Questions Further Pure Core AS (138 questions)

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OCR Further Pure Core AS 2018 June Q1
1
  1. Find a vector which is perpendicular to both \(\left( \begin{array} { r } 1
    3
    - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 3
    - 6
    4 \end{array} \right)\).
  2. The cartesian equation of a line is \(\frac { x } { 2 } = y - 3 = 2 z + 4\). Express the equation of this line in vector form.
OCR Further Pure Core AS 2018 June Q2
2 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
OCR Further Pure Core AS 2018 June Q3
3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 2 - 3 i\) and \(z _ { 2 } = a + 4 i\) where \(a\) is a real number.
  1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
  2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { id }\).
  3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
  4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\).
OCR Further Pure Core AS 2018 June Q4
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & 1 & 2
1 & - 1 & 1
2 & 2 & a \end{array} \right)\).
  1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
  2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
  3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
OCR Further Pure Core AS 2018 June Q5
5 In this question you must show detailed reasoning.
  1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
  2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
  3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.
OCR Further Pure Core AS 2018 June Q6
6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6
t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4
t & - 2 \end{array} \right)\) where \(t\) is a constant.
  1. Show that \(| \mathbf { A } | = | \mathbf { B } |\).
  2. Verify that \(| \mathbf { A B } | = | \mathbf { A } \| \mathbf { B } |\).
  3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\).
OCR Further Pure Core AS 2018 June Q7
7 Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geqslant 1\).
OCR Further Pure Core AS 2018 June Q8
8 The \(2 \times 2\) matrix A represents a transformation T which has the following properties.
  • The image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
  • An object shape whose area is 7 is transformed to an image shape whose area is 35 .
  • T has a line of invariant points.
    1. Find a possible matrix for \(\mathbf { A }\).
The transformation S is represented by the matrix \(\mathbf { B }\) where \(\mathbf { B } = \left( \begin{array} { l l } 3 & 1
2 & 2 \end{array} \right)\).
  • Find the equation of the line of invariant points of S .
  • Show that any line of the form \(y = x + c\) is an invariant line of S .
    •
    OCR Further Pure Core AS 2022 June Q1
    1
    1. Determine whether the point \(( 19 , - 12,17 )\) lies on the line \(\mathbf { r } = \left( \begin{array} { r } 4
      - 2
      7 \end{array} \right) + \lambda \left( \begin{array} { r } 3
      - 2
      4 \end{array} \right)\). Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { r } 1
      - 2
      2 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 3
      6
      2 \end{array} \right)\).
      1. Find, in degrees, the angle between \(\mathbf { a }\) and \(\mathbf { b }\).
      2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
    OCR Further Pure Core AS 2022 June Q2
    2 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } a & 1
    - 1 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } - 2 & 5
    - 1 & 0 \end{array} \right)\) where \(a\) is a constant.
    1. Find the following matrices.
      • \(\mathbf { A } + \mathbf { B }\)
      • AB
      • \(\mathbf { A } ^ { 2 }\)
        1. Given that the determinant of \(\mathbf { A }\) is 25 find the value of \(a\).
        2. You are given instead that the following system of equations does not have a unique solution.
      $$\begin{array} { r } a x + y = - 2
      - x + 3 y = - 6 \end{array}$$ Determine the value of \(a\).
    OCR Further Pure Core AS 2022 June Q4
    4 Prove that \(3 ^ { n } > 10 n\) for all integers \(n \geqslant 4\).
    OCR Further Pure Core AS 2022 June Q5
    5 In this question you must show detailed reasoning.
    1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
    2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).
    OCR Further Pure Core AS 2022 June Q6
    6 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \frac { 1 } { 13 } \left( \begin{array} { r r } 5 & 12
    12 & - 5 \end{array} \right)\). You are given that \(\mathbf { A }\) represents the transformation T which is a reflection in a certain straight line. You are also given that this straight line, the mirror line, passes through the origin, \(O\).
