OCR FP1 AS (Further Pure 1 AS) 2021 June

1. Find a vector which is perpendicular to both \(\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ -6 \\ 4 \end{pmatrix}\). [2]
2. The cartesian equation of a line is \(\frac{x}{2} = y - 3 = \frac{z + 4}{4}\). Express the equation of this line in vector form. [3]

In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.

1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]

In this question you must show detailed reasoning.

1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]

Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]