Edexcel F1 (Further Pure Mathematics 1) 2022 January

1. $$\mathbf { M } = \left( \begin{array} { r r } 3 x & 7
4 x + 1 & 2 - x \end{array} \right)$$ Find the range of values of \(x\) for which the determinant of the matrix \(\mathbf { M }\) is positive.
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2. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = 3 + 5 i \text { and } z _ { 2 } = - 2 + 6 i$$

1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
2. Without using your calculator and showing all stages of your working,
   1. determine the value of \(\left| z _ { 1 } \right|\)
   2. express \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are fully simplified fractions.
3. Hence determine the value of \(\arg \frac { Z _ { 1 } } { Z _ { 2 } }\) Give your answer in radians to 2 decimal places.

3. The parabola \(C\) has equation \(y ^ { 2 } = 18 x\) The point \(S\) is the focus of \(C\)

1. Write down the coordinates of \(S\) The point \(P\), with \(y > 0\), lies on \(C\) The shortest distance from \(P\) to the directrix of \(C\) is 9 units.
2. Determine the exact perimeter of the triangle \(O P S\), where \(O\) is the origin. Give your answer in simplest form.

4. The equation $$x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225 = 0$$ where \(A , B\) and \(C\) are real constants, has

• a complex root \(4 + 3 \mathrm { i }\)
• a repeated positive real root
  1. Write down the other complex root of this equation.
  2. Hence determine a quadratic factor of \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225\)
  3. Deduce the real root of the equation.
  4. Hence determine the value of each of the constants \(A , B\) and \(C\)

5. $$\mathbf { P } = \left( \begin{array} { r r } \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$$ The matrix \(\mathbf { P }\) represents the transformation \(U\)

1. Give a full description of \(U\) as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { Q }\) The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf { R }\)
3. Determine the matrix \(\mathbf { R }\) The transformation \(W\) is represented by the matrix \(3 \mathbf { R }\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T ^ { \prime }\) The transformation \(W ^ { \prime }\) maps the triangle \(T ^ { \prime }\) back to the original triangle \(T\)
4. Determine the matrix that represents \(W ^ { \prime }\)

6. The quadratic equation $$A x ^ { 2 } + 5 x - 12 = 0$$ where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)

1. Write down an expression in terms of \(A\) for
   1. \(\alpha + \beta\)
   2. \(\alpha \beta\) The equation $$4 x ^ { 2 } - 5 x + B = 0$$ where \(B\) is a constant, has roots \(\alpha - \frac { 3 } { \beta }\) and \(\beta - \frac { 3 } { \alpha }\)
2. Determine the value of \(A\)
3. Determine the value of \(B\) The rectangular hyperbola \(H\) has equation \(x y = 36\) The point \(P ( 4,9 )\) lies on \(H\)
4. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4 x - 9 y + 65 = 0$$ The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
5. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are rational constants. \section*{7. In this question you must show all stages of your working.
   Solutions relying entirely on calculator technology are not acceptable.
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8. $$\mathrm { f } ( x ) = 2 x ^ { - \frac { 2 } { 3 } } + \frac { 1 } { 2 } x - \frac { 1 } { 3 x - 5 } - \frac { 5 } { 2 } \quad x \neq \frac { 5 } { 3 }$$ The table below shows values of \(\mathrm { f } ( x )\) for some values of \(x\), with values of \(\mathrm { f } ( x )\) given to 4 decimal places where appropriate.

1. Complete the table giving the values to 4 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has exactly one positive root, \(\alpha\). Using the values in the completed table and explaining your reasoning,
2. determine an interval of width one that contains \(\alpha\).
3. Hence use interval bisection twice to obtain an interval of width 0.25 that contains \(\alpha\). Given also that the equation \(\mathrm { f } ( x ) = 0\) has a negative root, \(\beta\), in the interval \([ - 1 , - 0.5 ]\)
4. use linear interpolation once on this interval to find an approximation for \(\beta\). Give your answer to 3 significant figures.

9. (a) Prove by induction that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the standard summation formulae, show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + A ) ( n + B ) ( n + C )$$ where \(A , B\) and \(C\) are constants to be determined.
(c) Determine the value of \(n\) for which $$3 \sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 17 \sum _ { r = n } ^ { 2 n } r ^ { 2 }$$