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OCR Further Additional Pure 2020 November Q7
10 marks Challenging +1.2
7 Throughout this question, \(n\) is a positive integer.
  1. Explain why \(n ^ { 5 } \equiv n ( \bmod 5 )\).
  2. By proving that \(n ^ { 5 } \equiv n ( \bmod 2 )\), show that \(n ^ { 5 } \equiv n ( \bmod 10 )\).
    1. Prove that \(n ^ { 5 } - n\) is divisible by 30 for all positive integers \(n\).
    2. Is there an integer \(N\), greater than 30 , such that \(n ^ { 5 } - n\) is divisible by \(N\) for all positive integers \(n\) ? Justify your answer.
OCR Further Additional Pure 2020 November Q8
12 marks Challenging +1.2
8 The sequence \(\left\{ u _ { n } \right\}\) of positive real numbers is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }\) for \(n \geqslant 1\).
  1. Prove by induction that \(u _ { n } ^ { 2 } - 3 < 0\) for all positive integers \(n\).
  2. By considering \(u _ { n + 1 } - u _ { n }\), use the result of part (a) to show that \(u _ { n + 1 } > u _ { n }\) for all positive integers \(n\). The sequence \(\left\{ u _ { n } \right\}\) has a limit for \(n \rightarrow \infty\).
  3. Find the limit of the sequence \(\left\{ u _ { n } \right\}\) as \(n \rightarrow \infty\).
  4. Describe as fully as possible the behaviour of the sequence \(\left\{ u _ { n } \right\}\). \section*{END OF QUESTION PAPER}
OCR Further Additional Pure 2021 November Q2
5 marks Challenging +1.3
2 The following Cayley table is for \(G\), a group of order 6. The identity element is \(e\) and the group is generated by the elements \(a\) and \(b\).
G\(e\)\(а\)\(a ^ { 2 }\)\(b\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)
\(e\)\(e\)\(а\)\(a ^ { 2 }\)\(b\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)
\(a\)\(а\)\(a ^ { 2 }\)\(e\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(b\)
\(a ^ { 2 }\)\(a ^ { 2 }\)\(e\)\(a\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(b\)\(a b\)
\(b\)b\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a b\)\(e\)\(a ^ { 2 }\)\(a\)
\(a b\)\(a b\)b\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a\)\(e\)\(a ^ { 2 }\)
\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a b\)b\(a ^ { 2 }\)\(a\)\(e\)
  1. List all the proper subgroups of \(G\).
  2. State another group of order 6 to which \(G\) is isomorphic.
OCR Further Additional Pure 2021 November Q3
6 marks Standard +0.8
3 The points \(P , Q\) and \(R\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } , \mathbf { q } = \mathbf { i } - \mathbf { j } + \mathbf { k }\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + t \mathbf { k }\) respectively, relative to the origin \(O\). Determine the value(s) of \(t\) in each of the following cases.
  1. The line \(O R\) is parallel to \(\mathbf { p } \times \mathbf { q }\).
  2. The volume of tetrahedron \(O P Q R\) is 13 .
OCR Further Additional Pure 2021 November Q4
6 marks Challenging +1.2
4 Solve the simultaneous linear congruences \(x \equiv 1 ( \bmod 3 ) , x \equiv 5 ( \bmod 11 ) , 2 x \equiv 5 ( \bmod 17 )\).
OCR Further Additional Pure 2021 November Q5
8 marks Challenging +1.8
5 The surface \(S\) has equation \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = x y z - 1\).
  1. Show that \(( 2 z - x y ) \left( x \frac { \partial z } { \partial x } + y \frac { \partial z } { \partial y } \right) = 2 \left( 1 + z ^ { 2 } \right)\).
  2. Deduce that \(S\) has no stationary point.
OCR Further Additional Pure 2021 November Q6
10 marks Challenging +1.8
6 The binary operation ◇ is defined on the set \(\mathbb { C }\) of complex numbers by
\(( a + i b ) \diamond ( c + i d ) = a c + i ( b + a d )\)
where \(a , b , c\) and \(d\) are real numbers.
