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OCR Further Additional Pure AS 2020 November Q6
9 marks Challenging +1.2
6 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = - 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k }\) respectively.
  1. Determine the area of triangle \(O A B\), giving your answer in an exact form. The point \(C\) lies on the line \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { O }\) such that the area of triangle \(O A C\) is half the area of triangle \(O A B\).
  2. Determine the two possible position vectors of \(C\).
OCR Further Additional Pure AS 2020 November Q7
10 marks Standard +0.3
7 In a conservation project, a batch of 100000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previous research suggests that for every 1 million tadpoles hatched only 3550 will live to maturity at 12 weeks, when they become adult frogs. It is assumed that the steady decline in the population of tadpoles, from all causes, can be explained by a weekly death-rate factor, \(r\), which is constant across each week of this twelve-week period. Let \(\mathrm { T } _ { \mathrm { k } }\) denote the total number of tadpoles alive at the end of \(k\) weeks after the start of this project.
    1. Explain why a recurrence system for \(\mathrm { T } _ { \mathrm { k } }\) is given by \(T _ { 0 } = 100000\) and \(\mathrm { T } _ { \mathrm { k } + 1 } = ( 1 - \mathrm { r } ) \mathrm { T } _ { \mathrm { k } }\) for \(0 \leqslant k \leqslant 12\).
    2. Show that \(r = 0.375\), correct to 3 significant figures. The proportion of females within each batch of tadpoles is \(p\), where \(0 < p < 1\). In a simple model of the frog population the following assumptions are made.
      • The death rate factor for adult frogs is also \(r\) and is the same for males and females.
  2. The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
    1. Find the smallest value of \(p\) for which the frog population will survive according to the model.
    2. Write down one assumption that has been made in order to obtain this result.
    4. Each surviving female will then lay a batch of eggs from which 2500 tadpoles are hatched.
  4. By considering the total number of tadpoles hatched, give one criticism of the assumption that the frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project. \section*{END OF QUESTION PAPER}
OCR Further Additional Pure AS Specimen Q1
3 marks Standard +0.3
1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { 12 } { 1 + u _ { n } }\) for \(n \geq 1\).
Given that the sequence converges, with limit \(\alpha\), determine the value of \(\alpha\).
OCR Further Additional Pure AS Specimen Q2
5 marks Standard +0.3
2 The points \(A ( 1,2,2 ) , B ( 8,2,5 ) , C ( - 3,6,5 )\) and \(D ( - 10,6,2 )\) are the vertices of parallelogram \(A B C D\). Determine the area of \(A B C D\).
OCR Further Additional Pure AS Specimen Q3
5 marks Challenging +1.8
3 A non-commutative group \(G\) consists of the six elements \(\left\{ e , a , a ^ { 2 } , b , a b , b a \right\}\) where \(e\) is the identity element, \(a\) is an element of order 3 and \(b\) is an element of order 2 .
By considering the row in \(G\) 's group table in which each of the above elements is pre-multiplied by \(b\), show that \(b a ^ { 2 } = a b\).
OCR Further Additional Pure AS Specimen Q4
9 marks Challenging +1.2
4 Let \(S\) be the set \(\{ 16,36,56,76,96 \}\) and \(\times _ { H }\) the operation of multiplication modulo 100 .
  1. Given that \(a\) and \(b\) are odd positive integers, show that \(( 10 a + 6 ) ( 10 b + 6 )\) can also be written in the form \(10 n + 6\) for some odd positive integer \(n\).
  2. Construct the Cayley table for \(\left( S , \times _ { H } \right)\)
  3. Show that \(\left( S , \times _ { H } \right)\) is a group.
    [0pt] [You may use the result that \(\times _ { H }\) is associative on \(S\).]
  4. Write down all generators of \(\left( S , \times _ { H } \right)\).
OCR Further Additional Pure AS Specimen Q5
15 marks Standard +0.3
5 Let \(\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1\). The surface \(S\) has equation \(z = \mathrm { f } ( x , y )\).
  1. (a) Find \(f _ { x }\).
    (b) Find \(\mathrm { f } _ { y }\).
    (c) Show that \(S\) has a stationary point at ( \(0,0,1\) ).
    (d) Find the coordinates of the second stationary point of \(S\).
