Questions — Edexcel (9685 questions)

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Edexcel FP1 2016 June Q3
5 marks Standard +0.3
  1. (a) Using the formula for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) write down, in terms of \(n\) only, an expression for
$$\sum _ { r = 1 } ^ { 3 n } r ^ { 2 }$$ (b) Show that, for all integers \(n\), where \(n > 0\) $$\sum _ { r = 2 n + 1 } ^ { 3 n } r ^ { 2 } = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where the values of the constants \(a\), \(b\) and \(c\) are to be found.
Edexcel FP1 2016 June Q4
7 marks Moderate -0.3
4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)
  1. \(Z\)
  2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
  3. find the value of \(p\) and the value of \(q\).
Edexcel FP1 2016 June Q5
10 marks Standard +0.8
5. Points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p ^ { 2 } \neq q ^ { 2 }\), lie on the parabola \(y ^ { 2 } = 4 a x\).
  1. Show that the chord \(P Q\) has equation $$y ( p + q ) = 2 x + 2 a p q$$ Given that this chord passes through the focus of the parabola,
  2. show that \(p q = - 1\)
  3. Using calculus find the gradient of the tangent to the parabola at \(P\).
  4. Show that the tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) are perpendicular.
Edexcel FP1 2016 June Q6
10 marks Standard +0.3
6. $$\mathbf { P } = \left( \begin{array} { c c } - \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\ \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
  2. Find the value of \(p\) and the value of \(q\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
  3. 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 \(\mathbf { R }\).
  4. Find the matrix \(\mathbf { R }\).
  5. Deduce that the transformation \(T\) is self-inverse.
Edexcel FP1 2016 June Q7
13 marks Standard +0.3
7. A complex number \(z\) is given by $$z = a + 2 i$$ where \(a\) is a non-zero real number.
  1. Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\). Given that \(z ^ { 2 } + 2 z\) is real,
  2. find the value of \(a\). Using this value for \(a\),
  3. find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
  4. Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
  5. Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).
Edexcel FP1 2016 June Q8
10 marks Standard +0.3
8. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$ (ii) A sequence of positive rational numbers is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = \frac { 1 } { 3 } u _ { n } + \frac { 8 } { 9 } , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 5 \times \left( \frac { 1 } { 3 } \right) ^ { n } + \frac { 4 } { 3 }$$
Edexcel FP1 2016 June Q9
11 marks Standard +0.3
9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)
  1. Show that an equation of the normal to \(H\) at the point \(P \left( 5 p , \frac { 5 } { p } \right) , p \neq 0\), is $$y - p ^ { 2 } x = \frac { 5 } { p } - 5 p ^ { 3 }$$ This normal meets the line with equation \(y = - x\) at the point \(A\).
  2. Show that the coordinates of \(A\) are $$\left( - \frac { 5 } { p } + 5 p , \frac { 5 } { p } - 5 p \right)$$ The point \(M\) is the midpoint of the line segment \(A P\).
    Given that \(M\) lies on the positive \(x\)-axis,
  3. find the exact value of the \(x\) coordinate of point \(M\).
Edexcel FP1 2017 June Q2
5 marks Standard +0.3
2. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 1 \\ 4 & 3 \end{array} \right) , \quad \mathbf { P } = \left( \begin{array} { r r } 3 & 6 \\ 11 & - 8 \end{array} \right)$$
  1. Find \(\mathbf { A } ^ { - 1 }\) (2) The transformation represented by the matrix \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\) is equivalent to the transformation represented by the matrix \(\mathbf { P }\).
  2. Find \(\mathbf { B }\), giving your answer in its simplest form.
Edexcel FP1 2017 June Q3
7 marks Standard +0.3
3. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t } \quad t \neq 0$$ The points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 4 }\) and \(t = 2\) respectively.
The line \(l\) passes through the origin \(O\) and is perpendicular to the line \(P Q\).
  1. Find an equation for \(l\).
  2. Find a cartesian equation for \(H\).
  3. Find the exact coordinates of the two points where \(l\) intersects \(H\). Give your answers in their simplest form.
Edexcel FP1 2017 June Q4
8 marks Standard +0.3
4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.
  1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
  2. find the value of \(p\).
    (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
    Give your answers as exact values in their simplest form.
    II
Edexcel FP1 2017 June Q5
11 marks Standard +0.3
5. (i) $$\mathbf { A } = \left( \begin{array} { l l } p & 2 \\ 3 & p \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 5 & 4 \\ 6 & - 5 \end{array} \right)$$ where \(p\) is a constant.
  1. Find, in terms of \(p\), the matrix \(\mathbf { A B }\) Given that $$\mathbf { A B } + 2 \mathbf { A } = k \mathbf { I }$$ where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(p\) and the value of \(k\).
    (ii) $$\mathbf { M } = \left( \begin{array} { r r } a & - 9 \\ 1 & 2 \end{array} \right) , \text { where } a \text { is a real constant }$$ Triangle \(T\) has an area of 15 square units.
    Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of triangle \(T ^ { \prime }\) is 270 square units, find the possible values of \(a\).
Edexcel FP1 2017 June Q6
6 marks Moderate -0.3
6. Given that 4 and \(2 \mathrm { i } - 3\) are roots of the equation $$x ^ { 3 } + a x ^ { 2 } + b x - 52 = 0$$ where \(a\) and \(b\) are real constants,
  1. write down the third root of the equation,
  2. find the value of \(a\) and the value of \(b\).
Edexcel FP1 2017 June Q7
10 marks Standard +0.8
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant and \(a > 0\) The point \(Q \left( a q ^ { 2 } , 2 a q \right) , q > 0\), lies on the parabola \(C\).
