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

1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).
\(2 \mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D.
P is a reflection in the \(x\)-axis. \(\mathbf { A }\) is the matrix that represents P .

1. Write down the matrix \(\mathbf { A }\). Q is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). \(\mathbf { B }\) is the
   matrix that represents Q . matrix that represents Q.
2. Find the matrix \(\mathbf { B }\). T is P followed by Q. C is the matrix that represents T.
3. Determine the matrix \(\mathbf { C }\).
   \(L\) is the line whose equation is \(y = x\).
4. Explain whether or not \(L\) is a line of invariant points under \(T\). An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
5. Explain what the value of the determinant of \(\mathbf { C }\) means about
   • the area of \(N\) compared to the area of \(M\),
   • the orientation of \(N\) compared to the orientation of \(M\).

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

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

4 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1
x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.

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

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

6 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).

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

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

8 Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\}
& C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 i ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ 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 \(C _ { 2 }\).
   [0pt] [You may assume that \(C _ { 1 } \cap C _ { 2 } \neq \varnothing\).]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). \section*{END OF QUESTION PAPER} \section*{OCR
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