  2. The section \(z = \mathrm { f } ( a , y )\), where \(a\) is a constant, has exactly one stationary point. Determine the equation of the section. A customer takes out a loan of \(\pounds P\) from a bank at an annual interest rate of \(4.9 \%\). Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of \(\pounds M\) in order to reduce the outstanding amount of the loan. Let \(L _ { n }\) denote the outstanding amount of the loan at the end of month \(n\) after the fixed payment has been made, with \(L _ { 0 } = P\).
  3. Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation $$L _ { n + 1 } = 1.004 L _ { n } - M$$ with \(L _ { 0 } = P , n \geq 0\).
  4. Solve, in terms of \(n , M\) and \(P\), the first order recurrence relation given in part (i).
  5. The loan amount is \(\pounds 100000\) and will be fully repaid after 10 years. Find, to the nearest pound, the value of the monthly repayment.
  6. The bank's procedures only allow for calculations using integer amounts of pounds. When each monthly amount of the outstanding \(\operatorname { debt } \left( L _ { n } \right)\) is calculated it is always rounded up to the nearest pound before the monthly repayment ( \(M\) ) is subtracted.
    Rewrite (*) to take this into account.
  7. Let \(N = 10 a + b\) and \(M = a - 5 b\) where \(a\) and \(b\) are integers such that \(a \geq 1\) and \(0 \leq b \leq 9\). \(N\) is to be tested for divisibility by 17 .
    (a) Prove that \(17 \mid N\) if and only if \(17 \mid M\).
    (b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that \(17 \mid 4097\).
  8. (a) Show that, for \(n \geq 2\), any number of the form \(1001 _ { n }\) is composite.
    (b) Given that \(n\) is a positive even number, provide a counter-example to show that the statement "any number of the form \(10001 _ { n }\) is prime" is false. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the 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.
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OCR Further Pure Core 1 2019 June Q1
4 marks Standard +0.8
1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\).
OCR Further Pure Core 1 2019 June Q2
3 marks Standard +0.3
2 Indicate by shading on an Argand diagram the region \(\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \}\).
OCR Further Pure Core 1 2019 June Q3
4 marks Moderate -0.3
3 In this question you must show detailed reasoning.
You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.
OCR Further Pure Core 1 2019 June Q4
3 marks Easy -1.2
4 Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
OCR Further Pure Core 1 2019 June Q5
7 marks Standard +0.3
5 The diagram shows part of the curve \(y = 5 \cosh x + 3 \sinh x\). \includegraphics[max width=\textwidth, alt={}, center]{a6d9b3ec-5170-4f06-a8a3-b854efe36f07-3_496_771_315_246}
  1. Solve the equation \(5 \cosh x + 3 \sinh x = 4\) giving your solution in exact form.
  2. In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x\) giving your answer in the form \(a \mathrm { e } + \frac { b } { \mathrm { e } }\) where \(a\) and \(b\) are integers to be determined.
OCR Further Pure Core 1 2019 June Q6
6 marks Challenging +1.2
6 You are given that \(y = \tan ^ { - 1 } \sqrt { 2 x }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 2 } } \frac { \sqrt { x } } { \left( x + 2 x ^ { 2 } \right) } \mathrm { d } x = k \pi\) where \(k\) is a number to be determined in exact form.
OCR Further Pure Core 1 2019 June Q7
6 marks Standard +0.8
7 The function sech \(x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).
  1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
  2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).
OCR Further Pure Core 1 2019 June Q8
6 marks Standard +0.8
8 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\).
OCR Further Pure Core 1 2019 June Q9
12 marks Challenging +1.2
9 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).
  1. Show that \(\omega\) is a root of the equation.
  2. Write down the other four roots of the equation.
  3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
  4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
  5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found.
OCR Further Pure Core 1 2019 June Q10
11 marks Standard +0.8
10 You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
  2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
  3. For each of the values of \(a\) found in part (b) determine whether the equations have
    • a unique solution, which should be found, or
    • an infinite set of solutions or
    • no solution.
OCR Further Pure Core 1 2019 June Q11
13 marks Standard +0.8
11 A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t\) where \(x\) is the displacement of the particle below the equilibrium position at time \(t\).
When \(t = 0\) the particle is stationary and its displacement is 2 .
  1. Find the particular solution of the differential equation.
  2. Write down an approximate equation for the displacement when \(t\) is large.
OCR Further Pure Core 1 2022 June Q2
9 marks Moderate -0.8
2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ 1 & 3 \end{array} \right)\).
  1. Calculate \(\operatorname { det } \mathbf { A }\).
  2. Write down \(\mathbf { A } ^ { - 1 }\).
  3. Hence solve the equation \(\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }\).
  4. Write down the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = 4 \mathbf { I }\). Matrices \(\mathbf { C }\) and \(\mathbf { D }\) are given by \(\mathbf { C } = \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)\) where \(p\) is a constant.
