SPS SPS SM (SPS SM) 2021 November

1. Do not use a calculator for this question

Find the value of \(x\) for which \(\sqrt { 3 } \times 3 ^ { x } = \frac { 1 } { 9 }\)
[0pt] [2 marks]

2. You must show detailed working in this question Determine whether the line with equation \(2 x + 3 y + 4 = 0\) is parallel to the line through the points with coordinates \(( 9,4 )\) and \(( 3,8 )\).
[0pt] [4 marks]

3. An arithmetic sequence has first term \(a\) and common difference \(d\).
The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. Show that \(4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }\)
[0pt] [4 marks]

4. Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
[0pt] [2 marks]

5.
\(\mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)
Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\)
[0pt] [2 marks]
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L Factorise \(\mathrm { p } ( x )\) completely.
[0pt] [3 marks]
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6. You are not allowed to use a calculator for this question. Show detailed reasoning. Show that \(\frac { 5 \sqrt { 2 } + 2 } { 3 \sqrt { 2 } + 4 }\) can be expressed in the form \(m + n \sqrt { 2 }\), where \(m\) and \(n\) are integers.
[0pt] [3 marks]

7. The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
Find the possible values of the constant \(k\)
Fully justify your answer.
[0pt] [4 marks]

8. The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\).

1. 1. Write down the radius of the circle.
      [0pt] [1 mark]
2. (ii) Write down the coordinates of \(C\).
   [0pt] [1 mark]
3. The point \(P ( 5 , - 1 )\) lies on the circle. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = m x + c\)
   [0pt] [4 marks]
4. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
   [0pt] [4 marks]

9. David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.

1. Using David's model:
2. 1. state the population of rabbits on the island on 1 January 2016;
3. (ii) predict the population of rabbits on 1 January 2021.
4. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
5. Give one reason why David's model may not be appropriate.
   [0pt] [1 mark]
6. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where \(t\) is the time in years after 1 January 2016.
   Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
   [0pt] [3 marks]