SPS SPS SM (SPS SM) 2022 October

Simplify \(\left(\frac{x^{12}}{16}\right)^{-\frac{3}{4}}\) [2]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = -3x^2 + 12x + 8$$

1. Write \(f(x)\) in the form $$a(x + b)^2 + c$$ where \(a\), \(b\) and \(c\) are constants to be found. [3]

The curve \(C\) has a maximum turning point at \(M\).

1. Find the coordinates of \(M\). [2]

In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Simplify $$\frac{\sqrt{32} + \sqrt{18}}{3 + \sqrt{2}}$$ giving your answer in the form \(b\sqrt{2} + c\), where \(b\) and \(c\) are integers. [5]

The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
2. Hence find the set of possible values of \(k\). [2]

1. Given that $$y = \log_3 x$$ find expressions in terms of \(y\) for
   1. \(\log_3\left(\frac{x}{9}\right)\)
   2. \(\log_3 \sqrt{x}\)
   Write each answer in its simplest form. [3]
2. Hence or otherwise solve $$2\log_3\left(\frac{x}{9}\right) - \log_3 \sqrt{x} = 2$$ [4]

An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54

1. show that $$a + 4d = 6$$ [2]

Given also that the 8th term is half the 7th term,

1. find the values of \(a\) and \(d\). [4]

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

1. Using algebra, find all solutions of the equation $$3x^3 - 17x^2 - 6x = 0$$ [3]
2. Hence find all real solutions of $$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]

\includegraphics{figure_2} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = pm^q$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg. Figure 2 illustrates the linear relationship between \(\log_{10} h\) and \(\log_{10} m\) The line meets the vertical \(\log_{10} h\) axis at 2.25 and has a gradient of \(-0.235\)

1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). [3]

A particular mammal has a mass of 5kg and a resting heart rate of 119 beats per minute.

1. Comment on the suitability of the model for this mammal. [3]
2. With reference to the model, interpret the value of the constant \(p\). [1]

A sequence of numbers \(a_1, a_2, a_3, \ldots\) is defined by $$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, \(n \in \mathbb{N}\) where \(k\) is a constant. Given that

• the sequence is a periodic sequence of order 3
• \(a_1 = 2\)

1. show that $$k^2 + k - 2 = 0$$ [3]
2. For this sequence explain why \(k \neq 1\) [1]
3. Find the value of $$\sum_{r=1}^{80} a_r$$ [3]

A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]

Given also that \(l\) is a tangent to \(C\),

1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]