SPS SPS FM (SPS FM) 2026 November

1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]

\(f(x) = x^3 + 4x^2 + x - 6\).

1. Use the factor theorem to show that \((x + 2)\) is a factor of \(f(x)\). [2]
2. Factorise \(f(x)\) completely. [4]
3. Write down all the solutions to the equation $$x^3 + 4x^2 + x - 6 = 0.$$ [1]

The curve \(C\) has equation $$y = \frac{1}{2}x^3 - 9x^2 + \frac{8}{x} + 30, \quad x > 0$$

1. Find \(\frac{dy}{dx}\). [4]
2. Show that the point \(P(4, -8)\) lies on \(C\). [2]
3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

1. The curves \(e^x - 2e^y = 1\) and \(2e^x + 3e^{2y} = 41\) intersect at the point \(P\). Show that the \(y\)-coordinate of \(P\) satisfies the equation \(3e^{2y} + 4e^y - 39 = 0\). [1]
2. In this question you must show detailed reasoning. Hence find the exact coordinates of \(P\). [5]

1. Show that the equation $$4\cos\theta - 1 = 2\sin\theta\tan\theta$$ can be written in the form $$6\cos^2\theta - \cos\theta - 2 = 0$$ [4]
2. Hence solve, for \(0 \leq x < 90°\) $$4\cos 3x - 1 = 2\sin 3x\tan 3x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [4]

Find the values of \(x\) such that $$2\log_3 x - \log_3(x - 2) = 2$$ [5]

\(f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}\)

1. Write \(f(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers to be found. [3]
2. Sketch the curve with equation \(y = f(x)\) showing any points of intersection with the coordinate axes and the coordinates of any turning point. [3]
3. 1. Describe fully the transformation that maps the curve with equation \(y = f(x)\) onto the curve with equation \(y = g(x)\) where $$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
   2. Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$ [4]

Prove by induction that \(7 \times 9^n - 15\) is divisible by \(4\), for all integers \(n \geq 0\). [5]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio \(r\) and first term \(a\). Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]

Given also that \(S_{10}\) is four times \(S_5\)

1. find the exact value of \(r\). [4]

1. Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$(1 + kx)^{10}$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. [3] Given that in the expansion of \((1 + kx)^{10}\) the coefficient \(x^3\) is 3 times the coefficient of \(x\),
2. find the possible values of \(k\). [3]