SPS SPS SM (SPS SM) 2024 October

1. A is inversely proportional to B . B is inversely proportional to the square of C . When A is \(2 , \mathrm { C }\) is 8 . Find C when A is 12 .

2.

1. Write \(3 x ^ { 2 } + 24 x + 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants to be determined. The finite region \(R\) is enclosed by the curve \(y = 3 x ^ { 2 } + 24 x + 5\) and the \(x\)-axis.
2. State the inequalities that define \(R\), including its boundaries.

3. The 11th term of an arithmetic progression is 1 . The sum of the first 10 terms is 120 . Find the 4th term.

4. The quadratic equation \(k x ^ { 2 } + 2 k x + 2 k = 3 x - 1\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$4 k ^ { 2 } + 16 k - 9 > 0 .$$
2. Hence find the set of possible values of \(k\). Give your answer in set notation.

5. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(y = \frac { 3 } { 5 } x + 6\)
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(B ( 8,0 )\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l _ { 2 }\) is $$5 x + 3 y = 40$$ Given that
   • lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\)
   • line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\)
   • find the exact area of triangle \(A B C\), giving your answer as a fully simplified fraction in the form \(\frac { p } { q }\)

6. In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
2. Find the time when the mass is 30 grams.
3. Sketch the graph of \(m\) against \(t\).

7. Express \(\frac { a ^ { \frac { 7 } { 2 } } - a ^ { \frac { 5 } { 2 } } } { a ^ { \frac { 3 } { 2 } } - a }\) in the form \(a ^ { m } + \sqrt { a ^ { n } }\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1 .

8. A circle, \(C\), has equation \(x ^ { 2 } - 6 x + y ^ { 2 } = 16\).
A second circle, \(D\), has the following properties:

• The line through the centres of circle \(C\) and circle \(D\) has gradient 1 .
• Circle \(D\) touches circle \(C\) at exactly one point.
• The centre of circle \(D\) lies in the first quadrant.
• Circle \(D\) has the same radius as circle \(C\).

Find the coordinates of the centre of circle \(D\). \section*{9. In this question you must show detailed reasoning.} The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$

1. (a) Show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
   (b) Hence find the exact roots of the equation \(\mathrm { f } ( x ) = 0\).
2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( x ) = 0\).
   (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.

10. The first three terms of a geometric sequence are $$u _ { 1 } = 3 k + 4 \quad u _ { 2 } = 12 - 3 k \quad u _ { 3 } = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of k , giving a reason for your answer.
2. Find the value of \(\sum _ { r = 2 } ^ { \infty } u _ { r }\)