Questions Further Pure Core AS (138 questions)

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OCR MEI Further Pure Core AS 2024 June Q4
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
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 \(\mathrm { x } + \mathrm { z } = 5\).
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OCR MEI Further Pure Core AS 2020 November Q1
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
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 MEI Further Pure Core AS Specimen Q8
8 In this question you must show detailed reasoning.
  1. Explain why all cubic equations with real coefficients have at least one real root.
  2. Points representing the three roots of the equation \(z ^ { 3 } + 9 z ^ { 2 } + 27 z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices.
OCR Further Pure Core AS 2022 June Q3
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
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 2020 November Q1
1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).
\(2 \mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D.
P is a reflection in the \(x\)-axis. \(\mathbf { A }\) is the matrix that represents P .
  1. Write down the matrix \(\mathbf { A }\). Q is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). \(\mathbf { B }\) is the
    matrix that represents Q . matrix that represents Q.
  2. Find the matrix \(\mathbf { B }\). T is P followed by Q. C is the matrix that represents T.
  3. Determine the matrix \(\mathbf { C }\).
    \(L\) is the line whose equation is \(y = x\).
  4. Explain whether or not \(L\) is a line of invariant points under \(T\). An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  5. Explain what the value of the determinant of \(\mathbf { C }\) means about
    • the area of \(N\) compared to the area of \(M\),
    • the orientation of \(N\) compared to the orientation of \(M\).
OCR Further Pure Core AS 2020 November Q3
3 In this question you must show detailed reasoning. The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3 z - 4 z ^ { * }\)
    2. \(( z + 1 - 3 i ) ^ { 2 }\)
    3. \(\frac { z + 1 } { z - 1 }\)
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
  3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.
    • \(| \omega |\)
    • \(\arg ( \omega )\), giving your answer correct to 3 significant figures
OCR Further Pure Core AS 2020 November Q5
5 In this question you must show detailed reasoning. The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
OCR MEI Further Pure Core AS 2019 June Q8
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
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
1 You are given that \(z = 3 - 4 \mathrm { i }\).
  1. Find
    • \(| z |\),
    • \(\arg ( z )\),
    • \(Z ^ { * }\).
    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
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
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
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
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
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 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 \(x = 5\)
    • find the volume of \(H ^ { \prime }\),
    • determine whether A preserves the orientation of \(H\).
    • Find the values of \(x\) for which \(O , P ^ { \prime } , Q ^ { \prime }\) and \(R ^ { \prime }\) are coplanar (i.e. the four points lie in the same plane).
OCR Further Pure Core AS 2019 June Q8
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
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
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
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 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
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 \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).