OCR FP1 (Further Pure Mathematics 1) Specimen

1 Use formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$

2 The cubic equation \(x ^ { 3 } - 6 x ^ { 2 } + k x + 10 = 0\) has roots \(p - q , p\) and \(p + q\), where \(q\) is positive.

1. By considering the sum of the roots, find \(p\).
2. Hence, by considering the product of the roots, find \(q\).
3. Find the value of \(k\).

3 The complex number \(2 + \mathrm { i }\) is denoted by \(z\), and the complex conjugate of \(z\) is denoted by \(z ^ { * }\).

1. Express \(z ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
2. Show that \(4 z - z ^ { 2 }\) simplifies to a real number, and verify that this real number is equal to \(z z ^ { * }\).
3. Express \(\frac { z + 1 } { z - 1 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 3 ^ { 2 n } - 1$$

1. Write down the value of \(u _ { 1 }\).
2. Show that \(u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }\).
3. Hence prove by induction that each term of the sequence is a multiple of 8 .

5

1. Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
2. Hence find an expression in terms of \(n\) for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
3. State the value of
   (a) \(\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\),
   (b) \(\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\).

6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).

1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
   (a) the real part of \(z\) is zero,
   (b) \(\quad \arg z = \arg a\).

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & - 2
2 & 1 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { A }\).
2. The value of \(\operatorname { det } \mathbf { A }\) is 5 . Show clearly how this value relates to your diagram in part (i). A represents a sequence of two elementary geometrical transformations, one of which is a rotation \(R\).
3. Determine the angle of \(R\), and describe the other transformation fully.
4. State the matrix that represents \(R\), giving the elements in an exact form.

8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1
2 & 3 & - 1
2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.

1. Show that the determinant of \(\mathbf { M }\) is \(2 a\).
2. Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
3. Hence or otherwise solve the simultaneous equations $$\begin{array} { r } x + 2 y - z = 1
   2 x + 3 y - z = 2
   2 x - y + z = 0 \end{array}$$
4. Find the value of \(k\) for which the simultaneous equations $$\begin{array} { r } 2 y - z = k
   2 x + 3 y - z = 2
   2 x - y + z = 0 \end{array}$$ have solutions.
5. Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.