Questions F1 (198 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel F1 Specimen Q4
12 marks Standard +0.3
  1. The quadratic equation
$$5 x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }\)
  3. Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$
Edexcel F1 Specimen Q5
6 marks Standard +0.3
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3 \text { is divisible by } 4$$
Edexcel F1 Specimen Q6
10 marks Standard +0.8
6. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k )$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\)
Edexcel F1 Specimen Q7
12 marks Challenging +1.2
  1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is
$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).
Edexcel F1 Specimen Q8
11 marks Standard +0.3
8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$
  1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
  2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
  3. Find the matrix \(\mathbf { R }\).
    (ii) $$S = \left( \begin{array} { r r } 1 & - 3 \\ 3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
    1. find the scale factor of the enlargement,
    2. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.
Edexcel F1 2017 January Q2
7 marks Standard +0.3
The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
  3. find a quadratic equation which has roots $$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel F1 2021 June Q1
10 marks Moderate -0.3
1.(i) $$f ( x ) = x ^ { 3 } + 4 x - 6$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1,1.5]
  2. Taking 1.5 as a first approximation,apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\) .Give your answer to 3 decimal places. Show your working clearly.
    (ii) $$g ( x ) = 4 x ^ { 2 } + x - \tan x$$ where \(x\) is measured in radians. The equation \(\mathrm { g } ( x ) = 0\) has a single root \(\beta\) in the interval[1.4,1.5]
    Use linear interpolation on the values at the end points of this interval to obtain an approximation to \(\beta\) .Give your answer to 3 decimal places.
Edexcel F1 2021 June Q2
7 marks Standard +0.3
2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.
  1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
  2. find the possible values of \(p\).
Edexcel F1 2021 June Q3
10 marks Moderate -0.8
  1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).
The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
  1. Find the coordinates of the vertices of \(T ^ { \prime }\)
  2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
  3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
  4. Find the matrix \(\mathbf { R Q }\)
  5. Give a full geometrical description of the single transformation represented by the answer to part (d).
Edexcel F1 2021 June Q4
8 marks Standard +0.8
  1. A rectangular hyperbola \(H\) has equation \(x y = 25\)
The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).
  1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
  2. Find the coordinates of the points \(Q\) and \(R\).
Edexcel F1 2021 June Q5
7 marks Moderate -0.3
5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.
  1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
  2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
  3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}
Edexcel F1 2021 June Q6
16 marks Standard +0.8
6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).
  1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
  2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
  3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
  4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)
Edexcel F1 2021 June Q7
11 marks Moderate -0.3
7. (a) Prove by induction that for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
(c) Using your answers to part (b), find the value of $$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$
Edexcel F1 2021 June Q8
6 marks Standard +0.8
8. Prove by induction that \(4 ^ { n + 2 } + 5 ^ { 2 n + 1 }\) is divisible by 21 for all positive integers \(n\).
\includegraphics[max width=\textwidth, alt={}]{d7689f4a-a41e-45be-911b-4a74e81997eb-32_2644_1837_118_114}
Edexcel F1 2022 January Q1
5 marks Moderate -0.3
$$\mathbf{M} = \begin{pmatrix} 3x & 7 \\ 4x + 1 & 2 - x \end{pmatrix}$$ Find the range of values of \(x\) for which the determinant of the matrix \(\mathbf{M}\) is positive. [5]
Edexcel F1 2022 January Q2
8 marks Moderate -0.8
The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$
  1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
  2. Without using your calculator and showing all stages of your working,
    1. determine the value of \(|z_1|\) [1]
    2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
  3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]
Edexcel F1 2022 January Q3
5 marks Standard +0.3
The parabola \(C\) has equation \(y^2 = 18x\) The point \(S\) is the focus of \(C\)
  1. Write down the coordinates of \(S\) [1]
The point \(P\), with \(y > 0\), lies on \(C\) The shortest distance from \(P\) to the directrix of \(C\) is 9 units.
  1. Determine the exact perimeter of the triangle \(OPS\), where \(O\) is the origin. Give your answer in simplest form. [4]
Edexcel F1 2022 January Q4
8 marks Standard +0.8
The equation $$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$ where \(A\), \(B\) and \(C\) are real constants, has
  • a complex root \(4 + 3\text{i}\)
  • a repeated positive real root
  1. Write down the other complex root of this equation. [1]
  2. Hence determine a quadratic factor of \(x^4 + Ax^3 + Bx^2 + Cx + 225\) [2]
  3. Deduce the real root of the equation. [2]
  4. Hence determine the value of each of the constants \(A\), \(B\) and \(C\) [3]
Edexcel F1 2022 January Q5
8 marks Standard +0.3
$$\mathbf{P} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$$ The matrix \(\mathbf{P}\) represents the transformation \(U\)
  1. Give a full description of \(U\) as a single geometrical transformation. [2]
The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf{Q}\), is a reflection in the line \(y = -x\)
  1. Write down the matrix \(\mathbf{Q}\) [1]
The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf{R}\)
  1. Determine the matrix \(\mathbf{R}\) [2]
The transformation \(W\) is represented by the matrix \(3\mathbf{R}\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T'\) The transformation \(W'\) maps the triangle \(T'\) back to the original triangle \(T\)
  1. Determine the matrix that represents \(W'\) [3]
Edexcel F1 2022 January Q6
8 marks Standard +0.8
The quadratic equation $$Ax^2 + 5x - 12 = 0$$ where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)
  1. Write down an expression in terms of \(A\) for
    1. \(\alpha + \beta\)
    2. \(\alpha\beta\)
    [2]
The equation $$4x^2 - 5x + B = 0$$ where \(B\) is a constant, has roots \(\alpha - \frac{3}{\beta}\) and \(\beta - \frac{3}{\alpha}\)
  1. Determine the value of \(A\) [3]
  2. Determine the value of \(B\) [3]
Edexcel F1 2022 January Q7
9 marks Standard +0.8
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The rectangular hyperbola \(H\) has equation \(xy = 36\) The point \(P(4, 9)\) lies on \(H\)
  1. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4x - 9y + 65 = 0$$ [4]
The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
  1. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are rational constants. [5]
Edexcel F1 2022 January Q8
10 marks Standard +0.3
$$f(x) = 2x^{-\frac{2}{3}} + \frac{1}{2}x - \frac{1}{3x - 5} - \frac{5}{2} \quad x \neq \frac{5}{3}$$ The table below shows values of \(f(x)\) for some values of \(x\), with values of \(f(x)\) given to 4 decimal places where appropriate.
\(x\)12345
\(f(x)\)0.5\(-0.2885\)0.5834
  1. Complete the table giving the values to 4 decimal places. [2]
The equation \(f(x) = 0\) has exactly one positive root, \(\alpha\). Using the values in the completed table and explaining your reasoning,
  1. determine an interval of width one that contains \(\alpha\). [2]
  2. Hence use interval bisection twice to obtain an interval of width 0.25 that contains \(\alpha\). [3]
Given also that the equation \(f(x) = 0\) has a negative root, \(\beta\), in the interval \([-1, -0.5]\)
  1. use linear interpolation once on this interval to find an approximation for \(\beta\). Give your answer to 3 significant figures. [3]
Edexcel F1 2022 January Q9
14 marks Standard +0.8
  1. Prove by induction that, for \(n \in \mathbb{N}\) $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$$ [5]
  2. 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. [4]
  3. Determine the value of \(n\) for which $$3\sum_{r=1}^{n} r(r+1)(r-1) = 17\sum_{r=n}^{2n} r^2$$ [5]