OCR Further Pure Core 2 (Further Pure Core 2) 2021 June

In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]

The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
2. Determine the exact value of the length of \(QR\). [7]

A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geq 0\), \(Q\) can be modelled by the differential equation $$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$ Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.

1. Determine the charge on the capacitor when \(t = 0.5\). [6]
2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. [1]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).

1. Find \(\det \mathbf{A}\). [1]

The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.

1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
3. Find the angle of rotation. [2]

Two thin poles, \(OA\) and \(BC\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \((0, 3)\), \((5, 0)\) and \((2, 0)\). \includegraphics{figure_5} It is required to find the height of pole \(BC\) by modelling the shape of the curve that the chain forms. Jofra models the curve using the equation \(y = k \cosh(ax - b) - 1\) where \(k\), \(a\) and \(b\) are positive constants.

1. Determine the value of \(k\). [2]
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. [5]

Holly models the curve using the equation \(y = \frac{1}{4}x^2 - 3x + 3\).

1. Write down the coordinates of the point, \((u, v)\) where \(u\) and \(v\) are both non-zero, at which the two models will agree. [1]
2. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(BC\) by \(3.32\)m to 3 significant figures. [3]