SPS SPS SM (SPS SM) 2025 November

Do not use a calculator for this question

1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
2. 1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
   2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]

1. Write \(4x^2 - 24x + 27\) in the form \(a(x - b)^2 + c\). [4]
2. State the coordinates of the minimum point on the curve \(y = 4x^2 - 24x + 27\). [2]
3. Solve the equation \(4x^2 - 24x + 27 = 0\). [3]
4. Sketch the graph of the curve \(y = 4x^2 - 24x + 27\). [3]

The equation \(kx^2 + 4x + (5 - k) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\). Write your answer using set notation. [5]

Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of

1. \(\log_2 (16x)\), [2]
2. \(\log_2 \left(\frac{x^4}{2}\right)\) [3]
3. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]

(Total 9 marks)

An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 29 terms is 1102.

1. Show that \(a + 14d = 38\). (3 marks)
2. The sum of the second term and the seventh term is 13. Find the value of \(a\) and the value of \(d\). (4 marks)

A sequence \(t_1, t_2, t_3, t_4, t_5, \ldots\) is given by $$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$ where \(a\) is a non zero constant.

1. Given that \(\sum_{r=1}^{3} (r^3 + t_r) = 12\), determine the possible values of \(a\). [4]
2. Evaluate \(\sum_{r=8}^{31} (t_{r+1} - at_r)\). [4]

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows. These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.

1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]

The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$

1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
2. Determine the coordinates of \(P\). [3]
3. Find the equation of the common tangent to the circles at P. [4]