Questions — Edexcel FP1 (269 questions)

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Edexcel FP1 2018 June Q3
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
  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
  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. $$\mathbf { M } = \left( \begin{array} { r r } 8 & - 1
- 4 & 2 \end{array} \right)$$
  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
  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
  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
    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.
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Edexcel FP1 Q1
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 \(\mathbf { R S }\).
Edexcel FP1 Q2
2. A point \(P\) with coordinates \(( x , y )\) moves so that its distance from the point \(( - 3,0 )\) is equal to its distance from the line \(x = 3\). Find a cartesian equation for the locus of \(P\).
Edexcel FP1 Q3
3. \(z = 1 + \mathrm { i } \sqrt { 3 }\) Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
  1. \(z ^ { 2 } + z\),
  2. \(\frac { z } { 3 - z }\),
    giving the exact values of \(a\) and \(b\) in each part.
Edexcel FP1 Q4
4. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 3\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval ( 2,3 ).
  1. Use linear interpolation on the end points of this interval to obtain an approximation for \(\alpha\).
  2. Taking 2.5 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 2 decimal places.
Edexcel FP1 Q5
5. Given that \(a\) and \(b\) are non-zero constants and that $$\mathbf { X } = \left( \begin{array} { r r } a & 2 b
- a & 3 b \end{array} \right) ,$$
  1. find \(\mathbf { X } ^ { - 1 }\), giving your answer in terms of \(a\) and \(b\). Given also that \(\mathbf { Z X } = \mathbf { Y }\), where \(\mathbf { Y } = \left( \begin{array} { c c } 3 a & b
    a & 2 b \end{array} \right)\),
  2. find \(\mathbf { Z }\), simplifying your answer.
Edexcel FP1 Q6
6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$ (b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)\).
Edexcel FP1 Q7
7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).
  1. Find each of these roots in the form \(a + b \mathrm { i }\).
  2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
  3. Illustrate the two roots on a single Argand diagram.
  4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).
Edexcel FP1 Q8
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\). The point ( \(3 t , \frac { 3 } { t }\) ) is a general point on this hyperbola.
  1. Find the value of \(c ^ { 2 }\).
  2. Show that an equation of the normal to \(H\) at the point ( \(3 t , \frac { 3 } { t }\) ) is $$y = t ^ { 2 } x + \left( \frac { 3 } { t } - 3 t ^ { 3 } \right)$$ The point \(P\) on \(H\) has coordinates (6, 1.5). The tangent to \(H\) at \(P\) meets the curve again at the point \(Q\).
  3. Find the coordinates of the point \(Q\).
Edexcel FP1 Q9
9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0
9 & 1 \end{array} \right)$$
  1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0
    3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
  2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0
    3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).
Edexcel FP1 Specimen Q1
1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
  2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 Specimen Q2
2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).
  1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4
    - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
  3. find the value of \(a\).
Edexcel FP1 Specimen Q3
3. The matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
- \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)\)
  1. Find \(\mathbf { R } ^ { 2 }\).
  2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
  3. Describe the geometrical transformation represented by \(\mathbf { R }\).
Edexcel FP1 Specimen Q4
4. $$f ( x ) = 2 ^ { x } - 6 x$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4,5]. Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\).
Edexcel FP1 Specimen Q5
5. (a) Show that \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 1 \right) = \frac { 1 } { 3 } ( n - 2 ) n ( n + 2 )\).
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 40 } \left( r ^ { 2 } - r - 1 \right)\).
Edexcel FP1 Specimen Q6
6. Given that \(z = - 3 + 4 \mathrm { i }\),
  1. find the modulus of \(z\),
  2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
  3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
  4. Show the points \(A\) and \(B\) on an Argand diagram.
Edexcel FP1 Specimen Q7
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  1. Find the value of \(a\).
  2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 Specimen Q8
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),
  1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\),
  3. find the value of \(p\).
Edexcel FP1 Specimen Q9
9. Use the method of mathematical induction to prove that, for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\left( \begin{array} { c c } 2 & 1
    - 1 & 0 \end{array} \right) ^ { n } = \left( \begin{array} { c c } n + 1 & n
    - n & 1 - n \end{array} \right)\)
  2. \(\mathrm { f } ( n ) = 4 ^ { n } + 6 n - 1\) is divisible by 3 .