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

6. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\).
3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)

1. (i) Given that

$$\log _ { a } x + \log _ { a } 3 = \log _ { a } 27 - 1 , \text { where } a \text { is a positive constant }$$ find, in its simplest form, an expression for \(x\) in terms of \(a\).
(ii) Solve the equation $$\left( \log _ { 5 } y \right) ^ { 2 } - 7 \left( \log _ { 5 } y \right) + 12 = 0$$ showing each step of your working.

10. (a) Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { ( x - 3 ) ^ { 2 } ( x + 4 ) } { 2 } , \quad x \in \mathbb { R }$$ The graph cuts the \(y\)-axis at the point \(P\) and meets the positive \(x\)-axis at the point \(R\), as shown in Figure 5.

1. 1. State the \(y\) coordinate of \(P\).
   2. State the \(x\) coordinate of \(R\). The line segment \(P Q\) is parallel to the \(x\)-axis. Point \(Q\) lies on \(y = \mathrm { f } ( x ) , x > 0\)
2. Use algebra to show that the \(x\) coordinate of \(Q\) satisfies the equation $$x ^ { 2 } - 2 x - 15 = 0$$
3. Use part (b) to find the coordinates of \(Q\). The region \(S\), shown shaded in Figure 5, is bounded by the curve \(y = \mathrm { f } ( x )\) and the line segment \(P Q\).
4. Use calculus to find the exact area of \(S\).

2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5 \\ x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.
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4. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{aligned} x _ { 1 } & = 1 \\ x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(a\) is a constant.

1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
3. find the possible values of \(a\).

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 , \\ x ^ { 2 } + y ^ { 2 } = 29 . \end{gathered}$$

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

1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0 .$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

8. The equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\), where \(p\) is a positive constant, has equal roots.

1. Find the value of \(p\).
2. For this value of \(p\), solve the equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).

6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$ (b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.

4. $$\mathrm { f } ( x ) = 3 x + x ^ { 3 } , \quad x > 0 .$$

1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 15\),
2. find the value of \(x\).

5. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 , \\ x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 , \end{gathered}$$ where \(a\) is a constant.

1. Find an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } - 3 a - 3\). Given that \(x _ { 3 } = 7\),
3. find the possible values of \(a\).

4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2 \\ 4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$

7. Each year, Abbie pays into a savings scheme. In the first year she pays in \(\pounds 500\). Her payments then increase by \(\pounds 200\) each year so that she pays \(\pounds 700\) in the second year, \(\pounds 900\) in the third year and so on.

1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
3. Hence find the number of years that Abbie pays into the savings scheme.

9. (a) On separate axes sketch the graphs of

1. \(y = - 3 x + c\), where \(c\) is a positive constant,
2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
   (b) show that \(( 5 - c ) ^ { 2 } > 12\) (c) Hence find the range of possible values for \(c\).

3. $$f ( x ) = x ^ { 2 } - 10 x + 23$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
2. Hence, or otherwise, find the exact solutions to the equation $$x ^ { 2 } - 10 x + 23 = 0$$
3. Use your answer to part (b) to find the larger solution to the equation $$y - 10 y ^ { 0.5 } + 23 = 0$$ Write your solution in the form \(p + q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point \(A\) lies on \(C\) and has coordinates \(\left( 3 , - \frac { 3 } { 2 } \right)\).

1. Show that the equation of the normal to \(C\) at \(A\) can be written as \(10 y = 4 x - 27\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
2. Use algebra to find the coordinates of \(B\).

5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.

7. (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 90 ^ { \circ }\), the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where \(k\) is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,

1. show that \(k ^ { 2 } - 7 k - 60 = 0\)
2. predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the \(N\) th month.
3. Calculate the value of \(N\) according to the model.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n } \\ u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Find, in terms of \(a\) and \(b\),
   1. \(u _ { 2 }\)
   2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
2. \(b = a + 9\)
3. show that
$$a ^ { 2 } - 5 a - 66 = 0$$
4. Hence find the larger possible value of \(u _ { 2 }\)

4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where \(k\) is a constant.

1. State the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) Given that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\)
2. show that \(k = 11\)
3. Hence
   1. fully factorise \(\mathrm { f } ( x )\),
   2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
   Given that
   • 4 years after monitoring began there were 3690 voles on the island
   • 7 years after monitoring began there were 3470 voles on the island
   • the number of voles on the island at the end of each year is modelled as a geometric sequence
   • find the equation of this model in the form
   $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.