  1. Is \(\mathbb { C }\) closed under △ ? Justify your answer.
  2. Prove that ◇ is associative on \(\mathbb { C }\).
  3. Determine the identity element of \(\mathbb { C }\) under \(\diamond\).
  4. Determine the largest subset S of \(\mathbb { C }\) such that \(( \mathrm { S } , \diamond )\) is a group.
OCR Further Additional Pure 2021 November Q7
8 marks Challenging +1.2
7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { \mathrm { n } } \mathrm { xdx }\) for integers \(n \geqslant 0\).
  1. Show that, for \(n \geqslant 2 , \quad \mathrm { nl } _ { \mathrm { n } } = ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } - 2 }\).
  2. Use this reduction formula to deduce the exact value of \(I _ { 8 }\).
  3. Use the results of parts (a) and (b) to determine the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x\).
OCR Further Additional Pure 2021 November Q8
12 marks Hard +2.3
8
  1. Solve the second-order recurrence system \(\mathrm { H } _ { \mathrm { n } + 2 } = 5 \mathrm { H } _ { \mathrm { n } + 1 } - 4 \mathrm { H } _ { \mathrm { n } }\) with \(H _ { 0 } = 3 , H _ { 1 } = 7\) for \(n \geqslant 0\).
    1. Write down the quadratic residues modulo 10 .
    2. By considering the sequence \(\left\{ \mathrm { H } _ { \mathrm { n } } \right\}\) modulo 10, prove that \(\mathrm { H } _ { \mathrm { n } }\) is never a perfect square.
OCR Further Additional Pure 2021 November Q9
10 marks Challenging +1.8
9 For each value of \(k\) the sequence of real numbers \(\left\{ u _ { n } \right\}\) is given by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { k } { 6 + u _ { n } }\). For each of the following cases, either determine a value of \(k\) or prove that one does not exist.
  1. \(\left\{ \mathrm { u } _ { n } \right\}\) is constant.
  2. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 2 .
  3. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 4 .
OCR Further Additional Pure 2021 November Q10
7 marks Standard +0.8
10
\includegraphics[max width=\textwidth, alt={}, center]{df94bc38-5187-4349-9005-f9b72691c70d-4_519_770_251_242} A student wishes to model the saddle of a horse. They use a surface described by a function of the form \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { y } )\) with a saddle point at the origin \(O\). The z -axis is vertically upwards. The \(x\) - and \(y\)-axes lie in a horizontal plane, with the \(x\)-axis across the horse and the \(y\)-axis along the length of the horse (see diagram). The arc \(A O B\) is part of a parabola which lies in the \(y z\)-plane. The arc \(C O D\) is part of a parabola which lies in the \(x z\)-plane. The saddle is symmetric in both the \(x z\)-plane and \(y z\)-plane. The length of the saddle, the distance \(A B\), is to be 0.6 m with both \(A\) and \(B\) at a height of 0.27 m above \(O\). The width of the saddle, the distance \(C D\), is to be 0.5 m with both \(C\) and \(D\) at a depth of 0.4 m below \(O\).
  1. On separate diagrams, sketch the sections \(x = 0\) and \(y = 0\).
    [0pt]
  2. Determine a function f that describes the saddle. [You do not need to state the domain of function f .] \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Further Additional Pure Specimen Q1
4 marks
1 A curve is given by \(x = t ^ { 2 } - 2 \ln t , y = 4 t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
Given that \(A = k \pi\), where \(k\) is an integer,
  • write down an integral which gives \(A\) and
  • find the value of \(k\).
OCR Further Additional Pure Specimen Q2
3 marks Standard +0.8
2 Find the volume of tetrahedron OABC , where O is the origin, \(\mathrm { A } = ( 2,3,1 ) , \mathrm { B } = ( - 4,2,5 )\) and \(\mathrm { C } = ( 1,4,4 )\).
