SPS SPS FM (SPS FM) 2024 October

1. The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(f ( x )\).

Use differentiation from first principles to determine the value of \(f ^ { \prime } ( 2 )\).
[0pt] [BLANK PAGE]

2. The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
[0pt] [BLANK PAGE]

3. (i) Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
(ii) In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
[0pt] [BLANK PAGE]

4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
[0pt] [BLANK PAGE]

5. A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).

1. Show that the line does not meet the circle.
2. 1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
   2. Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
      [0pt] [BLANK PAGE]

6. A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0
& \log _ { 3 } x - 2 = 0
& \log _ { 3 } x = 2
& x = 8 \end{aligned}$$

1. Identify the two mistakes that the student has made.
2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
   [0pt] [BLANK PAGE]

7. In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).
\(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
Calculate the possible values of the angle \(A D B\).
[0pt] [BLANK PAGE]

8. In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac { 5 } { 3 x - 4 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 2 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 3 } + \ldots$$ is convergent.

1. Find the set of possible values of \(x\), giving your answer in set notation.
2. Given that the sum to infinity of the series is \(\frac { 2 } { 3 }\), find the value of \(x\).
   [0pt] [BLANK PAGE]

9. Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]