OCR MEI FP1 (Further Pure Mathematics 1) 2005 June

1

1. Find the inverse of the matrix \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} 4 x + 3 y & = 5 \\ x + 2 y & = - 4 \end{aligned}$$ showing your working clearly.

2 Find the roots of the quadratic equation \(x ^ { 2 } - 8 x + 17 = 0\) in the form \(a + b \mathrm { j }\).
Express these roots in modulus-argument form.

3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).

4 The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(2 \alpha\) and \(2 \beta\).

5

1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$

6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).

7 Find \(\sum _ { r = 1 } ^ { n } 3 r ( r - 1 )\), expressing your answer in a fully factorised form.

8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } }\).

1. Find the equations of the asymptotes.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
3. Sketch the curve.
4. Solve the inequality \(\frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } } \geqslant - 1\).

9 The quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers, has roots \(2 + \mathrm { j }\) and - 2 j .

1. Write down the other roots of the equation.
2. Find the values of \(A , B , C\) and \(D\).

10

1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$