OCR Further Pure Core 1 (Further Pure Core 1) 2021 November

1

1. Sketch on a single Argand diagram the loci given by
   1. \(\quad | z - 1 + 2 i | = 3\),
   2. \(\quad | z + 1 | = | z - 2 |\).
2. Indicate, by shading, the region of the Argand diagram for which \(| z - 1 + 2 i | \leqslant 3\) and \(| z + 1 | \leqslant | z - 2 |\).

2 You are given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )\).

1. 1. Find the value of \(f ( 0 )\).
   2. Determine the value of \(f ^ { \prime } ( 0 )\).
   3. Show that \(f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }\).
2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).

3 A function \(\mathrm { f } ( \mathrm { z } )\) is defined on all complex numbers z by \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 3 \mathrm { z } ^ { 2 } + \mathrm { k } \mathrm { z } - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta\) and \(\gamma\). You are given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5\).

1. Explain why \(f ( z ) = 0\) has only one real root.
2. Find the value of \(k\).
3. Find a cubic equation with integer coefficients that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

4 Points \(A , B\) and \(C\) have coordinates ( \(4,2,0\) ), ( \(1,5,3\) ) and ( \(1,4 , - 2\) ) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\).
   \(M\) is the point on \(l\) that is closest to \(C\).
2. Find the coordinates of \(M\).
3. Find the exact area of the triangle \(A B C\).

5 Use de Moivre's theorem to find the constants \(A , B\) and \(C\) in the identity \(\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta\).
\(6 O\) is the origin of a coordinate system whose units are cm .
The points \(A , B , C\) and \(D\) have coordinates ( 1,0 ), ( 1,4 ), ( 6,9 ) and ( 0,9 ) respectively.
The arc \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_641_1131_808_242} Find the volume of the funnel according to the model.

7 The diagram below shows the curve with polar equation \(r = \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_385_807_1834_260}

1. Find the values of \(\theta\) at the pole.
2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value.
3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve.
4. Given that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\), find a cartesian equation for the curve.

8 You are given that \(\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = f ( x )\) has no turning points.
2. Determine the exact solution of the equation \(\mathrm { f } ( x ) = 5\).

9 You are given that the matrix \(\left( \begin{array} { c c } 2 & 1
- 1 & 0 \end{array} \right)\) represents a transformation T .

1. You are given that the line with equation \(\mathrm { y } = \mathrm { kx }\) is invariant under T . Determine the value of \(k\).
2. Determine whether the line with equation \(\mathrm { y } = \mathrm { kx }\) in part (a) is a line of invariant points under T .

10 Using an algebraic method, determine the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } \geqslant 0.49\).

11 The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 3 \theta = 0\) where \(\lambda\) is a constant and \(\lambda \geqslant 0\).

1. 1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic?
   2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic.
2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position.
3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic.