Edexcel FP1 (Further Pure Mathematics 1) 2013 June

1. \(\mathbf { M } = \left( \begin{array} { c c } a & 1
   1 & 2 - a \end{array} \right)\), where \(a\) is a constant.
   1. Find det M in terms of \(a\).
      (2)
   A triangle \(T\) is transformed to \(T ^ { \prime }\) by the matrix M .
   Given that the area of \(T ^ { \prime }\) is 0 ,
2. find the value of \(a\).
   (3)

2. $$f ( z ) = z ^ { 3 } + 5 z ^ { 2 } + 11 z + 15$$ Given that \(z = 2 i - 1\) is a solution of the equation \(f ( z ) = 0\), use algebra to solve \(f ( z ) = 0\) completely.
(5)

3. $$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$

1. Express \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) giving exact values of \(r\) and \(\theta\).
   (4)
2. Find \(\left| z _ { 1 } z _ { 2 } \right|\).
3. Show and label \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
   (2)

4. The hyperbola \(H\) has equation $$x y = 3$$ The point \(Q ( 1,3 )\) is on \(H\).

1. Find the equation of the normal to \(H\) at \(Q\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
   (5) The normal at \(Q\) intersects \(H\) again at the point \(R\).
2. Find the coordinates of \(R\).
   (5)

5. Prove, by induction, that \(3 ^ { 2 n } + 7\) is divisible by 8 for all positive integers \(n\).

6. A curve \(C\) is in the form of a parabola with equation \(y ^ { 2 } = 4 x\).
\(P \left( p ^ { 2 } , 2 p \right)\) and \(Q \left( q ^ { 2 } , 2 q \right)\) are points on \(C\) where \(p > q\).

1. Find an equation of the tangent to \(C\) at \(P\).
   (5)
2. The tangent at \(P\) and the tangent at \(Q\) are perpendicular and intersect at the point \(R ( - 1,2 )\).
   1. Find the exact value of \(p\) and the exact value of \(q\).
   2. Find the area of the triangle \(P Q R\).

7. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { n ( n + 1 ) ( 3 n + 2 ) ( n - 1 ) } { 12 }$$ for all positive integers \(n\).
(b) Hence find the sum of the series $$10 ^ { 2 } \times 9 + 11 ^ { 2 } \times 10 + 12 ^ { 2 } \times 11 + \ldots + 50 ^ { 2 } \times 49$$

8. $$f ( x ) = x ^ { 3 } - 2 x - 3$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval \([ 1,2 ]\).
2. Starting with the interval \([ 1,2 ]\), use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
3. Using \(x _ { 0 } = 1.8\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures.

9. With reference to a fixed origin \(O\) and coordinate axes \(O x\) and \(O y\), a transformation from \(\mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(A\) where $$A = \left( \begin{array} { c c } 3 & 1
1 & - 2 \end{array} \right)$$

1. Find \(\mathrm { A } ^ { 2 }\).
2. Show that the matrix A is non-singular.
3. Find \(\mathrm { A } ^ { - 1 }\). The transformation represented by matrix A maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 1,2 - k )\), where \(k\) is a constant,
4. show that \(P\) lies on the line with equation \(y = 4 x - 1\)