SPS SPS FM (SPS FM) 2022 February

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & 1 \\ 0 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}\).

1. Find \(\mathbf{A} + 3\mathbf{B}\). [2]
2. Show that \(\mathbf{A} - \mathbf{B} = k\mathbf{I}\), where \(\mathbf{I}\) is the identity matrix and \(k\) is a constant whose value should be stated. [2]

The complex numbers \(3 - 2i\) and \(2 + i\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + iy\) and showing clearly how you obtain these answers,

1. \(2z - 3w\), [2]
2. \((iz)^2\), [3]
3. \(\frac{z}{w}\). [3]

The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_3}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]

The transformation \(S\) is a shear parallel to the \(x\)-axis in which the image of the point \((1, 1)\) is the point \((0, 1)\).

1. Draw a diagram showing the image of the unit square under \(S\). [2]
2. Write down the matrix that represents \(S\). [2]

1. Sketch the curve \(y = \left(\frac{1}{2}\right)^x\), and state the coordinates of any point where the curve crosses an axis. [3]
2. Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve \(y = \left(\frac{1}{2}\right)^x\), the axes, and the line \(x = 2\). [4]
3. The point \(P\) on the curve \(y = \left(\frac{1}{2}\right)^x\) has \(y\)-coordinate equal to \(\frac{1}{6}\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac{\log_{10} 3}{\log_{10} 2}.$$ [4]

In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\).

1. Find \(\mathbf{A}^2\) and \(\mathbf{A}^3\). [3]
2. Hence suggest a suitable form for the matrix \(\mathbf{A}^n\). [1]
3. Use induction to prove that your answer to part (ii) is correct. [4]

1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]