SPS SPS FM Pure (SPS FM Pure) 2023 June

You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).

1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
2. State the range of \(gf(x)\). [1]
3. Solve the inequality \(|3x - 1| > 1\). [3]

In this question you must show detailed reasoning.

1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]

You are given that \(f(x) = \ln(2x - 5) + 2x^2 - 30\), for \(x > 2.5\).

1. Show that \(f(x) = 0\) has a root \(\alpha\) in the interval \([3.5, 4]\). [2]

A student takes 4 as the first approximation to \(\alpha\). Given \(f(4) = 3.099\) and \(f'(4) = 16.67\) to 4 significant figures,

1. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures. [2]
2. Show that \(\alpha\) is the only root of \(f(x) = 0\). [2]

You are given that \(M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}\).

1. Show that \(M\) is non-singular. [2]

The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(M\). Given that the area of hexagon \(R\) is 5 square units,

1. find the area of hexagon \(S\). [1]

The matrix \(M\) represents an enlargement, with centre \((0,0)\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \((0,0)\).

1. Find the value of \(k\). [2]
2. Find the value of \(\theta\). [2]

\includegraphics{figure_5} Figure 1 shows a sketch of a triangle \(ABC\). Given \(\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{BC} = \mathbf{i} - 9\mathbf{j} + 3\mathbf{k}\), show that \(\angle BAC = 105.9°\) to one decimal place. [5]

A spherical balloon is inflated so that its volume increases at a rate of \(10\text{ cm}^3\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm. [For a sphere of radius \(r\), surface area \(= 4\pi r^2\) and volume \(= \frac{4}{3}\pi r^3\)]. [5]

Prove that for all \(n \in \mathbb{N}\) $$\begin{pmatrix} 3 & 4i \\ i & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & 4ni \\ ni & 1-2n \end{pmatrix}$$ [6]

1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]

The complex number \(w\) satisfies \(|w - 4i| = 3\).

1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]

1. Use the binomial expansion to show that \((1 - 2x)^{-\frac{1}{4}} \approx 1 + x + \frac{5}{8}x^2\) for sufficiently small values of \(x\). [2]
2. For what values of \(x\) is the expansion valid? [1]
3. Find the expansion of \(\sqrt{\frac{1+2x}{1-2x}}\) in ascending powers of \(x\) as far as the term in \(x^2\). [3]
4. Use \(x = \frac{1}{20}\) in your answer to part (iii) to find an approximate value for \(\sqrt{11}\). [2]

The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number. Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]

In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]

\includegraphics{figure_12} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\). [7]

1. Solve the differential equation $$\frac{dy}{dx} = y(1 + y)(1 - x),$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\), where \(f\) is a function to be determined. [7]
2. By considering the sign of \(\frac{dy}{dx}\) near \((1, 1)\), or otherwise, show that this point is a maximum point on the curve \(y = f(x)\). [3]

A curve \(C\) has equation $$x^3 + y^3 = 3xy + 48$$ Prove that \(C\) has two stationary points and find their coordinates. [7]

In this question you must use detailed reasoning.

1. Show that \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1+\sin 2x}{-\cos 2x} dx = \ln(\sqrt{a} + b)\), where \(a\) and \(b\) are integers to be determined. [6]
2. Show that \(\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1+\sin 2x}{-\cos 2x} dx\) is undefined, explaining your reasoning clearly. [2]