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

395 questions

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OCR C3 Q4
5 marks Moderate -0.3
\includegraphics{figure_4} The function f is defined by \(f(x) = 2 - \sqrt{x}\) for \(x \geq 0\). The graph of \(y = f(x)\) is shown above.
  1. State the range of f. [1]
  2. Find the value of ff(4). [2]
  3. Given that the equation \(|f(x)| = k\) has two distinct roots, determine the possible values of the constant \(k\). [2]
OCR C3 Q2
5 marks Standard +0.8
Solve the inequality \(|2x - 3| < |x + 1|\). [5]
OCR C3 Q7
8 marks Standard +0.3
The curve \(y = \ln x\) is transformed to the curve \(y = \ln(\frac{1}{2}x - a)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.
  1. Give full details of the translation and stretch involved. [2]
  2. Sketch the graph of \(y = \ln(\frac{1}{2}x - a)\). [2]
  3. Sketch, on another diagram, the graph of \(y = |\ln(\frac{1}{2}x - a)|\). [2]
  4. State, in terms of \(a\), the set of values of \(x\) for which \(|\ln(\frac{1}{2}x - a)| = -\ln(\frac{1}{2}x - a)\). [2]
OCR C3 Q2
5 marks Standard +0.8
Solve the inequality \(|4x - 3| < |2x + 1|\). [5]
OCR C3 2010 January Q4
8 marks Moderate -0.8
\includegraphics{figure_4} The function \(f\) is defined for all real values of \(x\) by $$f(x) = 2 - \sqrt{x + 1}.$$ The diagram shows the graph of \(y = f(x)\).
  1. Evaluate \(f(-126)\). [2]
  2. Find the set of values of \(x\) for which \(f(x) = |f(x)|\). [2]
  3. Find an expression for \(f^{-1}(x)\). [3]
  4. State how the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are related geometrically. [1]
OCR C3 2013 January Q3
7 marks Standard +0.8
  1. Given that \(|t| = 3\), find the possible values of \(|2t - 1|\). [3]
  2. Solve the inequality \(|x - t^2| > |x + 3\sqrt{2}|\). [4]
OCR C3 2009 June Q5
10 marks Moderate -0.8
The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which
  1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
  2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
  3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]
OCR C3 2010 June Q5
7 marks Standard +0.8
  1. Solve the inequality \(|2x + 1| \leqslant |x - 3|\). [5]
  2. Given that \(x\) satisfies the inequality \(|2x + 1| \leqslant |x - 3|\), find the greatest possible value of \(|x + 2|\). [2]
OCR MEI C3 Q1
Easy -1.2
Solve the equation \(|3x + 2| = 1\).
OCR MEI C3 2011 January Q2
4 marks Moderate -0.8
Solve the inequality \(|2x + 1| \geqslant 4\). [4]
OCR MEI C3 2013 January Q3
2 marks Easy -1.2
Express \(1 < x < 3\) in the form \(|x - a| < b\), where \(a\) and \(b\) are to be determined. [2]
OCR MEI C3 2011 June Q1
4 marks Moderate -0.8
Solve the equation \(|2x - 1| = |x|\). [4]
OCR MEI C3 2014 June Q3
4 marks Standard +0.3
Solve the equation \(|3 - 2x| = 4|x|\). [4]
OCR MEI C3 2016 June Q4
4 marks Moderate -0.8
By sketching the graphs of \(y = |2x + 1|\) and \(y = -x\) on the same axes, show that the equation \(|2x + 1| = -x\) has two roots. Find these roots. [4]
OCR MEI C3 Q3
6 marks Moderate -0.8
  1. Sketch the graph of \(y = |3x - 6|\). [2]
  2. Solve the equation \(|3x - 6| = x + 4\) and illustrate your answer on your graph. [4]
Edexcel C3 Q6
10 marks Standard +0.8
  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [6]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
OCR C3 Q6
9 marks Standard +0.8
  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
OCR C3 Q2
5 marks Standard +0.8
Find the set of values of \(x\) such that $$|3x + 1| \leq |x - 2|.$$ [5]
OCR H240/03 2019 June Q1
2 marks Easy -1.2
\includegraphics{figure_1} The diagram shows triangle \(ABC\), with \(AC = 13.5\) cm, \(BC = 8.3\) cm and angle \(ABC = 32°\). Find angle \(CAB\). [2]
OCR H240/03 2019 June Q2
8 marks Moderate -0.8
A circle with centre \(C\) has equation \(x^2 + y^2 - 6x + 4y + 4 = 0\).
  1. Find
    1. the coordinates of \(C\), [2]
    2. the radius of the circle. [1]
  2. Determine the set of values of \(k\) for which the line \(y = kx - 3\) does not intersect or touch the circle. [5]
OCR H240/03 2021 November Q4
5 marks Moderate -0.3
  1. Sketch, on a single diagram, the following graphs.
    [2]
  2. Hence explain why the equation \(x|x - 1| = k\) has exactly one real root for any negative value of \(k\). [1]
  3. Determine the real root of the equation \(x|x - 1| = -6\). [2]
OCR H240/03 2022 June Q1
3 marks Easy -1.8
Solve the equation \(|2x - 3| = 9\). [3]
AQA Paper 1 2024 June Q17
6 marks Moderate -0.8
The function f is defined by $$f(x) = |x| + 1 \quad \text{for } x \in \mathbb{R}$$ The function g is defined by $$g(x) = \ln x$$ where g has its greatest possible domain.
  1. Using set notation, state the range of f [2 marks]
  2. State the domain of g [1 mark]
  3. The composite function h is given by $$h(x) = g f(x) \quad \text{for } x \in \mathbb{R}$$
    1. Write down an expression for \(h(x)\) in terms of \(x\) [1 mark]
    2. Determine if h has an inverse. Fully justify your answer. [2 marks]
AQA Paper 2 2019 June Q1
1 marks Easy -1.8
Identify the graph of \(y = 1 - |x + 2|\) from the options below. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_1}
AQA Paper 3 2018 June Q4
3 marks Easy -1.2
Sketch the graph of \(y = |2x + a|\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes. [3 marks] \includegraphics{figure_4}