  5. Find, in terms of \(p\),
    • the matrix CD
    • the matrix DC.
    It is observed that \(\mathbf { C D } \neq \mathbf { D C }\).
  6. The result that \(\mathbf { C D } \neq \mathbf { D C }\) is a counter example to the claim that matrix multiplication has a particular property. Name this property.
OCR Further Pure Core 1 2022 June Q3
11 marks Standard +0.3
3 In this question you must show detailed reasoning.
  1. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\). The loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are given by \(| z | = | z - 2 \mathrm { i } |\) and \(| z - 2 | = \sqrt { 5 }\) respectively.
    1. Sketch on a single Argand diagram the loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\), showing any intercepts with the imaginary axis.
    2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
    1. Show that both of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfy \(| z - 2 | < \sqrt { 5 }\).
    2. State, with a reason, which root of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfies \(| z | < | z - 2 i |\).
  4. On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
OCR Further Pure Core 1 2022 June Q4
4 marks Moderate -0.5
4 Determine the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } - \sqrt { 3 } \\ 1 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \sqrt { 3 } \\ - \sqrt { 3 } \end{array} \right)\) and the \(y\)-axis.
OCR Further Pure Core 1 2022 June Q5
11 marks Challenging +1.2
5 The diagram below shows the curve \(C\) with polar equation \(r = 3 ( 1 - \sin 2 \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-5_728_963_303_239}
  1. Show that a cartesian equation of \(C\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 9 ( x - y ) ^ { 4 }\).
  2. Show that the line with equation \(\mathrm { y } = \mathrm { x }\) is a line of symmetry of \(C\).
  3. In this question you must show detailed reasoning. Find the exact area of each of the loops of \(C\).
OCR Further Pure Core 1 2022 June Q6
6 marks Challenging +1.2
6 Let \(\mathrm { y } = \mathrm { x } \cosh \mathrm { x }\).
Prove by induction that, for all integers \(n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x\).
OCR Further Pure Core 1 2022 June Q7
10 marks Challenging +1.2
7
  1. Determine the values of \(A , B , C\) and \(D\) such that \(\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }\).
  2. In this question you must show detailed reasoning. Hence determine the exact value of \(\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).
OCR Further Pure Core 1 2022 June Q8
13 marks
8 A biologist is studying the effect of pesticides on crops. On a certain farm pesticide is regularly applied to a particular crop which grows in soil. Over time, pesticide is transferred between the crop and the soil at a rate which depends on the amount of pesticide in both the crop and the soil. The amount of pesticide in the crop after \(t\) days is \(x\) grams. The amount of pesticide in the soil after \(t\) days is \(y\) grams. Initially, when \(t = 0\), there is no pesticide in either the crop or the soil. At first it is assumed that no pesticide is lost from the system. The biologist further assumes that pesticide is added to the crop at a constant rate of \(k\) grams per day, where \(k > 6\). After collecting some initial data, the biologist suggests that for \(t \geqslant 0\), this situation can be modelled by the following pair of first order linear differential equations. \(\frac { d x } { d t } = - 2 x + 78 y + k\) \(\frac { d y } { d t } = 2 x - 78 y\)
    1. Show that \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 80 \frac { d x } { d t } = 78 \mathrm { k }\).
    2. Determine the particular solution for \(x\) in terms of \(k\) and \(t\). If more than 250 grams of pesticide is found in the crop, then it will fail food safety standards.
    3. The crop is tested 50 days after the pesticide is first added to it. Explain why, according to this model, the crop will fail food safety standards as a result of this test. Further data collection suggests that some pesticide decays in the soil and so is lost from the system. The model is refined in light of this data. The particular solution for \(x\) for this refined model is \(\mathrm { x } = \mathrm { k } \left( 20 - \mathrm { e } ^ { - 41 \mathrm { t } } \left( 20 \cosh ( \sqrt { 1677 } \mathrm { t } ) + \frac { 819 } { \sqrt { 1677 } } \sinh ( \sqrt { 1677 } \mathrm { t } ) \right) \right.\).
  2. Given now that \(k < 12\), determine whether the crop will fail food safety standards in the long run according to this refined model. In the refined model, it is still assumed that pesticide is added to the crop at a constant rate.
  3. Suggest a reason why it might be more realistic to model the addition of pesticide as not being at a constant rate.