SPS SPS FM (SPS FM) 2025 October

Determine the equation of the line that passes through the point \((1, 3)\) and is perpendicular to the line with equation \(3x + 6y - 5 = 0\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]

In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.

1. Show that \(\cos CAB = \frac{2}{3}\). [2]
2. Hence find the exact value of \(\sin CAB\). [2]
3. Find the exact area of triangle \(ABC\). [2]

Given the function \(f(x) = 3x^3 - 7x - 1\), defined for all real values of \(x\), prove from first principles that \(f'(x) = 9x^2 - 7\). [4]

The cubic polynomial \(2x^3 - kx^2 + 4x + k\), where \(k\) is a constant, is denoted by f(x). It is given that f'(2) = 16.

1. Show that \(k = 3\). [3]

For the remainder of the question, you should use this value of \(k\).

1. Use the factor theorem to show that \((2x + 1)\) is a factor of f(x). [2]
2. Hence show that the equation f(x) = 0 has only one real root. [3]

In this question you must show detailed reasoning. Consider the expansion of \(\left(\frac{x^2}{2} + \frac{a}{x}\right)^6\). The constant term is 960. Find the possible values of \(a\). [4]

The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$

1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
2. Find the nature of the stationary point, justifying your answer. [2]

The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]

A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]

\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)

1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
2. Find the equation of C. You may leave your answer in factorised form. [3]

The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,

1. find the range of possible values for \(k\). [2]

The graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis followed by a translation.

1. 1. State the scale factor of the stretch. [1]
   2. Give full details of the translation. [2]

Alternatively the graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.

1. State the scale factor of the stretch parallel to the \(y\)-axis. [1]

The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$

1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
2. Find the range of the function fg. [2]
3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]

Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]

In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0° < x < 180°\) $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]