Edexcel F1 (Further Pure Mathematics 1) 2021 October

1. $$\mathbf { A } = \left( \begin{array} { r r } 3 & a
- 2 & - 2 \end{array} \right)$$ where \(a\) is a non-zero constant and \(a \neq 3\)

1. Determine \(\mathbf { A } ^ { - 1 }\) giving your answer in terms of \(a\). Given that \(\mathbf { A } + \mathbf { A } ^ { - 1 } = \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. determine the value of \(a\).

2. $$f ( x ) = 7 \sqrt { x } - \frac { 1 } { 2 } x ^ { 3 } - \frac { 5 } { 3 x } \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [2.8, 2.9]
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence, using \(x _ { 0 } = 2.8\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to calculate a second approximation to \(\alpha\), giving your answer to 3 decimal places.
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3. Use linear interpolation once on the interval [2.8, 2.9] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.

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3. The quadratic equation $$2 x ^ { 2 } - 5 x + 7 = 0$$ has roots \(\alpha\) and \(\beta\) Without solving the equation,

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. determine, giving each answer as a simplified fraction, the value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
3. find a quadratic equation that has roots $$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.

4. $$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are integers. The complex number \(- 1 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Determine the value of \(a\) and the value of \(b\).
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

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5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n - 1 ) ( 3 n - 10 )$$ (b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n + 1 } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

1. In part (i), the elements of each matrix should be expressed in exact numerical form.
   1. (a) Write down the \(2 \times 2\) matrix that represents a rotation of \(210 ^ { \circ }\) anticlockwise about the origin.
      (b) Write down the \(2 \times 2\) matrix that represents a stretch parallel to the \(y\)-axis with scale factor 5
   The transformation \(T\) is a rotation of \(210 ^ { \circ }\) anticlockwise about the origin followed by a stretch parallel to the \(y\)-axis with scale factor 5
   (c) Determine the \(2 \times 2\) matrix that represents \(T\)
2. $$\mathbf { M } = \left( \begin{array} { r r } k & k + 3
   - 5 & 1 - k \end{array} \right) \quad \text { where } k \text { is a constant }$$ (a) Find det \(\mathbf { M }\), giving your answer in simplest form in terms of \(k\). A closed shape \(R\) is transformed to a closed shape \(R ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of \(R\) is 2 square units and that the area of \(R ^ { \prime }\) is \(16 k\) square units,
   (b) determine the possible values of \(k\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\)

The point \(P\) on \(C\) has coordinates ( \(5 p ^ { 2 } , 10 p\) ) where \(p\) is a non-zero constant.

1. Use calculus to show that the tangent to \(C\) at \(P\) has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Write down the coordinates of \(A\). The point \(S\) is the focus of \(C\).
3. Write down the coordinates of \(S\). The straight line \(l _ { 1 }\) passes through \(A\) and \(S\).
   The straight line \(l _ { 2 }\) passes through \(O\) and \(P\), where \(O\) is the origin. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\),
4. show that the coordinates of \(B\) satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

9. (i) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 \quad u _ { 2 } = - 6
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 \times 2 ^ { n } - 2 \times 3 ^ { n }$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 4 n - 1 }$$ is divisible by 11