  1. Show that an equation of the tangent to \(C\) at \(Q\) is $$q y = x + a q ^ { 2 }$$ The tangent to \(C\) at the point \(Q\) meets the \(x\)-axis at the point \(X \left( - \frac { 1 } { 4 } a , 0 \right)\) and meets the directrix of \(C\) at the point \(D\).
  2. Find, in terms of \(a\), the coordinates of \(D\). Given that the point \(F\) is the focus of the parabola \(C\),
  3. find the area, in terms of \(a\), of the triangle \(F X D\), giving your answer in its simplest form.
Edexcel FP1 2017 June Q8
9 marks Standard +0.3
8. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 8 r + 3 \right) = \frac { 1 } { 2 } n ( 2 n + 5 ) ( n + 3 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 3 r ^ { 2 } + 8 r + 3 + k \left( 2 ^ { r - 1 } \right) \right) = 3520$$ (b) find the exact value of the constant \(k\).
Edexcel FP1 2017 June Q9
12 marks Standard +0.3
9. (i) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 6 , \quad u _ { 2 } = 27 \\ u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 ^ { n } ( n + 1 )$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 3 n + 1 } \text { is divisible by } 19$$ \includegraphics[max width=\textwidth, alt={}, center]{536d7ec7-91b0-4fda-a485-2ac4a72c7d59-29_56_20_109_1950}
Edexcel FP1 2018 June Q1
6 marks Moderate -0.5
1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that \(\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)\), where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence use algebra to find the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
Edexcel FP1 2018 June Q2
10 marks Standard +0.3
2. $$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } + \frac { 4 } { 3 x } + 2 x - 5 , \quad x < 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\).
  1. Show that \(\alpha\) lies in the interval \([ - 3 , - 2.5 ]\)
  2. Taking - 3 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
  3. Use linear interpolation once on the interval \([ - 3 , - 2.5 ]\) to find another approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2018 June Q3
9 marks Standard +0.3
3. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } - 2 & 3 \\ 1 & 1 \end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r } - 1 & 5 & 12 \\ 3 & - 5 & - 1 \end{array} \right)$$
  1. find \(\mathbf { A } ^ { - 1 }\)
  2. Hence, or otherwise, find the matrix \(\mathbf { B }\), giving your answer in its simplest form.
    (ii) Given that $$\mathbf { C } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
  3. describe fully the single geometrical transformation represented by the matrix \(\mathbf { C }\).
  4. Hence find the matrix \(\mathbf { C } ^ { 39 }\)
Edexcel FP1 2018 June Q4
9 marks Standard +0.3
  1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = \frac { 1 } { 3 } n ( n - a ) ( n + a )$$ where \(a\) is a positive integer to be determined.
(b) Hence, or otherwise, state the positive value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = 0$$ Given that $$\sum _ { r = 3 } ^ { 17 } \left( k r ^ { 3 } + r ^ { 2 } - r - 8 \right) = 6710 \quad \text { where } k \text { is a constant }$$ (c) find the exact value of \(k\).
Edexcel FP1 2018 June Q5
9 marks Challenging +1.2
  1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant.
Given that \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\),
  1. use calculus to show that the equation of the tangent to \(H\) at \(P\) can be written as $$t ^ { 2 } y + x = 2 c t$$ The points \(A\) and \(B\) lie on \(H\).
    The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 8 c } { 5 } , \frac { 3 c } { 5 } \right)\).
    Given that the \(x\) coordinate of \(A\) is positive,
  2. find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 2018 June Q6
6 marks Standard +0.3
6. $$\mathbf { M } = \left( \begin{array} { r r } 8 & - 1 \\ - 4 & 2 \end{array} \right)$$
  1. Find the value of \(\operatorname { det } \mathbf { M }\) The triangle \(T\) has vertices at the points \(( 4,1 ) , ( 6 , k )\) and \(( 12,1 )\), where \(k\) is a constant.
    The triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of triangle \(T ^ { \prime }\) is 216 square units,
  2. find the possible values of \(k\).
Edexcel FP1 2018 June Q7
8 marks Standard +0.8
  1. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\).
The straight line \(l\) passes through the point \(S\) and meets the directrix of \(C\) at the point \(D\).
Given that the \(y\) coordinate of \(D\) is \(\frac { 24 a } { 5 }\),
  1. show that an equation of the line \(l\) is $$12 x + 5 y = 12 a$$ The point \(P \left( a k ^ { 2 } , 2 a k \right)\), where \(k\) is a positive constant, lies on the parabola \(C\).
    Given that the line segment \(S P\) is perpendicular to \(l\),
  2. find, in terms of \(a\), the coordinates of the point \(P\).
Edexcel FP1 2018 June Q8
6 marks Standard +0.3
  1. Prove by induction that
$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7 for all positive integers \(n\).
Edexcel FP1 2018 June Q9
12 marks Standard +0.3
    1. Given that
$$\frac { 3 w + 7 } { 5 } = \frac { p - 4 \mathrm { i } } { 3 - \mathrm { i } } \quad \text { where } p \text { is a real constant }$$
  1. express \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that arg \(w = - \frac { \pi } { 2 }\)
  2. find the value of \(p\).
    (ii) Given that $$( z + 1 - 2 i ) ^ { * } = 4 i z$$ find \(z\), giving your answer in the form \(z = x + i y\), where \(x\) and \(y\) are real constants. \includegraphics[max width=\textwidth, alt={}, center]{89f82cd3-9afa-4431-bc74-a073909c903f-36_106_129_2469_1816}
Edexcel FP1 Q1
5 marks Moderate -0.8
1. $$\mathbf { R } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { S } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$
  1. Find \(\mathbf { R } ^ { 2 }\).
  2. Find \(\mathbf { R S }\).
  3. Describe the geometrical transformation represented by \(\mathbf { R S }\).