1.02g Inequalities: linear and quadratic in single variable

8 The quadratic equation \(( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0\) has real roots.

1. Show that \(4 k ^ { 2 } - 9 k - 9 \geqslant 0\).
2. Hence find the possible values of \(k\).

7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).

1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
2. The curve \(C\) and the line \(L\) intersect in two distinct points.
   1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
   2. Hence find the possible values of \(k\).

2

1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
2. Solve the equation \(x = 4 - | 2 x + 1 |\).
3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
   [0pt] [4 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}

4 The equation \(9 x ^ { 2 } + 4 x + p ^ { 2 } = 0\) has no real solutions for \(x\). Find the set of possible values of \(p\).
Fully justify your answer.
[0pt] [4 marks]

9 The diagram below shows the graphs of \(y = x ^ { 2 } - 4 x - 12\) and \(y = x + 2\) \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-10_933_912_358_566} 9

1. Write down three inequalities which together describe the shaded region.
   9
2. Find the coordinates of the points \(A , B\) and \(C\).
   9
3. Find the exact area of the shaded region.
   Fully justify your answer.
   [0pt] [6 marks]

1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

1 The graph of \(y = a x ^ { 2 } + b x + c\) has roots \(x = 2\) and \(x = 5\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_905_963_717_625} State the set of values of \(x\) which satisfy $$a x ^ { 2 } + b x + c > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 2 \} \cup \{ x : x > 5 \} \\ & \{ x : 0 < x < 2 \} \cap \{ x : x > 5 \} \\ & \{ x : 2 < x < 5 \} \\ & \{ x : 2 > x > 5 \} \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_118_115_1950_1087}
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16 Curve \(C\) has equation \(y = \frac { a x } { x + b }\) where \(a\) and \(b\) are constants.
The equations of the asymptotes to \(C\) are \(x = - 2\) and \(y = 3\) \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609} 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The gradient of \(C\) at the origin is \(\frac { 3 } { 2 }\) With reference to the graph, explain why there is exactly one root of the equation $$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$ 16
3. Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14

1. Write down the equation of the asymptote.
   14
2. Find the set of possible values of \(p\) □
   14
3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\) 14 (d) (i) Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14 (d) (ii) Hence, find the \(y\)-coordinate of point \(M\)

7 The function f is defined by $$\mathrm { f } ( x ) = \frac { a x - 5 } { 2 x + b } \quad x \in \mathbb { R } , x \neq \frac { 9 } { 2 }$$ where \(a\) and \(b\) are integers.
The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = \frac { 9 } { 2 }\) and \(y = 3\) 7

1. Find the value of \(a\) and the value of \(b\) 7
2. Solve the inequality $$\mathrm { f } ( x ) \leq x + 2$$ Fully justify your answer.

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | }\) and the line with equation \(y = 5 - 4 x\) Use algebra to determine the values of \(x\) for which $$\frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | } < 5 - 4 x$$

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 \(k\).

3. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$

1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$

1 Find the set of values of \(x\) for which \(x ^ { 2 } - x - 12 < 0\).

3 Solve \(3 x ^ { 2 } + 11 x - 20 > 0\).

1 Solve \(5 x + 3 < | 3 x - 1 |\).

A curve \(C\) has equation \(y = x ^ { 3 } - 3 x ^ { 2 }\). a) Find the stationary points of \(C\) and determine their nature.
b) Draw a sketch of \(C\), clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\) is positive or
negative, giving a reason for your answer. In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\ & \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true. The value of a car, \(\pounds V\), may be modelled as a continuous variable. At time \(t\) years, the value of the car is given by \(V = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. When the car is new, it is worth \(\pounds 30000\). When the car is two years old, it is worth \(\pounds 20000\). Determine the value of the car when it is six years old, giving your answer correct to the nearest \(\pounds 100\). The curve \(C\) has equation \(y = 7 + 13 x - 2 x ^ { 2 }\). The point \(P\) lies on \(C\) and is such that the tangent to \(C\) at \(P\) has equation \(y = x + c\), where \(c\) is a constant. Find the coordinates of \(P\) and the value of \(c\). a) Solve \(2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2\).
b) Solve \(3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }\).
c) Express \(4 ^ { x } - 10 \times 2 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\). Hence solve the equation \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\).

The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).

1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]

The function f is defined by \(\text{f}(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\).

1. Express \(x^2 - 4x + 8\) in the form \((x - a)^2 + b\). [2]
2. Hence find the set of values of \(x\) for which \(\text{f}(x) < 9\), giving your answer in exact form. [3]

The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]

Find the set of values of \(x\) for which

1. \(3(2x + 1) > 5 - 2x\), [2]
2. \(2x^2 - 7x + 3 > 0\), [4]
3. both \(3(2x + 1) > 5 - 2x\) and \(2x^2 - 7x + 3 > 0\). [2]

Find the set of values of \(x\) for which $$x^2 - 7x - 18 > 0.$$ [4]