OCR Further Additional Pure (Further Additional Pure) 2018 December

1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).

1. Find
   • \(\frac { \partial z } { \partial x }\),
   • \(\frac { \partial z } { \partial y }\).
   • Find in cartesian form, the equation of the tangent plane to the surface at the point where \(x = 1\) and \(y = \frac { 1 } { 4 } \pi\).

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 (*).
3. 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.

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 \(- 1 ( \bmod 47 )\).

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. 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.

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.

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 \(\sqrt { 3 }\).

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