OCR FP1 AS (Further Pure 1 AS) 2017 December

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} -3 & 3 & 2 \\ 5 & -4 & -3 \\ -1 & 1 & 1 \end{pmatrix}\).

1. Find \(\mathbf{A}^{-1}\). [1]
2. Solve the simultaneous equations $$-3x + 3y + 2z = 12a$$ $$5x - 4y - 3z = -6$$ $$-x + y + z = 7$$ giving your solution in terms of \(a\). [3]

The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.

1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

\(Q\) is a point on \(l_1\) which is 12 metres away from \(P\). \(R\) is the point on \(l_2\) such that \(QR\) is perpendicular to \(l_1\).

1. Determine the length \(QR\). [2]

In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).

1. Write down the value of \(\omega_1 \omega_2\). [1]
2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]

In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 \equiv (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
2. Given that \(\alpha^2\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^2\), \(\beta^3\) and \(\gamma^3\). [4]

Prove by induction that \(n! \geq 6n\) for \(n \geq 4\). [5]

A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of \(s\) the matrices which represent each of the shears. [7]

1. Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [2]
2. Find the shortest possible vector of the form \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [5]

1. Vector \(\mathbf{v}\) is perpendicular to both \(\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ p \\ p^2 \end{pmatrix}\) where \(p\) is a real number. Show that it is impossible for \(\mathbf{v}\) to be perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ p-1 \end{pmatrix}\). [6]