    1. Explain why there must be a line of invariant points for T . State the geometric significance of this line.
    2. By considering the line of invariant points for T , determine the equation of the mirror line. Give your answer in the form \(y = m x + c\). The coordinates of the point \(P\) are \(( 1,5 )\).
    3. By considering the image of \(P\) under the transformation T , or otherwise, determine the coordinates of the point on the mirror line which is closest to \(P\).
    4. The line with equation \(y = a x + 2\) is an invariant line for T. Determine the value of \(a\).
    OCR Further Pure Core AS 2022 June Q7
    7 In this question you must show detailed reasoning.
    Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined as follows.
    \(\mathrm { C } _ { 1 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } + 2 - \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}\) and \(\mathrm { C } _ { 2 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } - 2 - \sqrt { 3 } - 2 \mathrm { i } ) = \frac { 2 } { 3 } \pi \right\}\)
    By considering the representations of \(C _ { 1 }\) and \(C _ { 2 }\) on an Argand diagram, determine the locus \(C _ { 1 } \cap C _ { 2 }\).
    OCR Further Pure Core AS 2022 June Q8
    8 The line segment \(A B\) is a diameter of a sphere, \(S\). The point \(C\) is any point on the surface of \(S\).
    1. Explain why \(\overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0\) for all possible positions of \(C\). You are now given that \(A\) is the point ( \(11,12 , - 14\) ) and \(B\) is the point ( \(9,13,6\) ).
    2. Given that the coordinates of \(C\) have the form ( \(2 p , p , 1\) ), where \(p\) is a constant, determine the coordinates of the possible positions of \(C\). \section*{END OF QUESTION PAPER}
    OCR Further Pure Core AS 2024 June Q1
    1 Use a matrix method to determine the solution of the following simultaneous equations. $$\begin{aligned} 2 x - 3 y + z & = 1
    x - 2 y - 4 z & = 40
    5 x + 6 y - z & = 61 \end{aligned}$$
    OCR Further Pure Core AS 2024 June Q2
    2 In this question you must show detailed reasoning.
    1. Express \(\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }\) in the form \(\mathrm { a } + \mathrm { bi }\) where \(a\) and \(b\) are real.
    2. Solve the equation \(4 x ^ { 2 } - 8 x + 5 = 0\). Give your answer(s) in the form \(\mathrm { c } + \mathrm { di }\) where \(c\) and \(d\) are real.
    OCR Further Pure Core AS 2024 June Q3
    3
      1. Find \(\left( \begin{array} { c } 1
        2
        - 1 \end{array} \right) \times \left( \begin{array} { c } 3
        5
        - 2 \end{array} \right)\).
      2. State a geometrical relationship between the answer to part (a)(i) and the vectors \(\left( \begin{array} { c } 1
        2
        - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 3
        5
        - 2 \end{array} \right)\).
      3. Verify the relationship stated in part (a)(ii).
    2. Find the angle between the vectors \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 8 \mathbf { k }\).
    OCR Further Pure Core AS 2024 June Q4
    4 The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number \(4 - 2 \mathrm { i }\).
    \includegraphics[max width=\textwidth, alt={}, center]{4159328b-475e-4f29-91f2-f2f343573251-3_417_775_349_644}
    1. Use set notation to define the locus of complex numbers, \(z\), represented by points which lie on the circle. The locus \(L\) is defined by \(\mathrm { L } = \{ \mathrm { z } : \mathrm { z } \in \mathbb { C } , | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + 2 | \}\).
    2. On the Argand diagram in the Printed Answer Booklet, sketch and label the locus \(L\). You are given that the locus \(\left\{ z : z \in \mathbb { C } , \arg ( z - 1 ) = \frac { 1 } { 4 } \pi , \operatorname { Re } ( z ) = 3 \right\}\) contains only one number.
