Questions Further Additional Pure (91 questions)

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OCR Further Additional Pure Specimen Q1
4 marks Challenging +1.8
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 Further Additional Pure 2019 June Q8
11 marks Hard +2.3
8 In this question you must show detailed reasoning.
  1. Prove that \(2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )\), where \(p\) is an odd prime.
  2. Find two odd prime factors of the number \(N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Further Additional Pure 2021 November Q1
3 marks Moderate -0.8
1 In this question you must show detailed reasoning. Express the number \(\mathbf { 4 1 7 2 3 } _ { 10 }\) in hexadecimal (base 16).
OCR Further Additional Pure 2018 March Q1
5 marks Standard +0.3
1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$
OCR Further Additional Pure 2018 March Q2
5 marks Standard +0.8
2 Four points \(A , B , C\) and \(D\) have coordinates \(( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )\) and \(( 0,3,7 )\) respectively. Find the volume of tetrahedron \(A B C D\).
OCR Further Additional Pure 2018 March Q3
10 marks Challenging +1.2
3 The surface \(S\) has equation \(z = \frac { x } { y } \sin y + \frac { y } { x } \cos x\) where \(0 < x \leqslant \pi\) and \(0 < y \leqslant \pi\).
  1. Find
    • \(\frac { \partial z } { \partial x }\),
    • \(\frac { \partial z } { \partial y }\).
    • Determine the equation of the tangent plane to \(S\) at the point \(A\) where \(x = y = \frac { 1 } { 4 } \pi\). Give your answer in the form \(a x + b y + c z = d\) where \(a , b , c\) and \(d\) are exact constants.
    • Write down a normal vector to \(S\) at \(A\).
OCR Further Additional Pure 2018 March Q4
12 marks Hard +2.3
4
  1. (a) Find all the quadratic residues modulo 11.
    (b) Prove that the equation \(y ^ { 5 } = x ^ { 2 } + 2017\) has no solution in integers \(x\) and \(y\).
  2. In this question you must show detailed reasoning. The numbers \(M\) and \(N\) are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that \(M\) is divisible by \(N\).
OCR Further Additional Pure 2018 March Q5
15 marks Challenging +1.8
5
  1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
    (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
  2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
    (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
  3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
    (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
    (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.
OCR Further Additional Pure 2018 March Q6
14 marks Challenging +1.8
6 In this question you must show detailed reasoning. It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).
  1. Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
  2. Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\). The curve \(C\) is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
  3. Determine
    • the values of the integers \(k\) and \(m\) such that \(A = k \pi I _ { m }\),
    • the exact value of \(A\).
OCR Further Additional Pure 2018 March Q7
14 marks Challenging +1.8
7 The set \(M\) contains all matrices of the form \(\mathbf { X } ^ { n }\), where \(\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)\) and \(n\) is a positive integer.
  1. Show that \(M\) contains exactly 12 elements.
  2. Deduce that \(M\), together with the operation of matrix multiplication, form a cyclic group \(G\).
  3. Determine all the proper subgroups of \(G\). \section*{END OF QUESTION PAPER}
OCR Further Additional Pure 2018 December Q1
7 marks Standard +0.8
1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).
  1. Find
OCR Further Additional Pure 2018 December Q2
13 marks Challenging +1.2
2 A sequence \(\left\{ u _ { n } \right\}\) is given by \(u _ { n + 1 } = 4 u _ { n } + 1\) for \(n \geqslant 1\) and \(u _ { 1 } = 3\).
  1. Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Solve the recurrence system (*).
    1. Prove by induction that each term of the sequence can be written in the form \(( 10 m + 3 )\) where \(m\) is an integer.
    2. Show that no term of the sequence is a square number.
OCR Further Additional Pure 2018 December Q3
9 marks Challenging +1.8
3
  1. Show that \(10 ^ { 2 } \equiv 6 ( \bmod 47 )\).
  2. Determine the integer \(r\), with \(0 < r < 47\), such that \(6 r \equiv 1 ( \bmod 47 )\).
  3. Determine the least positive integer \(n\) for which \(10 ^ { n } \equiv 1\) or \(- 1 ( \bmod 47 )\).
OCR Further Additional Pure 2018 December Q4
12 marks Challenging +1.2
4 The set \(L\) consists of all points \(( x , y )\) in the cartesian plane, with \(x \neq 0\). The operation ◇ is defined by \(( a , b ) \diamond ( c , d ) = ( a c , b + a d )\) for \(( a , b ) , ( c , d ) \in L\).
    1. Show that \(L\) is closed under ◇.
    2. Prove that \(\diamond\) is associative on \(L\).
    3. Find the identity element of \(L\) under ◇ .
    4. Find the inverse element of \(( a , b )\) under ◇.
  2. Find a subgroup of \(( L , \diamond )\) of order 2.
OCR Further Additional Pure 2018 December Q5
10 marks Standard +0.3
5 Torque is a vector quantity that measures how much a force acting on an object causes that object to rotate. The torque (about the origin), \(\mathbf { T }\), exerted on an object is given by \(\mathbf { T } = \mathbf { p } \times \mathbf { F }\), where \(\mathbf { F }\) is the force acting on the object and \(\mathbf { p }\) is the position vector of the point at which \(\mathbf { F }\) is applied to the object. The points \(A\) and \(B\), with position vectors \(\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = 3 \mathbf { i } + 5 \mathbf { j } + \mathbf { k }\) are on the surface of a rock. The force \(\mathbf { F } _ { 1 } = 6 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }\) is applied to the rock at \(A\) while the force \(\mathbf { F } _ { 2 } = - 7 \mathbf { i } - 10 \mathbf { j } + 2 \mathbf { k }\) is applied to the rock at \(B\).
  1. Find the torque (about the origin) exerted on the rock by \(\mathbf { F } _ { 1 }\).
  2. Determine which of the two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) exerts a torque (about the origin) of greater magnitude on the rock.
  3. Show that the torque (about the origin) is the same as your answer to part (a) when \(\mathbf { F } _ { 1 }\) acts on the rock at any point on the line \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { p }\), where \(\mathbf { p }\) is a vector in the same direction as \(\mathbf { F } _ { 1 }\). A third force \(\mathbf { F } _ { 3 }\) is now applied to the rock at \(A\), which exerts zero torque (about the origin).
  4. Show that \(\mathbf { F } _ { 3 }\) must act in the direction of the line through \(A\) and the origin.
OCR Further Additional Pure 2018 December Q6
13 marks Challenging +1.8
6 For positive integers \(n\), the integrals \(I _ { n }\) are given by \(I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } \sqrt { 2 + x ^ { 2 } } \mathrm {~d} x\).
  1. Show that \(I _ { 1 } = 26 \sqrt { 3 }\).
  2. Prove that, for \(n \geqslant 3 , ( n + 2 ) I _ { n } = 3 \sqrt { 3 } \left( 27 \times 5 ^ { n - 1 } - 1 \right) - 2 ( n - 1 ) I _ { n - 2 }\).
  3. Determine the exact value of \(I _ { 5 }\) as a rational multiple of \(\sqrt { 3 }\).
OCR Further Additional Pure 2018 December Q7
11 marks Challenging +1.2
7 For each value of \(t\), the surface \(S _ { t }\) has equation \(z = t x ^ { 2 } + y ^ { 2 } + 3 x y - y\).
  1. Verify that there are no stationary points on \(S _ { t }\) when \(t = \frac { 9 } { 4 }\).
  2. Determine, as \(t\) varies, the nature of any stationary point(s) of \(S _ { t }\).
    (You do not have to find the coordinates of the stationary points.) \section*{OCR} Oxford Cambridge and RSA