ^ { 3 } - x ^ { 2 } + 2 x - 2\) is divided by \(2 x - 1\).
  2. Given that \(\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }\), find the values of \(a\) and \(b\).
AQA C4 2006 June Q1
8 marks Moderate -0.3
1
  1. The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10\).
    1. Find \(\mathrm { p } ( 2 )\).
    2. Use the Factor Theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
    3. Write \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Hence simplify \(\frac { 3 x ^ { 2 } - 6 x } { 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10 }\).
AQA C4 2007 June Q1
5 marks Moderate -0.8
1
  1. Find the remainder when \(2 x ^ { 2 } + x - 3\) is divided by \(2 x + 1\).
    (2 marks)
  2. Simplify the algebraic fraction \(\frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - 1 }\).
    (3 marks)
AQA C4 2008 June Q1
9 marks Moderate -0.3
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
    1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$
AQA C4 2009 June Q1
5 marks Moderate -0.8
1
  1. Use the Remainder Theorem to find the remainder when \(3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5\) is divided by \(3 x - 1\).
  2. Express \(\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }\) in the form \(a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }\), where \(a , b\) and \(c\) are integers.
OCR MEI C4 2006 January Q1
5 marks Moderate -0.3
1 Solve the equation \(\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3\).
OCR MEI C4 2008 June Q1
3 marks Easy -1.2
1 Express \(\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }\) as a single fraction, simplifying your answer.