    3. Find this number.
    OCR Further Pure Core AS 2024 June Q5
    5 The line through points \(A ( 8 , - 7 , - 2 )\) and \(B ( 11 , - 9,0 )\) is denoted by \(L _ { 1 }\).
    1. Find a vector equation for \(L _ { 1 }\).
    2. Determine whether the point \(( 26 , - 19 , - 14 )\) lies on \(L _ { 1 }\). The line \(L _ { 2 }\) passes through the origin, \(O\), and intersects \(L _ { 1 }\) at the point \(C\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular.
    3. By using the fact that \(C\) lies on \(L _ { 1 }\), find a vector equation for \(L _ { 2 }\).
    4. Hence find the shortest distance from \(O\) to \(L _ { 1 }\).
    OCR Further Pure Core AS 2024 June Q6
    6 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
    0 & 1 \end{array} \right)\) where \(a\) is a constant.
    Prove by induction that \(\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \text { an }
    0 & 1 \end{array} \right)\) for all integers \(n \geqslant 1\).
    OCR Further Pure Core AS 2024 June Q7
    7 In this question you must show detailed reasoning.
    The roots of the equation \(2 x ^ { 3 } - 3 x ^ { 2 } - 3 x + 5 = 0\) are \(\alpha , \beta\) and \(\gamma\).
    By considering \(( \alpha + \beta + \gamma ) ^ { 2 }\) and \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), determine a cubic equation with integer coefficients whose roots are \(\frac { \alpha \beta } { \gamma } , \frac { \beta \gamma } { \alpha }\) and \(\frac { \gamma \alpha } { \beta }\).
    OCR Further Pure Core AS 2024 June Q8
    8 Three transformations, \(T _ { A } , T _ { B }\) and \(T _ { C }\), are represented by the matrices \(A , B\) and \(\mathbf { C }\) respectively. You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & 0
    2 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } 1 & 0
    0 & - 1 \end{array} \right)\).
    1. Find the matrix which represents the inverse transformation of \(T _ { A }\).
    2. By considering matrix multiplication, determine whether \(T _ { A }\) followed by \(T _ { B }\) is the same transformation as \(T _ { B }\) followed by \(T _ { A }\). Transformations R and S are each defined as being the result of successive transformations, as specified in the table.
      TransformationFirst transformationfollowed by
      R\(\mathrm { T } _ { \mathrm { A } }\) followed by \(\mathrm { T } _ { \mathrm { B } }\)\(\mathrm { T } _ { \mathrm { C } }\)
      S\(\mathrm { T } _ { \mathrm { A } }\)\(\mathrm { T } _ { \mathrm { B } }\) followed by \(\mathrm { T } _ { \mathrm { C } }\)
    3. Explain, using a property of matrix multiplication, why R and S are the same transformations. A quadrilateral, \(Q\), has vertices \(D , E , F\) and \(G\) in anticlockwise order from \(D\). Under transformation \(\mathrm { R } , Q ^ { \prime }\) s image, \(Q ^ { \prime }\), has vertices \(D ^ { \prime } , E ^ { \prime } , F ^ { \prime }\) and \(G ^ { \prime }\) (where \(D ^ { \prime }\) is the image of \(D\), etc). The area of \(Q\), in suitable units, is 5 . You are given that det \(\mathbf { C } = a ^ { 2 } + 1\) where \(a\) is a real constant.
      1. Determine the order of the vertices of \(Q ^ { \prime }\), starting anticlockwise from \(D ^ { \prime }\).
      2. Find, in terms of \(a\), the area of \(Q ^ { \prime }\).
      3. Explain whether the inverse transformation for R exists. Justify your answer.
    OCR Further Pure Core AS 2020 November Q4
    4 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1
    x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.
    1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\).
    2. Explain why the method used in part (a) works for all values of \(a\) and \(b\).
    OCR Further Pure Core AS 2020 November Q6
    6 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).