Questions — Edexcel F1 (198 questions)

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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. (a) 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.
(b) 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
  1. 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.
Edexcel F1 2018 June Q8
5 marks Standard +0.8
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { l l } a & 0 \\ 1 & b \end{array} \right) ^ { n } = \left( \begin{array} { c c } a ^ { n } & 0 \\ \frac { a ^ { n } - b ^ { n } } { a - b } & b ^ { n } \end{array} \right)$$ where \(a\) and \(b\) are constants and \(a \neq b\).
Edexcel F1 2018 June Q9
8 marks Moderate -0.3
9. Given that $$\frac { z - k \mathrm { i } } { z + 3 \mathrm { i } } = \mathrm { i } \text {, where } k \text { is a positive real constant }$$
  1. show that \(z = - \frac { ( k + 3 ) } { 2 } + \frac { ( k - 3 ) } { 2 } \mathrm { i }\)
  2. Using the printed answer in part (a),
    1. find an exact simplified value for the modulus of \(z\) when \(k = 4\)
    2. find the argument of \(z\) when \(k = 1\). Give your answer in radians to 3 decimal places, where \(- \pi < \arg z < \pi\)
Edexcel F1 2018 June Q10
13 marks Standard +0.8
10. The rectangular hyperbola \(H\) has equation \(x y = 144\). The point \(P\), on \(H\), has coordinates \(\left( 12 p , \frac { 12 } { p } \right)\), where \(p\) is a non-zero constant.
  1. Show, by using calculus, that the normal to \(H\) at the point \(P\) has equation $$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$ Given that the normal through \(P\) crosses the positive \(x\)-axis at the point \(Q\) and the negative \(y\)-axis at the point \(R\),
  2. find the coordinates of \(Q\) and the coordinates of \(R\), giving your answers in terms of \(p\).
  3. Given also that the area of triangle \(O Q R\) is 512 , find the possible values of \(p\).
    VIUV SIHI NI JIIIM ION OCVI4V SIHI NI JINM IONOOVJYV SIHI NI GLIYM LON OO
Edexcel F1 2020 June Q1
7 marks Standard +0.8
1. $$f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\) .
  3. Using \(x _ { 0 } = 1.4\) 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. \(f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0\)
    1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
    2. Determine \(\mathrm { f } ^ { \prime } ( x )\) .
Edexcel F1 2020 June Q2
9 marks Standard +0.3
2
2. The quadratic equation $$5 x ^ { 2 } - 2 x + 3 = 0$$ 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. determine a quadratic equation that has roots $$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel F1 2020 June Q3
9 marks Standard +0.3
3. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are integers.
The complex numbers \(3 + \mathrm { i }\) and \(- 1 - 2 \mathrm { i }\) are roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. Write down the other roots of this equation.
  2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
  3. Determine the values of \(a , b , c\) and \(d\).
    VILU SIHI NI JIIIM ION OCVIUV SIHI NI III M M I ON OOVIAV SIHI NI JIIIM I ION OC
Edexcel F1 2020 June Q4
9 marks Standard +0.3
4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ for all positive integers \(n\).
(b) Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500 \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-13_2255_50_314_34}
VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F1 2020 June Q5
9 marks Challenging +1.2
  1. The rectangular hyperbola \(H\) has equation \(x y = 64\)
The point \(P \left( 8 p , \frac { 8 } { p } \right)\), where \(p \neq 0\), lies on \(H\).
  1. Use calculus to show that the normal to \(H\) at \(P\) has equation $$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
  2. Determine, in terms of \(p\), the coordinates of \(Q\), giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-17_2255_50_314_34}
Edexcel F1 2020 June Q6
10 marks Standard +0.3
6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$
  1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(45 ^ { \circ }\) clockwise about the origin.
  2. Write down the matrix \(\mathbf { B }\), giving each element of the matrix in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
  3. Determine \(\mathbf { C }\).
    (ii) The trapezium \(T\) has vertices at the points \(( - 2,0 ) , ( - 2 , k ) , ( 5,8 )\) and \(( 5,0 )\), where \(k\) is a positive constant. Trapezium \(T\) is transformed onto the trapezium \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 5 & 1 \\ - 2 & 3 \end{array} \right)$$ Given that the area of trapezium \(T ^ { \prime }\) is 510 square units, calculate the exact value of \(k\).
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F1 2020 June Q7
10 marks Standard +0.8
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The line \(l\) with equation \(3 x - 4 y + 48 = 0\) is a tangent to \(C\) at the point \(P\).
  1. Show that \(a = 9\)
  2. Hence determine the coordinates of \(P\). Given that the point \(S\) is the focus of \(C\) and that the line \(l\) crosses the directrix of \(C\) at the point \(A\),
  3. determine the exact area of triangle \(P S A\). \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-25_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO