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Edexcel F1 2016 June Q3
7 marks Standard +0.3
3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).
  1. Taking - 1.5 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 2 decimal places.
  2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places.
    tion 3continued -
Edexcel F1 2016 June Q4
5 marks Moderate -0.8
4. Given that $$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$
  1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
  2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
Edexcel F1 2016 June Q5
5 marks Moderate -0.3
5. $$2 z + z ^ { * } = \frac { 3 + 4 i } { 7 + i }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.
Edexcel F1 2016 June Q6
10 marks Standard +0.8
6. The rectangular hyperbola \(H\) has equation \(x y = 25\)
  1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\). The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
  2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
  3. Find the coordinates of \(B\).
Edexcel F1 2016 June Q7
10 marks Standard +0.3
7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
  2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
  3. find the matrix \(\mathbf { R }\).
  4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\). \section*{II}
Edexcel F1 2016 June Q8
9 marks Standard +0.3
8. $$f ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants. Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. write down another complex root of this equation.
  2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  3. Show on a single Argand diagram all four roots of the equation \(f ( z ) = 0\)
Edexcel F1 2016 June Q9
9 marks Standard +0.8
9. The quadratic equation $$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation,
  1. find the exact value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-27_99_332_2622_1466}
Edexcel F1 2016 June Q10
11 marks Standard +0.3
10.
  1. A sequence of positive numbers is defined by $$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$
  2. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
Edexcel F1 2017 June Q1
4 marks Moderate -0.8
  1. The quadratic equation
$$3 x ^ { 2 } - 5 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation, find the exact value of $$\frac { \alpha } { \beta } + \frac { \beta } { \alpha }$$
Count coution \(\_\_\_\_\) T
Edexcel F1 2017 June Q2
4 marks Moderate -0.5
2. Given that $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 2 \\ - 1 & 0 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } 2 & 4 \\ - k & 2 k \\ 3 & 0 \end{array} \right) , \text { where } k \text { is a constant }$$
  1. find the matrix \(\mathbf { A B }\),
  2. find the exact value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)
Edexcel F1 2017 June Q3
5 marks Standard +0.8
3. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
Edexcel F1 2017 June Q4
7 marks Standard +0.3
4. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t }$$ The straight line with equation \(3 y - 2 x = 10\) intersects \(H\) at the points \(A\) and \(B\). Given that the point \(A\) is above the \(x\)-axis,
  1. find the coordinates of the point \(A\) and the coordinates of the point \(B\).
  2. Find the coordinates of the midpoint of \(A B\).
Edexcel F1 2017 June Q5
9 marks Moderate -0.3
5. $$f ( x ) = 30 + \frac { 7 } { \sqrt { x } } - x ^ { 5 } , \quad x > 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [2,2.1].
[0pt]
  1. Starting with the interval [2,2.1], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
  2. Taking 2 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 2 decimal places.
Edexcel F1 2017 June Q6
8 marks Standard +0.3
6.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { n } { a } ( n + 1 ) ( n + 2 ) ( 3 n + b )$$ where \(a\) and \(b\) are integers to be found.
  2. Hence find the value of $$\sum _ { r = 25 } ^ { 49 } \left( r ^ { 2 } ( r + 1 ) + 2 \right)$$
Edexcel F1 2017 June Q7
8 marks Moderate -0.3
7. $$f ( z ) = z ^ { 4 } + 4 z ^ { 3 } + 6 z ^ { 2 } + 4 z + a$$ where \(a\) is a real constant. Given that \(1 + 2 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. write down another complex root of this equation.
    1. Hence, find the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
    2. State the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{cfeb435a-03c2-4bcd-9c9f-6f62b4556cb3-15_31_33_205_2014}
      " " \includegraphics[max width=\textwidth, alt={}, center]{cfeb435a-03c2-4bcd-9c9f-6f62b4556cb3-15_42_53_317_1768}
Edexcel F1 2017 June Q8
11 marks Standard +0.3
8. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 36 x\). The point \(P \left( 9 p ^ { 2 } , 18 p \right)\), where \(p\) is a positive constant, lies on \(C\).
  1. Using calculus, show that an equation of the tangent to \(C\) at \(P\) is $$p y - x = 9 p ^ { 2 }$$ This tangent cuts the directrix of \(C\) at the point \(A ( - a , 6 )\), where \(a\) is a constant.