OCR Further Additional Pure Specimen Q3
5 marks Standard +0.3
3 Given \(z = x \sin y + y \cos x\), show that \(\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } + z = 0\).
OCR Further Additional Pure Specimen Q4
6 marks Standard +0.8
4
  1. Solve the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 4 u _ { n }\) for \(n \geq 0\), given that \(u _ { 0 } = 1\) and \(u _ { 1 } = 1\).
  2. Show that each term of the sequence \(\left\{ u _ { n } \right\}\) is an integer.
OCR Further Additional Pure Specimen Q5
9 marks Challenging +1.3
5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).
  1. Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
  2. (a) Evaluate \(I _ { 1 }\).
    (b) Use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).
OCR Further Additional Pure Specimen Q6
10 marks Challenging +1.2
6 A surface \(S\) has equation \(z = \mathrm { f } ( x , y )\), where \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y\). It is given that \(S\) has a single stationary point, \(P\).
  1. (a) Determine the coordinates of \(P\).
    (b) Determine the nature of \(P\).
  2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).
OCR Further Additional Pure Specimen Q7
11 marks Challenging +1.2
7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is an integer denoted by \(N _ { t }\). The initial number of breeding pairs is given by \(N _ { 0 }\). An initial discrete population model is proposed for \(N _ { t }\). $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$
  1. (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
    (b) Show that \(N _ { t + 1 } - N _ { t } < 150 - N _ { t }\) when \(N _ { t }\) lies between 0 and 150 .
    (c) Hence find the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N _ { 0 } \in ( 0,150 )\). An alternative discrete population model is proposed for \(N _ { t }\). $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
  2. (a) Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models.
    (b) Which of the two models gives values for \(N _ { t }\) with the more appropriate level of precision?
OCR Further Additional Pure Specimen Q8
13 marks Challenging +1.8
8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.
  1. (a) The matrices \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
    (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
  2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.]
  3. Determine a subgroup of \(G\) of order 17.
OCR Further Additional Pure Specimen Q9
14 marks Hard +2.3
9
  1. (a) Prove that \(p \equiv \pm 1 ( \bmod 6 )\) for all primes \(p > 3\).
    (b) Hence or otherwise prove that \(p ^ { 2 } - 1 \equiv 0 ( \bmod 24 )\) for all primes \(p > 3\).
  2. Given that \(p\) is an odd prime, determine the residue of \(2 ^ { p ^ { 2 } - 1 }\) modulo \(p\).
  3. Let \(p\) and \(q\) be distinct primes greater than 3 . Prove that \(p ^ { q - 1 } + q ^ { p - 1 } \equiv 1 ( \bmod p q )\). \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
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OCR M2 2007 June Q1
3 marks Easy -1.2
1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of \(35 ^ { \circ }\) with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m .
OCR M2 2007 June Q2
4 marks Moderate -0.8
2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(27 ^ { \circ }\).
OCR M2 2007 June Q3
8 marks Standard +0.3
3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a time \(t\) seconds.
  1. Calculate the value of \(t\).
  2. Calculate the acceleration of the rocket at the instant when its speed is \(120 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR M2 2007 June Q4
8 marks Moderate -0.5
4 A ball is projected from a point \(O\) on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. At time \(t\) seconds after projection the ball is at the point \(( x , y )\) referred to horizontal and vertically upward axes through \(O\). Air resistance may be neglected.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence show that \(y = 3 x - \frac { 1 } { 10 } x ^ { 2 }\). The ball hits the sea at a point which is 25 m below the level of \(O\).
  2. Find the horizontal distance between the cliff and the point where the ball hits the sea.
OCR M2 2007 June Q5
8 marks Moderate -0.3
5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find
  1. the increase in kinetic energy,
  2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
  3. Calculate the distance \(A B\).