SPS SPS FM (SPS FM) 2024 October

1. 1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
   2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]

1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]

1. Find and simplify the first three terms in the expansion of \((2-5x)^5\) in ascending powers of \(x\). [3]
2. In the expansion of \((1+ax)^2(2-5x)^5\), the coefficient of \(x\) is 48. Find the value of \(a\). [3]

The functions f and g are defined for all real values of \(x\) by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

In this question you must show detailed reasoning Find the equation of the normal to the curve \(y = \frac{x^2-32}{\sqrt{x}}\) at the point on the curve where \(x = 4\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]

In this question you must show detailed reasoning. A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]

Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]

1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.

[6 marks]