Questions — OCR Further Pure Core AS (68 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 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 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 \(a\).
OCR Further Pure Core AS 2023 June Q5
4 marks Moderate -0.3
5 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 2 } - 3 x + 12 = 0\) are \(\alpha\) and \(\beta\). By considering the symmetric functions of the roots, \(\alpha + \beta\) and \(\alpha \beta\), determine the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).
OCR Further Pure Core AS 2023 June Q6
6 marks Moderate -0.3
6 Prove by induction that \(4 \times 8 ^ { n } + 66\) is divisible by 14 for all integers \(n \geqslant 0\).
OCR Further Pure Core AS 2023 June Q7
6 marks Standard +0.8
7 In this question you must show detailed reasoning. Matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3 \\ a + 9 & a & 4 \\ 0 & - 13 & a - 1 \end{array} \right)\) where \(a\) is a constant.
Find all possible values of \(a\) for which \(\operatorname { det } \mathbf { A }\) has the same value as it has when \(a = 2\).
OCR Further Pure Core AS 2023 June Q8
9 marks Moderate -0.3
8
  1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
  2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
  3. The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
    1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
    2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.