SPS SPS FM Pure (SPS FM Pure) 2021 May

Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.

1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]

The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_2}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
   2. Find the value of \(r\) at the point \(P\). [1]
   3. Mark the point \(P\) on a copy of the graph. [1]
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).

1. Find \(\mathbf{A}^4\). [1]
2. Describe the transformation that \(\mathbf{A}\) represents. [2]

The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).

1. Write down the matrix \(\mathbf{B}\). [1]

The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).

1. Find the coordinates of \(P'\). [1]

Using the formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\), show that \(\sum_{r=1}^{10} r(3r - 2) = 1045\). [3]

Prove by induction that, for all positive integers \(n\), \(7^n + 3^{n-1}\) is a multiple of \(4\). [5]

\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.

1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]

A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),

1. determine the value of \(a\) and the value of \(k\). [3]

Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,

1. determine this value of \(k\). [3]

Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),

1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]

Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),

1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]

1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]

1. Using the Maclaurin series for \(\ln(1 + x)\), find the first four terms in the series expansion for \(\ln(1 + 3x^2)\). [2]
2. Find the range of \(x\) for which the expansion is valid. [1]
3. Find the exact value of the series $$\frac{3^1}{2 \times 2^2} - \frac{3^2}{3 \times 2^4} + \frac{3^3}{4 \times 2^6} - \frac{3^4}{5 \times 2^8} + \ldots$$ [3]

A particular radioactive substance decays over time. A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac{dx}{dt} + \frac{1}{10}x = e^{-0.1t}\cos t.$$

1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). [3]

Initially there was \(10\) g of the substance.

1. Find the particular solution of the differential equation. [2]
2. Find to \(6\) significant figures the amount of substance that would be predicted by the model at
   1. \(6\) hours, [1]
   2. \(6.25\) hours. [1]