1.02l Modulus function: notation, relations, equations and inequalities

395 questions

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CAIE P3 2017 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x + 1 | < 3 | x - 2 |\).
CAIE P3 2019 June Q1
3 marks Moderate -0.5
1 Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { 0 } ^ { 3 } \left| 2 ^ { x } - 4 \right| \mathrm { d } x$$
CAIE P3 2017 March Q2
4 marks Standard +0.8
2 Solve the inequality \(| x - 4 | < 2 | 3 x + 1 |\).
CAIE P3 2002 November Q1
3 marks Easy -1.2
1 Solve the inequality \(| 9 - 2 x | < 1\).
CAIE P3 2003 November Q1
4 marks Moderate -0.3
1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).
CAIE P3 2005 November Q1
4 marks Standard +0.3
1 Given that \(a\) is a positive constant, solve the inequality $$| x - 3 a | > | x - a |$$
CAIE P3 2006 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(\left| 3 ^ { x } - 8 \right| < 0.5\), giving 3 significant figures in your answer.
CAIE P3 2009 November Q1
4 marks Moderate -0.3
1 Solve the inequality \(2 - 3 x < | x - 3 |\).
CAIE P3 2010 November Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x - 3 | > | 3 x + 1 |\).
CAIE P3 2012 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(3 | x - 1 | < | 2 x + 1 |\).
CAIE P3 2013 November Q2
4 marks Standard +0.3
2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.
CAIE P3 2014 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 1 | < | 2 x + 5 |\).
CAIE P3 2015 November Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).
CAIE P3 2018 November Q1
4 marks Standard +0.8
1 Find the set of values of \(x\) satisfying the inequality \(2 | 2 x - a | < | x + 3 a |\), where \(a\) is a positive constant. [4]
CAIE P3 2019 November Q2
4 marks Standard +0.8
2 Solve the inequality \(| 2 x - 3 | > 4 | x + 1 |\).
CAIE P3 2019 November Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x + 2 | > | 3 x - 1 |\).
CAIE P3 Specimen Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).
CAIE Further Paper 1 2020 June Q1
6 marks Standard +0.8
1 Let \(a\) be a positive constant.
  1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
  2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).
CAIE Further Paper 1 2020 June Q6
15 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no turning points.
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).
CAIE Further Paper 1 2021 June Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2021 June Q7
14 marks Challenging +1.3
7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2022 June Q5
12 marks Challenging +1.2
5 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k\) has 4 distinct real solutions.
CAIE Further Paper 1 2022 June Q1
6 marks Standard +0.8
1
  1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { x } + 1 } { \mathrm { x } - 1 }\).
  2. Sketch the curve with equation \(\mathrm { y } = \frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 }\) and find the set of values of x for which \(\frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 } < - 2\).
CAIE Further Paper 1 2020 November Q6
12 marks Standard +0.8
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(1 < y < 5\).
  3. Find the coordinates of the intersections of \(C\) with the axes, and sketch \(C\).
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 } \right|\).
CAIE Further Paper 1 2020 November Q6
13 marks Standard +0.8
6 Let \(a\) be a positive constant.
  1. The curve \(C _ { 1 }\) has equation \(\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }\). Sketch \(C _ { 1 }\). The curve \(C _ { 2 }\) has equation \(\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }\). The curve \(C _ { 3 }\) has equation \(\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|\).
    1. Find the coordinates of any stationary points of \(C _ { 2 }\).
    2. Find also the coordinates of any points of intersection of \(C _ { 2 }\) and \(C _ { 3 }\).
  3. Sketch \(C _ { 2 }\) and \(C _ { 3 }\) on a single diagram, clearly identifying each curve. Hence find the set of values of \(x\) for which \(\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|\).