Questions Further Additional Pure (85 questions)

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OCR Further Additional Pure 2018 December Q3
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
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
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
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
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
OCR Further Additional Pure 2017 Specimen Q1
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 2017 Specimen Q5
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. Evaluate \(I _ { 1 }\) and 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 2017 Specimen Q6
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. Determine the coordinates, and 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 2017 Specimen Q7
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 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 determine 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. Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models and give a reason why Model II may be considered more suitable.
OCR Further Additional Pure 2017 Specimen Q8
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 \(G\).
    [0pt] [You may use the result that matrix multiplication is associative.]
  3. Determine a subgroup of \(G\) of order 17.