Questions Further Pure Core AS (141 questions)

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OCR MEI Further Pure Core AS 2021 November Q5
5 marks Standard +0.3
5 Prove by induction that \(\sum _ { r = 1 } ^ { n } r \times 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2021 November Q6
12 marks Standard +0.8
6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.
    1. Show that T reverses orientation.
    2. State, in terms of \(\lambda\), the area scale factor of T .
    1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
    2. Hence specify the transformation equivalent to two applications of T .
  3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
    1. Determine the matrix associated with S .
    2. Hence describe the transformation S .
OCR MEI Further Pure Core AS 2021 November Q7
9 marks Challenging +1.2
7
    1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
    2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
  2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.
OCR MEI Further Pure Core AS 2021 November Q9
9 marks Challenging +1.2
9
  1. On a single Argand diagram, sketch the loci defined by
    The point of intersection of the two loci in part (a) represents the complex number \(w\). Find \(w\), giving your answer in exact form. \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2023 June Q7
10 marks Standard +0.3
  1. By expanding \(( \sqrt { 3 } + \mathrm { i } ) ^ { 5 }\), express \(z ^ { 5 }\) in the form \(\mathrm { a } +\) bi where \(a\) and \(b\) are real and exact.
    1. Express \(z\) in modulus-argument form.
    2. Hence find \(z ^ { 5 }\) in modulus-argument form.
    3. Use this result to verify your answers to part (a).
OCR MEI Further Pure Core AS 2024 June Q4
7 marks Standard +0.8
4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
  2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).
OCR MEI Further Pure Core AS 2024 June Q9
8 marks Challenging +1.2
9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.
  • It is a unit vector.
  • It is parallel to the plane \(2 x + 2 y + z = 10\).
  • It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).
\section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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OCR MEI Further Pure Core AS 2020 November Q1
3 marks Moderate -0.3
1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.
OCR MEI Further Pure Core AS 2021 November Q8
7 marks Challenging +1.2
8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,
  • the roots of the equation,
  • the value of \(k\).
OCR Further Pure Core AS 2022 June Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
OCR Further Pure Core AS 2024 June Q9
8 marks Challenging +1.8
9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation \(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).
OCR MEI Further Pure Core AS 2019 June Q8
11 marks Standard +0.3
8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.
  1. Show that \(a = - 2\) and \(b = 2\).
  2. Find the other roots of this equation.
OCR MEI Further Pure Core AS 2022 June Q4
6 marks Standard +0.8
4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.
OCR Further Pure Core AS 2019 June Q1
5 marks Easy -1.2
1 You are given that \(z = 3 - 4 \mathrm { i }\).
  1. Find
    On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w ^ { * }\) is represented by the point \(B\).
  2. Describe the geometrical relationship between the points \(A\) and \(B\).
OCR Further Pure Core AS 2019 June Q2
4 marks Easy -1.3
2 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.
  1. State, with a reason, which one of A and B is true.
  2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).
OCR Further Pure Core AS 2019 June Q3
10 marks Challenging +1.2
3 The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).
  1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
  2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
  3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
  4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
  5. Find the value of \(p\).
OCR Further Pure Core AS 2019 June Q4
14 marks Standard +0.8
4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).
  1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
  2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Find the exact area of \(R\).
  4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).
OCR Further Pure Core AS 2019 June Q5
9 marks Standard +0.8
5 In this question you must show detailed reasoning. You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
  1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
  2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.
OCR Further Pure Core AS 2019 June Q6
5 marks Standard +0.8
6 A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\). Find the smallest possible value of the area of \(Q ^ { \prime }\).
OCR Further Pure Core AS 2019 June Q7
7 marks Standard +0.3
7 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .
  1. In the case where \(x = 5\)
OCR Further Pure Core AS 2019 June Q8
6 marks Standard +0.8
8 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\). \section*{END OF QUESTION PAPER}
OCR Further Pure Core AS 2023 June Q1
5 marks Standard +0.3
1 The roots of the equation \(4 x ^ { 4 } - 2 x ^ { 3 } - 3 x + 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). By using a suitable substitution, find a quartic equation whose roots are \(\alpha + 2 , \beta + 2 , \gamma + 2\) and \(\delta + 2\) giving your answer in the form \(a t ^ { 4 } + b t ^ { 3 } + c t ^ { 2 } + d t + e = 0\), where \(a , b , c , d\), and \(e\) are integers.
OCR Further Pure Core AS 2023 June Q2
8 marks Standard +0.3
2 The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the following equations. \(L _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 5 \\ 6 \\ 15 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ - 2 \\ - 2 \end{array} \right)\) \(L _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 24 \\ 1 \\ - 5 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ 1 \\ - 4 \end{array} \right)\)
  1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, giving the position vector of the point of intersection.
  2. Find the equation of the line which intersects \(L _ { 1 }\) and \(L _ { 2 }\) and is perpendicular to both. Give your answer in cartesian form.
OCR Further Pure Core AS 2023 June Q3
8 marks Standard +0.3
3 In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval \([ 0,2 \pi )\).
Complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are defined by \(z _ { 1 } = 3 + 4 \mathrm { i }\) and \(z _ { 2 } = - 5 + 12 \mathrm { i }\).
  1. Determine \(z _ { 1 } z _ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
  2. Express \(z _ { 2 }\) in modulus-argument form.
  3. Verify, by direct calculation, that \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).
OCR Further Pure Core AS 2023 June Q4
4 marks Moderate -0.3
4 The vector \(\mathbf { p }\), all of whose components are positive, is given by \(\mathbf { p } = \left( \begin{array} { c } a ^ { 2 } \\ a - 5 \\ 26 \end{array} \right)\) where \(a\) is a constant.
You are given that \(\mathbf { p }\) is perpendicular to the vector \(\left( \begin{array} { c } 2 \\ 6 \\ - 3 \end{array} \right)\).
Determine the value of \(a\).