Questions Further Pure Core AS (141 questions)

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OCR Further Pure Core AS Specimen Q3
9 marks Moderate -0.8
3
  1. You are given two matrices, A and B, where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2 \\ 2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
  2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5 \\ 1 & 1 & 3 \\ - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2 \\ - 5 & 9 & - 1 \\ 3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
  3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).
OCR Further Pure Core AS Specimen Q4
4 marks Standard +0.3
4 Draw the region of the Argand diagram for which \(| z - 3 - 4 i | \leq 5\) and \(| z | \leq | z - 2 |\).
OCR Further Pure Core AS Specimen Q5
9 marks Standard +0.3
5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).
  1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
  2. Find the equation of the line of invariant points of this transformation.
  3. (a) Find the determinant of \(\mathbf { M }\).
    (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).
OCR Further Pure Core AS Specimen Q6
6 marks Standard +0.3
6 At the beginning of the year John had a total of \(\pounds 2000\) in three different accounts. He has twice as much money in the current account as in the savings account.
  • The current account has an interest rate of \(2.5 \%\) per annum.
  • The savings account has an interest rate of \(3.7 \%\) per annum.
  • The supersaver account has an interest rate of \(4.9 \%\) per annum.
John has predicted that he will earn a total interest of \(\pounds 92\) by the end of the year.
  1. Model this situation as a matrix equation.
  2. Find the amount that John had in each account at the beginning of the year.
  3. In fact, the interest John will receive is \(\pounds 92\) to the nearest pound. Explain how this affects the calculations.
OCR Further Pure Core AS Specimen Q7
9 marks Challenging +1.2
7 In this question you must show detailed reasoning.
It is given that \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( z ) = 0\) are plotted on an Argand diagram.
Show that these points lie on the circle \(| z | = k\), where \(k\) is a real number to be determined.
OCR Further Pure Core AS Specimen Q8
5 marks Standard +0.3
8 Prove that \(n ! > 2 ^ { n }\) for \(n \geq 4\).
OCR Further Pure Core AS Specimen Q9
11 marks Standard +0.3
9
  1. Find the value of \(k\) such that \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 \\ 3 \\ k \end{array} \right)\) are perpendicular. Two lines have equations \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 3 \\ 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)\).
  2. Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  3. The vector \(\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)\) is perpendicular to the lines \(l _ { 1 }\) and \(l _ { 2 }\). Find the values of \(a\) and \(b\). \section*{END OF QUESTION PAPER} \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series.
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OCR MEI Further Pure Core AS 2019 June Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
Find \(\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\), giving your answer correct to 4 decimal places.
OCR MEI Further Pure Core AS 2019 June Q2
3 marks Standard +0.3
2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).
OCR MEI Further Pure Core AS 2019 June Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning. \(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1 \\ 0 & 1 \end{array} \right)\), find \(\mathbf { B }\).
OCR MEI Further Pure Core AS 2019 June Q4
8 marks Standard +0.3
4
  1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & 1 & 2 \\ - 2 & 1 & 2 \end{array} \right)\).
  2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19 \\ - x + y + 2 z & = 4 \\ - 2 x + y + 2 z & = k \end{aligned}$$
  3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).
OCR MEI Further Pure Core AS 2019 June Q5
6 marks Moderate -0.3
5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).
OCR MEI Further Pure Core AS 2019 June Q6
11 marks Standard +0.8
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
12 marks Standard +0.8
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
6 marks Moderate -0.3
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
7 marks Standard +0.3
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
5 marks Moderate -0.3
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 marks Standard +0.3
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
10 marks Moderate -0.3
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
9 marks Standard +0.3
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.
OCR MEI Further Pure Core AS 2022 June Q8
12 marks Standard +0.8
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} }{www.ocr.org.uk}) after the live examination series.
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OCR MEI Further Pure Core AS 2023 June Q1
3 marks Easy -1.2
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
4 marks Moderate -0.3
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
6 marks Standard +0.3
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
6 marks Standard +0.3
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 \(b\).