1.02f Solve quadratic equations: including in a function of unknown

1. The equation

$$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
Find the range of possible values of \(c\).

11. (a) On Diagram 1 sketch the graphs of

1. \(y = x ( 3 - x )\)
2. \(y = x ( x - 2 ) ( 5 - x )\) showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\) coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
   (c) find, using algebra and showing your working, the exact coordinates of \(P\).
   
   \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
   \section*{Diagram 1}

5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$

8. The straight line \(l\) has equation \(y = k ( 2 x - 1 )\), where \(k\) is a constant. The curve \(C\) has equation \(y = x ^ { 2 } + 2 x + 11\) Find the set of values of \(k\) for which \(l\) does not cross or touch \(C\).
(6)

10. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 4 x - 3 ) ( x - 5 ) ^ { 2 }$$

1. Sketch \(C _ { 1 }\) showing the coordinates of any point where the curve touches or crosses the coordinate axes.
2. Hence or otherwise
   1. find the values of \(x\) for which \(\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0\)
   2. find the value of the constant \(p\) such that the curve with equation \(y = \mathrm { f } ( x ) + p\) passes through the origin. A second curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )\)
3. 1. Find, in simplest form, \(\mathrm { g } ( x )\). You may leave your answer in a factorised form.
   2. Hence, or otherwise, find the \(y\) intercept of curve \(C _ { 2 }\)

7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = 2 x - 3 \sqrt { x } - 5 \quad x > 0$$

1. Solve the equation $$f ( x ) = 9$$
2. Solve the equation $$\mathrm { f } ^ { \prime \prime } ( x ) = 6$$

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
2. Expand f(x) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found. The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis. The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
3. Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
   (Solutions relying entirely on calculator technology are not acceptable.)

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

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$

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

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\) Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\) Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)

1. (a) Find, using algebra, all real solutions of

$$2 x ^ { 3 } + 3 x ^ { 2 } - 35 x = 0$$ (b) Hence find all real solutions of $$2 ( y - 5 ) ^ { 6 } + 3 ( y - 5 ) ^ { 4 } - 35 ( y - 5 ) ^ { 2 } = 0$$

2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = a x ^ { 3 } + ( 6 a + 8 ) x ^ { 2 } - a ^ { 2 } x$$ where \(a\) is a positive constant. Given \(\mathrm { f } ( - 1 ) = 32\)

1. 1. show that the only possible value for \(a\) is 3
   2. Using \(a = 3\) solve the equation $$\mathrm { f } ( x ) = 0$$
2. Hence find all real solutions of
   1. \(3 y + 26 y ^ { \frac { 2 } { 3 } } - 9 y ^ { \frac { 1 } { 3 } } = 0\)
   2. \(3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0\)

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

Solutions relying on calculator technology are not acceptable. Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle \(A B C\) with

• \(A B = p\) metres
• \(A C = q\) metres
• \(B C = 2 \sqrt { 2 }\) metres
• angle \(B A C = 60 ^ { \circ }\)
  1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side \(A C\) is 2 metres longer than side \(A B\), use algebra to find

1. the exact value of \(p\),
2. the exact value of \(q\). Using the answers to part (b),

calculate the exact area of the flower bed.

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
• \(C\) cuts the negative \(x\)-axis at - 5
  1. find \(\mathrm { f } ( x )\)

The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)

find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)

Define the region \(R\), shown shaded in Figure 2, using inequalities.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,

• has gradient \(\frac { 1 } { 2 }\)
• intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
• find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
• show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
• Hence, or otherwise, find the value of \(b\)

1. The curve \(C _ { 1 }\) has equation

$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)

1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
2. Hence find the coordinates of \(P\)

1. The curve \(C _ { 1 }\) has equation

$$y = x \left( 4 - x ^ { 2 } \right)$$

1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.

8. Solve, using algebra, the equation $$x - 6 x ^ { \frac { 1 } { 2 } } + 4 = 0$$ Fully simplify your answers, writing them in the form \(a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers to be found.
(5)

2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form
   2. \(\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }\)
2. Express \(\mathrm { f } ( x )\) in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 } \\ & \text { where } a , b \text { and } c \text { are integers to be found. } \\ & \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses } \\ & \text { te } y \text {-axis at the point } R \text {. } \\ & \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and } \\ & n \text { are integers to be found. } \end{aligned}$$ \(\_\_\_\_\) "

7. The curve \(C\) has equation $$y = \frac { 1 } { 2 - x }$$

1. Sketch the graph of \(C\). On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line \(l\) has equation \(y = 4 x + k\), where \(k\) is a constant. Given that \(l\) meets \(C\) at two distinct points,
2. show that $$k ^ { 2 } + 16 k + 48 > 0$$
3. Hence find the range of possible values for \(k\).

1. The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was \(\pounds 1.05\) Exactly 5 years after monitoring began, the share price was \(\pounds 1.65\) The share price, \(\pounds V\), of the company is modelled by the equation $$V = p t + q$$ where \(t\) is the number of years after monitoring began and \(p\) and \(q\) are constants.

1. Find the value of \(p\) and the value of \(q\). Exactly \(T\) years after monitoring began, the share price was \(\pounds 2.50\)
2. Find the value of \(T\), according to the model, giving your answer to one decimal place.

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

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.

1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
2. Hence, using algebra, find the range of possible values of \(p\)

9. Diagram NOT accurately drawn \begin{figure}[h] \end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle \(C\) and throw a ball as far as possible into the target area, \(P Q R S\), shown shaded in Figure 3. Given that

• circle \(C\) has centre \(O\)
• \(P\) and \(S\) are points on \(C\)
• \(O P Q R S O\) is a sector of a circle with centre \(O\)
• the length of arc \(P S\) is 0.72 m
• the size of angle \(P O S\) is 0.6 radians
  1. show that \(O P = 1.2 \mathrm {~m}\)

Given also that

$$5 x ^ { 2 } + 12 x - 1500 = 0$$

Hence calculate the total perimeter of the target area, \(P Q R S\), giving your answer to the nearest metre.

6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$