  2. Write down the value of \(a\).
  3. Find the exact value of \(p\).
  4. Hence find the exact coordinates of the point \(P\), giving each coordinate as a simplified surd.
Edexcel F1 2017 June Q9
10 marks Moderate -0.3
9. $$z = \frac { 1 } { 5 } - \frac { 2 } { 5 } \mathrm { i }$$
  1. Find the modulus and the argument of \(z\), giving the modulus as an exact answer and giving the argument in radians to 2 decimal places. Given that $$\mathrm { zw } = \lambda \mathrm { i }$$ where \(\lambda\) is a real constant,
  2. find \(w\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. Give your answer in terms of \(\lambda\).
  3. Given that \(\lambda = \frac { 1 } { 10 }\)
    1. find \(\frac { 4 } { 3 } ( z + w )\),
    2. plot the points \(A , B , C\) and \(D\), representing \(z , z w , w\) and \(\frac { 4 } { 3 } ( z + w )\) respectively, on a single Argand diagram.
Edexcel F1 2017 June Q10
9 marks Standard +0.3
10. In your answers to this question, the elements of each matrix should be expressed in exact form in surds where necessary. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(45 ^ { \circ }\) anticlockwise about the origin.
  1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(60 ^ { \circ }\) anticlockwise about the origin.
  2. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
  3. Use your matrices from parts (a) and (b) to find the matrix \(\mathbf { R }\).
  4. Give a full geometric description of \(T\) as a single transformation.
  5. Deduce from your answers to parts (c) and (d) that \(\sin 75 ^ { \circ } = \frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }\) and find the
    exact value of \(\cos 75 ^ { \circ }\), explaining your answers fully.
Edexcel F1 2018 June Q1
4 marks Moderate -0.5
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { n } { a } ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are integers to be found.
Edexcel F1 2018 June Q2
7 marks Moderate -0.8
The transformation represented by the \(2 \times 2\) matrix \(\mathbf { P }\) is an anticlockwise rotation about the origin through 45 degrees.
  1. Write down the matrix \(\mathbf { P }\), giving the exact numerical value of each element. $$\mathbf { Q } = \left( \begin{array} { c c } k \sqrt { 2 } & 0 \\ 0 & k \sqrt { 2 } \end{array} \right) \text {, where } k \text { is a constant and } k > 0$$
  2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { Q }\). The combined transformation represented by the matrix \(\mathbf { P Q }\) transforms the rhombus \(R _ { 1 }\) onto the rhombus \(R _ { 2 }\). The area of the rhombus \(R _ { 1 }\) is 6 and the area of the rhombus \(R _ { 2 }\) is 147
  3. Find the value of the constant \(k\).
Edexcel F1 2018 June Q3
5 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b3a6bed4-2d9c-48a3-8831-efb5ba09baa4-08_536_533_221_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the parabola \(C\) which has cartesian equation \(y ^ { 2 } = 6 x\). The point \(S\) is the focus of \(C\).
  1. Find the coordinates of the point \(S\). The point \(P\) lies on the parabola \(C\), and the point \(Q\) lies on the directrix of \(C\). \(P Q\) is parallel to the \(x\)-axis with distance \(P Q = 14\)
  2. State the distance \(S P\). Given that the point \(P\) is above the \(x\)-axis,
  3. find the exact coordinates of \(P\).
Edexcel F1 2018 June Q4
7 marks Standard +0.3
4. $$\mathbf { A } = \left( \begin{array} { c c } 2 p & 3 q \\ 3 p & 5 q \end{array} \right)$$ where \(p\) and \(q\) are non-zero real constants.
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\) and \(q\). Given \(\mathbf { X A } = \mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { c c } p & q \\ 6 p & 11 q \\ 5 p & 8 q \end{array} \right)$$
  2. find the matrix \(\mathbf { X }\), giving your answer in its simplest form.
Edexcel F1 2018 June Q5
8 marks Moderate -0.3
5. Given that $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real numbers,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence find the exact roots of the equation $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
  3. Show your roots on a single Argand diagram.
Edexcel F1 2018 June Q6
9 marks Standard +0.3
6. $$f ( x ) = \frac { 2 \left( x ^ { 3 } + 3 \right) } { \sqrt { x } } - 9 , \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has two real roots \(\alpha\) and \(\beta\), where \(0.4 < \alpha < 0.5\) and \(1.2 < \beta < 1.3\)
  1. Taking 0.45 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 decimal places.
    [0pt]
  2. Use linear interpolation once on the interval [1.2, 1.3] to find an approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel F1 2018 June Q7
9 marks Standard +0.3
7. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(5 x ^ { 2 } - 4 x + 3 = 0\) Without solving the quadratic equation,
  1. find the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\)
  2. find a quadratic equation which has roots \(\frac { 3 } { \alpha ^ { 2 } }\) and \(\frac { 3 } { \beta ^ { 2 } }\) giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.