Edexcel F1 (Further Pure Mathematics 1) 2014 January

1. $$\mathrm { f } ( x ) = 6 \sqrt { x } - x ^ { 2 } - \frac { 1 } { 2 x } , \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 3,4 ]\).
2. Taking 3 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
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3. Use linear interpolation once on the interval [3,4] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.

2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
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3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4
1 & 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular. The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\).
   Given that the area of triangle \(R\) is 10 square units,
2. find the area of triangle \(S\). Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle \(R\) is transformed to the triangle \(T\) by the matrix \(\mathbf { B }\),
3. find, without evaluating \(\mathbf { B }\), the area of triangle \(T\).

4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$

1. Given that \(x = - 4\) and \(x = 3\) are roots of the equation \(\mathrm { f } ( x ) = 0\), use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$ (b) find the exact value of the constant \(k\).

6.

1. $$\mathbf { B } = \left( \begin{array} { r r } - 1 & 2
   3 & - 4 \end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r } 4 & - 2
   1 & 0 \end{array} \right)$$ (a) Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) is equivalent to the transformation represented by \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\).
   (b) Find \(\mathbf { A }\).
2. $$\mathbf { M } = \left( \begin{array} { r r } - \sqrt { 3 } & - 1
   1 & - \sqrt { 3 } \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement scale factor \(k\), centre ( 0,0 ), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
   (a) Find the value of \(k\).
   (b) Find the value of \(\theta\).

7. (i) Given that $$\frac { 2 w - 3 } { 10 } = \frac { 4 + 7 i } { 4 - 3 i }$$ find \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show your working.
(ii) Given that $$z = ( 2 + \lambda i ) ( 5 + i )$$ where \(\lambda\) is a real constant, and that $$\arg z = \frac { \pi } { 4 }$$ find the value of \(\lambda\).
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8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).

1. Show that an equation of the tangent to \(C\) at \(P\) is $$p y = x + a p ^ { 2 }$$ The tangent to \(C\) at the point \(P\) intersects the directrix of \(C\) at the point \(B\) and intersects the \(x\)-axis at the point \(D\). Given that the \(y\)-coordinate of \(B\) is \(\frac { 5 } { 6 } a\) and \(p > 0\),
2. find, in terms of \(a\), the \(x\)-coordinate of \(D\). Given that \(O\) is the origin,
3. find, in terms of \(a\), the area of the triangle \(O P D\), giving your answer in its simplest form.

1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),

$$f ( n ) = 7 ^ { n } - 2 ^ { n } \text { is divisible by } 5$$