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

395 questions

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CAIE Further Paper 1 2021 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } }\).
  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 \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    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 November Q6
14 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(0 < y < 12\).
  3. Sketch C.
    1. Sketch the graphs of \(y = \left| \frac { x ^ { 2 } } { x - 3 } \right|\) and \(y = | x | - 3\) on a single diagram, stating the coordinates of the intersections with the axes.
    2. Use your sketch to find the set of values of \(c\) for which \(\left| \frac { x ^ { 2 } } { x - 3 } \right| \leqslant | x | + c\) has no solution. [1]
CAIE Further Paper 1 2021 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { 4 \mathrm { x } + 5 } { 4 - 4 \mathrm { x } ^ { 2 } }\).
  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 \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    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 November Q7
16 marks Standard +0.8
7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2022 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } - \mathrm { x } } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the exact 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 \(y = \left| \frac { x ^ { 2 } - x } { x + 1 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x } { x + 1 } \right| < 6\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2020 Specimen Q7
17 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\).
  1. State the equations of the asymptotes of \(C\).
  2. Show that \(y \leqslant \frac { 25 } { 12 }\) at all points on \(C\).
  3. Find the coordinates of any stationary points of \(C\).
  4. Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
  5. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\).
CAIE P2 2019 June Q2
6 marks Standard +0.3
2
  1. Solve the inequality \(| 3 x - 5 | < | x + 3 |\).
  2. Hence find the greatest integer \(n\) satisfying the inequality \(\left| 3 ^ { 0.1 n + 1 } - 5 \right| < \left| 3 ^ { 0.1 n } + 3 \right|\).
CAIE P2 2019 June Q2
5 marks Moderate -0.3
2
  1. Solve the equation \(| 4 + 2 x | = | 3 - 5 x |\).
  2. Hence solve the equation \(\left| 4 + 2 e ^ { 3 y } \right| = \left| 3 - 5 e ^ { 3 y } \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2016 March Q2
4 marks Standard +0.3
2 Solve the inequality \(| x - 5 | < | 2 x + 3 |\).
CAIE P2 2017 March Q3
6 marks Standard +0.3
3
  1. Solve the inequality \(| 2 x - 5 | < | x + 3 |\).
  2. Hence find the largest integer \(y\) satisfying the inequality \(| 2 \ln y - 5 | < | \ln y + 3 |\).
CAIE P2 2019 March Q2
4 marks Standard +0.3
2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$
CAIE P2 2002 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 2 x - 1 | < | 3 x |\).
CAIE P2 2004 November Q1
3 marks Moderate -0.5
1 Solve the inequality \(| x + 1 | > | x |\).
CAIE P2 2006 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 2 x - 1 | > | x |\).
CAIE P2 2007 November Q3
5 marks Standard +0.3
3
  1. Solve the inequality \(| y - 5 | < 1\).
  2. Hence solve the inequality \(\left| 3 ^ { x } - 5 \right| < 1\), giving 3 significant figures in your answer.
CAIE P2 2008 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| x - 3 | > | 2 x |\).
CAIE P2 2009 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| 2 x + 3 | < | x - 3 |\).
CAIE P2 2009 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| x + 3 | > | 2 x |\).
CAIE P2 2010 November Q1
3 marks Standard +0.3
1 Solve the inequality \(| x + 1 | > | x - 4 |\).
CAIE P2 2010 November Q1
3 marks Easy -1.2
1 Solve the inequality \(| 3 x + 1 | > 8\).
CAIE P2 2011 November Q1
3 marks Easy -1.2
1 Solve the inequality \(| 4 - 5 x | < 3\).
CAIE P2 2011 November Q1
4 marks Standard +0.3
1 Solve the inequality \(| x + 2 | > \left| \frac { 1 } { 2 } x - 2 \right|\).
CAIE P2 2011 November Q2
4 marks Standard +0.3
2 Solve the inequality \(| 2 x - 3 | \leqslant | 3 x |\).
CAIE P2 2012 November Q1
3 marks Standard +0.3
1 Solve the inequality \(| x - 2 | \geqslant | x + 5 |\).
CAIE P2 2012 November Q1
3 marks Standard +0.3
1 Solve the inequality \(| 2 x + 1 | < | 2 x - 5 |\).