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OCR Further Pure Core AS 2022 June Q7
7 marks Standard +0.8
7 In this question you must show detailed reasoning.
Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined as follows. \(\mathrm { C } _ { 1 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } + 2 - \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}\) and \(\mathrm { C } _ { 2 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } - 2 - \sqrt { 3 } - 2 \mathrm { i } ) = \frac { 2 } { 3 } \pi \right\}\) By considering the representations of \(C _ { 1 }\) and \(C _ { 2 }\) on an Argand diagram, determine the locus \(C _ { 1 } \cap C _ { 2 }\).
OCR Further Pure Core AS 2022 June Q8
9 marks Standard +0.8
8 The line segment \(A B\) is a diameter of a sphere, \(S\). The point \(C\) is any point on the surface of \(S\).
  1. Explain why \(\overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0\) for all possible positions of \(C\). You are now given that \(A\) is the point ( \(11,12 , - 14\) ) and \(B\) is the point ( \(9,13,6\) ).
  2. Given that the coordinates of \(C\) have the form ( \(2 p , p , 1\) ), where \(p\) is a constant, determine the coordinates of the possible positions of \(C\). \section*{END OF QUESTION PAPER}
OCR Further Pure Core AS 2024 June Q1
4 marks Standard +0.3
1 Use a matrix method to determine the solution of the following simultaneous equations. $$\begin{aligned} 2 x - 3 y + z & = 1 \\ x - 2 y - 4 z & = 40 \\ 5 x + 6 y - z & = 61 \end{aligned}$$
OCR Further Pure Core AS 2024 June Q2
4 marks Moderate -0.3
2 In this question you must show detailed reasoning.
  1. Express \(\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }\) in the form \(\mathrm { a } + \mathrm { bi }\) where \(a\) and \(b\) are real.
  2. Solve the equation \(4 x ^ { 2 } - 8 x + 5 = 0\). Give your answer(s) in the form \(\mathrm { c } + \mathrm { di }\) where \(c\) and \(d\) are real.
OCR Further Pure Core AS 2024 June Q3
7 marks Standard +0.3
3
    1. Find \(\left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) \times \left( \begin{array} { c } 3 \\ 5 \\ - 2 \end{array} \right)\).
    2. State a geometrical relationship between the answer to part (a)(i) and the vectors \(\left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 3 \\ 5 \\ - 2 \end{array} \right)\).
    3. Verify the relationship stated in part (a)(ii).
  2. Find the angle between the vectors \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 8 \mathbf { k }\).
OCR Further Pure Core AS 2024 June Q4
6 marks Standard +0.3
4 The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number \(4 - 2 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{4159328b-475e-4f29-91f2-f2f343573251-3_417_775_349_644}
  1. Use set notation to define the locus of complex numbers, \(z\), represented by points which lie on the circle. The locus \(L\) is defined by \(\mathrm { L } = \{ \mathrm { z } : \mathrm { z } \in \mathbb { C } , | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + 2 | \}\).
  2. On the Argand diagram in the Printed Answer Booklet, sketch and label the locus \(L\). You are given that the locus \(\left\{ z : z \in \mathbb { C } , \arg ( z - 1 ) = \frac { 1 } { 4 } \pi , \operatorname { Re } ( z ) = 3 \right\}\) contains only one number.
  3. Find this number.
OCR Further Pure Core AS 2024 June Q5
10 marks Standard +0.3
5 The line through points \(A ( 8 , - 7 , - 2 )\) and \(B ( 11 , - 9,0 )\) is denoted by \(L _ { 1 }\).
  1. Find a vector equation for \(L _ { 1 }\).
  2. Determine whether the point \(( 26 , - 19 , - 14 )\) lies on \(L _ { 1 }\). The line \(L _ { 2 }\) passes through the origin, \(O\), and intersects \(L _ { 1 }\) at the point \(C\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular.
  3. By using the fact that \(C\) lies on \(L _ { 1 }\), find a vector equation for \(L _ { 2 }\).
  4. Hence find the shortest distance from \(O\) to \(L _ { 1 }\).
OCR Further Pure Core AS 2024 June Q6
5 marks Standard +0.3
6 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right)\) where \(a\) is a constant.
Prove by induction that \(\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \text { an } \\ 0 & 1 \end{array} \right)\) for all integers \(n \geqslant 1\).
