Questions Further Pure Core AS (138 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
OCR MEI Further Pure Core AS 2019 June Q6
6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k
1 & \lambda - k \end{array} \right)\), and \(\lambda\)
and \(k\) are real constants. and \(k\) are real constants.
  1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\).
    2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
  2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
  3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
    Find the possible values of \(\lambda\).
OCR MEI Further Pure Core AS 2019 June Q7
7
  1. Sketch on a single Argand diagram
    1. the set of points for which \(| z - 1 - 3 i | = 3\),
    2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
  2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
OCR MEI Further Pure Core AS 2022 June Q1
1
    1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7
      2 x - 4 y - 3 z & = - 5
      - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
    2. Hence solve the equations.
  2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1
    2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.
OCR MEI Further Pure Core AS 2022 June Q2
2
  1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
  2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).
OCR MEI Further Pure Core AS 2022 June Q3
3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
OCR MEI Further Pure Core AS 2022 June Q5
5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\).
\includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.
OCR MEI Further Pure Core AS 2022 June Q6
6
  1. Using standard summation formulae, show that \(\sum _ { r = 1 } ^ { n } r ( r + 2 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 7 )\).
  2. Use induction to prove the result in part (a).
OCR MEI Further Pure Core AS 2022 June Q7
7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\frac { 1 } { 3 } \pi\). The point B represents \(\frac { 1 } { z }\).
  1. Sketch an Argand diagram showing the origin O and the points A and B .
  2. The point C is such that OACB is a parallelogram. C represents the complex number \(w\). Determine each of the following.
    • The modulus of \(w\), giving your answer in exact form.
    • The argument of \(w\), giving your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Further Pure Core AS 2022 June Q8
8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta
\sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).
  1. Show that T leaves areas unchanged for all values of \(\theta\).
  2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2
    0 & 1 \end{array} \right)\).
    1. Find \(\mathbf { M N } ^ { - 1 }\).
    2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
      If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
      For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
      OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.
OCR MEI Further Pure Core AS 2023 June Q1
1 The transformation R of the plane is reflection in the line \(x = 0\).
  1. Write down the matrix \(\mathbf { M }\) associated with R .
  2. Find \(\mathbf { M } ^ { 2 }\).
  3. Interpret the result of part (b) in terms of the transformation \(R\).
OCR MEI Further Pure Core AS 2023 June Q2
2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.
OCR MEI Further Pure Core AS 2023 June Q3
3 In this question you must show detailed reasoning.
The function \(\mathrm { f } ( \mathrm { z } )\) is given by \(\mathrm { f } ( \mathrm { z } ) = 2 \mathrm { z } ^ { 3 } - 7 \mathrm { z } ^ { 2 } + 16 \mathrm { z } - 15\).
By first evaluating \(\mathrm { f } \left( \frac { 3 } { 2 } \right)\), find the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\).
OCR MEI Further Pure Core AS 2023 June Q4
4 You are given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } ) = \mathrm { n } ^ { 2 }\) for all \(n\), where \(a\) and \(b\) are constants.
By finding \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } )\) in terms of \(a , b\) and \(n\), determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core AS 2023 June Q5
5 The Argand diagram below shows the points representing 1 and \(z\), where \(| z | = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{26cec6f9-78a7-4f0b-969a-13ad02510c25-3_577_595_312_242} Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.
  • \(\mathrm { Z } ^ { * }\)
  • \(\frac { 1 } { z }\)
  • \(1 + z\)
  • iz
OCR MEI Further Pure Core AS 2023 June Q6
6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1
- 1 & 0 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
  2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
  3. Prove your conjecture.
OCR MEI Further Pure Core AS 2023 June Q8
8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3
3 x - y - 2 z = 2
- 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.
  1. By considering a suitable determinant, show that the planes do not meet at a single point.
  2. Given that the planes form a sheaf, determine the value of \(k\).
OCR MEI Further Pure Core AS 2023 June Q9
9 A transformation T of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { c c } k + 1 & - 1
1 & k \end{array} \right)\), where \(k\) is a
constant. constant. Show that, for all values of \(k , \mathrm {~T}\) has no invariant lines through the origin.
OCR MEI Further Pure Core AS 2023 June Q10
10 The plane P has normal vector \(2 \mathbf { i } + a \mathbf { j } - \mathbf { k }\), where \(a\) is a positive constant, and the point ( \(3 , - 1,1\) ) lies in P . The plane \(\mathrm { x } - \mathrm { z } = 3\) makes an angle of \(45 ^ { \circ }\) with P . Find the cartesian equation of P . \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2024 June Q1
1 The quadratic equation \(\mathrm { x } ^ { 2 } + \mathrm { ax } + \mathrm { b } = 0\), where \(a\) and \(b\) are real constants, has a root 2-3.
  1. Write down the other root.
  2. Hence or otherwise determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core AS 2024 June Q2
2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & a
- 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & 0
1 & - 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r r } - 1 & 0
2 & 1 \end{array} \right)\), where \(a\) is
a constant. a constant.
  1. By multiplying out the matrices on both sides of the equation, verify that \(\mathbf { A } ( \mathbf { B C } ) = ( \mathbf { A B } ) \mathbf { C }\).
  2. State the property of matrix multiplication illustrated by this result.
OCR MEI Further Pure Core AS 2024 June Q3
3
  1. Using standard summation formulae, write down an expression in terms of \(n\) for \(\sum _ { r = 1 } ^ { 2 n } r ^ { 3 }\).
  2. Hence show that \(\sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } \mathrm { r } ^ { 3 } = \frac { 1 } { 4 } \mathrm { n } ^ { 2 } ( \mathrm { an } + \mathrm { b } ) ( \mathrm { cn } + \mathrm { d } )\), where \(a , b , c\) and \(d\) are integers to be determined.
OCR MEI Further Pure Core AS 2024 June Q5
5
  1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0
    0 & 3 & - 1
    - 1 & 0 & 2 \end{array} \right)\).
  2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
    1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3
      - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
    2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).
OCR MEI Further Pure Core AS 2024 June Q6
6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9
1 & - 2 \end{array} \right)\).
  1. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n
    n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\).
  2. A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively. Determine whether the student is correct.
OCR MEI Further Pure Core AS 2024 June Q7
7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0
- x + 3 y - 2 z = 0
x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.
  1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
  2. You are now given that \(k = 1\).
    1. Show that there are no points that lie on all three planes.
    2. Describe the geometrical arrangement of the three planes.
OCR MEI Further Pure Core AS 2024 June Q8
8 In an Argand diagram, the point P representing the complex number \(w\) lies on the locus defined by \(\left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\}\). You are given that \(\operatorname { Re } ( w ) = 1\).
  1. Find \(w\). The point P also lies on the locus defined by \(\{ \mathrm { z } : | \mathrm { z } + 3 - 9 \mathrm { i } | = \mathrm { k } \}\), where \(k\) is a constant.
  2. Find the complex number represented by the other point of intersection of the loci defined by $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$