Solve modulus equation then apply exponential/log substitution

Solve a modulus equation with linear expressions, then use the result to solve a related equation involving exponential or logarithmic substitution, e.g. |3^(y+1)-5| = 2×3^y+7.

13 questions · Standard +0.1

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CAIE P2 2020 November Q4
5 marks Standard +0.3
4
  1. Solve the equation \(| 2 x - 5 | = | x + 6 |\).
  2. Hence find the value of \(y\) such that \(\left| 2 ^ { 1 - y } - 5 \right| = \left| 2 ^ { - y } + 6 \right|\). Give your answer correct to 3 significant figures.
CAIE P2 2021 November Q2
7 marks Standard +0.3
2
  1. Sketch, on the same diagram, the graphs of \(y = x + 3\) and \(y = | 2 x - 1 |\).
  2. Solve the equation \(x + 3 = | 2 x - 1 |\).
  3. Find the value of \(y\) such that \(5 ^ { \frac { 1 } { 2 } y } + 3 = \left| 2 \times 5 ^ { \frac { 1 } { 2 } y } - 1 \right|\). Give your answer correct to 3 significant figures.
CAIE P2 2023 November Q4
7 marks Standard +0.8
4
  1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = 2 x + 7\).
  2. Solve the equation \(| 3 x - 5 | = 2 x + 7\).
  3. Hence solve the equation \(\left| 3 ^ { y + 1 } - 5 \right| = 2 \times 3 ^ { y } + 7\), giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q1
4 marks Moderate -0.3
1
  1. Solve the equation \(| x + 2 | = | x - 13 |\).
  2. Hence solve the equation \(\left| 3 ^ { y } + 2 \right| = \left| 3 ^ { y } - 13 \right|\), giving your answer correct to 3 significant figures.
CAIE P2 2015 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(| 3 x + 4 | = | 3 x - 11 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 2 ^ { y } + 4 \right| = \left| 3 \times 2 ^ { y } - 11 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2016 June Q3
5 marks Standard +0.3
3
  1. Solve the equation \(| 3 u + 1 | = | 2 u - 5 |\).
  2. Hence solve the equation \(| 3 \cot x + 1 | = | 2 \cot x - 5 |\) for \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2013 June Q4
6 marks Standard +0.3
4
  1. Solve the equation \(| 4 x - 1 | = | x - 3 |\).
  2. Hence solve the equation \(\left| 4 ^ { y + 1 } - 1 \right| = \left| 4 ^ { y } - 3 \right|\) correct to 3 significant figures.
CAIE P3 2016 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(2 | x - 1 | = 3 | x |\).
  2. Hence solve the equation \(2 \left| 5 ^ { x } - 1 \right| = 3 \left| 5 ^ { x } \right|\), giving your answer correct to 3 significant figures.
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 2015 November Q1
5 marks Moderate -0.8
1
  1. Solve the equation \(| 3 x - 2 | = 5\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.
CAIE P2 2015 November Q2
5 marks Standard +0.3
2
  1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.
CAIE P2 2018 November Q1
5 marks Standard +0.3
1
  1. Solve the equation \(| 9 x - 2 | = | 3 x + 2 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 ^ { y + 2 } - 2 \right| = \left| 3 ^ { y + 1 } + 2 \right|\), giving your answer correct to 3 significant figures.
CAIE P2 2019 November Q2
5 marks Standard +0.3
2
  1. Solve the equation \(| 4 x + 5 | = | x - 7 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 2 } + 5 \right| = \left| 2 ^ { y } - 7 \right|\), giving the answer correct to 3 significant figures.