OCR Further Pure Core AS (Further Pure Core AS) 2020 November

In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]

P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.

1. Write down the matrix A. [1]

Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.

1. Find the matrix B. [2]

T is P followed by Q. C is the matrix that represents T.

1. Determine the matrix C. [2]

\(L\) is the line whose equation is \(y = x\).

1. Explain whether or not \(L\) is a line of invariant points under T. [2]

An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).

1. Explain what the value of the determinant of C means about
   [3]

In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).

1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3z - 4z^*\) [2]
   2. \((z + 1 - 3\text{i})^2\) [2]
   3. \(\frac{z + 1}{z - 1}\) [2]
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
   [3]

You are given the system of equations $$a^2x - 2y = 1$$ $$x + b^2y = 3$$ where \(a\) and \(b\) are real numbers.

1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\). [4]
2. Explain why the method used in part (a) works for all values of \(a\) and \(b\). [2]

In this question you must show detailed reasoning. The cubic equation \(5x^3 + 3x^2 - 4x + 7 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta\), \(\beta + \gamma\) and \(\gamma + \alpha\). [7]

Prove that \(n! > 2^{2n}\) for all integers \(n \geq 9\). [5]

The equations of two intersecting lines are $$\mathbf{r} = \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix}$$ where \(a\) is a constant.

1. Find a vector, \(\mathbf{b}\), which is perpendicular to both lines. [2]
2. Show that \(\mathbf{b} \cdot \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} = \mathbf{b} \cdot \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\). [2]
3. Hence, or otherwise, find the value of \(a\). [2]

Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ where \(d\) is a real number.

1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C_1\) and \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]