Find range of k for number of roots

5. \begin{figure}[h] \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 \(q\).

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 $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$
   1. Sketch the curve \(y = \mathrm { g } ( x )\) and state the range of g .
   2. 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 \(k\).

9
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 \(a\).

11. \begin{figure}[h] \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 \(k\).

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Sketch the curve \(C\) with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
   • the equation of the vertical asymptote
   • the coordinates of the intersection of \(C\) with the \(y\)-axis
   The straight line \(l\) has equation \(y = k x - 4\), where \(k\) is a constant.
   Given that \(l\) cuts \(C\) at least once,
2. 1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
   2. find the range of possible values for \(k\).

$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).

1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]

Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),

1. find the value of \(k\). [2]
2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]

The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.

1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$

1. State the range of \(f\) [1]
2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
3. state the set of possible values for \(k\). [2]