SPS SPS SM (SPS SM) 2020 October

Simplify fully the following expressions:

1. \(\frac{7y^{13}}{35y^7}\) [1]
2. \(6x^{-2} \div x^{-5}\) [1]

A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).

1. State what type of sequence this is. [1]
2. Find \(u_{17}\). [2]

1. Write \(3x^2 - 6x + 1\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [2]
2. Solve \(3x^2 - 6x + 1 \leq 0\), giving your answer in set notation. [4]

In this question you must show detailed reasoning.

1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]

Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]

1. A student was asked to solve the equation \(2^{2x+4} - 9(2^x) = 0\). The student's attempt is written out below. $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ $$\text{Let } y = 2^x$$ $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$ Identify the two mistakes that the student has made. [2]
2. Solve the equation \(2^{2x+4} - 9(2^x) = 0\), giving your answer in exact form. [3]

1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]

The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).

1. Find the centre and radius of the circle. [3]
2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]

In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]