OCR Further Pure Core AS 2024 June Q7
6 marks Challenging +1.8
7 In this question you must show detailed reasoning.
The roots of the equation \(2 x ^ { 3 } - 3 x ^ { 2 } - 3 x + 5 = 0\) are \(\alpha , \beta\) and \(\gamma\).
By considering \(( \alpha + \beta + \gamma ) ^ { 2 }\) and \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), determine a cubic equation with integer coefficients whose roots are \(\frac { \alpha \beta } { \gamma } , \frac { \beta \gamma } { \alpha }\) and \(\frac { \gamma \alpha } { \beta }\).
OCR Further Pure Core AS 2024 June Q8
10 marks Standard +0.3
8 Three transformations, \(T _ { A } , T _ { B }\) and \(T _ { C }\), are represented by the matrices \(A , B\) and \(\mathbf { C }\) respectively. You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)\).
  1. Find the matrix which represents the inverse transformation of \(T _ { A }\).
  2. By considering matrix multiplication, determine whether \(T _ { A }\) followed by \(T _ { B }\) is the same transformation as \(T _ { B }\) followed by \(T _ { A }\). Transformations R and S are each defined as being the result of successive transformations, as specified in the table.
    TransformationFirst transformationfollowed by
    R\(\mathrm { T } _ { \mathrm { A } }\) followed by \(\mathrm { T } _ { \mathrm { B } }\)\(\mathrm { T } _ { \mathrm { C } }\)
    S\(\mathrm { T } _ { \mathrm { A } }\)\(\mathrm { T } _ { \mathrm { B } }\) followed by \(\mathrm { T } _ { \mathrm { C } }\)
  3. Explain, using a property of matrix multiplication, why R and S are the same transformations. A quadrilateral, \(Q\), has vertices \(D , E , F\) and \(G\) in anticlockwise order from \(D\). Under transformation \(\mathrm { R } , Q ^ { \prime }\) s image, \(Q ^ { \prime }\), has vertices \(D ^ { \prime } , E ^ { \prime } , F ^ { \prime }\) and \(G ^ { \prime }\) (where \(D ^ { \prime }\) is the image of \(D\), etc). The area of \(Q\), in suitable units, is 5 . You are given that det \(\mathbf { C } = a ^ { 2 } + 1\) where \(a\) is a real constant.
    1. Determine the order of the vertices of \(Q ^ { \prime }\), starting anticlockwise from \(D ^ { \prime }\).
    2. Find, in terms of \(a\), the area of \(Q ^ { \prime }\).
    3. Explain whether the inverse transformation for R exists. Justify your answer.
OCR Further Pure Core AS Specimen Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
The equation \(x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
Find the values of \(p\) and \(q\).
OCR Further Pure Core AS Specimen Q2
4 marks Moderate -0.5
2 In this question you must show detailed reasoning.
Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = - 1 - \mathrm { i }\), find the following, giving each in the form \(a + b \mathrm { i }\).
  1. \(z _ { 1 } ^ { * } z _ { 2 }\)
  2. \(\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }\)
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 Statistics AS 2018 June Q1
8 marks Moderate -0.8
1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the Xth weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
  2. Find \(\mathrm { P } ( X = 11 )\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
  4. Find \(\operatorname { Var } ( X )\).
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.
OCR Further Statistics AS 2018 June Q2
8 marks Moderate -0.5
2 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\).
OCR Further Statistics AS 2018 June Q3
6 marks Moderate -0.8
3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).
  1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
  3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.
OCR Further Statistics AS 2018 June Q4
8 marks Standard +0.3
4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$
  1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
  2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
  3. Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
  4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
OCR Further Statistics AS 2018 June Q5
8 marks Standard +0.3
5
  1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
    Find the probability that the team contains no more than 3 girls.
  2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).
OCR Further Statistics AS 2018 June Q7
8 marks Moderate -0.8
7 An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.
\(c\)1.941.781.621.511.521.4
\(m\)6.57.27.47.68.39.7
  1. State which, if either, of \(m\) and \(c\) is an independent variable.
  2. Calculate the equation of the least squares regression line of \(c\) on \(m\).
  3. State what effect, if any, there would be on your answer to part (ii) if the masses of the fish had been recorded in kilograms rather than pounds. ( \(1 \mathrm {~kg} \approx 2.2\) pounds.)
  4. The data is illustrated in the scatter diagram. Explain what is meant by 'least squares', illustrating your answer using the copy of this diagram in the Printed Answer Booklet.
    [diagram]