Find range of k for number of roots

Given equation |f(x)| = k or f(x) = k, find values of constant k for which equation has exactly one, two, or specified number of roots.

5 questions · Standard +0.4

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CAIE P2 2024 March Q2
4 marks Standard +0.3
2
  1. Sketch the graph of \(y = | 3 x - 7 |\), stating the coordinates of the points where the graph meets the axes.
  2. Hence find the set of values of the constant \(k\) for which the equation \(| 3 \mathrm { x } - 7 | = \mathrm { k } ( \mathrm { x } - 4 )\) has exactly two real roots.
Edexcel C3 2018 June Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42aff260-e734-48ff-a92a-674032cb0377-16_561_848_214_699} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly one root,
  1. state the set of possible values of \(k\).
  2. Solve the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10\) The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\). The vertex on the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\) has coordinates \(( p , q )\).
  3. State the value of \(p\) and the value of \(q\).
OCR C3 2014 June Q8
11 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516} The diagram shows the curve \(y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the two stationary points.
  2. The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$ (a) Sketch the curve \(y = \mathrm { g } ( x )\) and state the range of g .
    (b) It is given that the equation \(\mathrm { g } ( x ) = k\), where \(k\) is a constant, has exactly two distinct real roots. Write down the set of possible values of \(k\).
OCR H240/01 2020 November Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-6_391_606_1672_244} The diagram shows the graph of \(y = | 2 x - 3 |\).
  1. State the coordinates of the points of intersection with the axes.
  2. Given that the graphs of \(y = | 2 x - 3 |\) and \(y = a x + 2\) have two distinct points of intersection, determine
    1. the set of possible values of \(a\),
    2. the \(x\)-coordinates of the points of intersection of these graphs, giving your answers in terms of \(a\).
Edexcel Paper 2 Specimen Q11
6 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-14_570_556_205_758} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$
  1. State the range of f
  2. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots, (c) state the set of possible values for \(k\).