OCR Further Pure Core 1 (Further Pure Core 1) 2024 June

1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).

2 The locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}\).

1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
2. Determine whether the complex number \(1.2 + 0.8\) is is \(C _ { 1 }\). The locus \(C _ { 2 }\) is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus \(C _ { 2 }\) is illustrated on the Argand diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
3. Use set notation to define \(C _ { 2 }\).
4. Determine whether the complex number \(1.2 + 0.8\) is in \(C _ { 2 }\).

3 A transformation T is represented by the matrix \(\mathbf { N } = \left( \begin{array} { l l l } a & 4 & 2
5 & 1 & 0
3 & 6 & 3 \end{array} \right)\), where \(a\) is a constant.

1. Find \(\mathbf { N } ^ { 2 }\) in terms of \(a\).
2. Find det \(\mathbf { N }\) in terms of \(a\). The value of \(a\) is 13 to the nearest integer.
   A shape \(S _ { 1 }\) has volume 11.6 to 1 decimal place. Shape \(S _ { 1 }\) is mapped to shape \(S _ { 2 }\) by the transformation T . A student claims that the volume of \(S _ { 2 }\) is less than 400 .
3. Comment on the student's claim.

4 In this question you must show detailed reasoning.
The equation \(2 x ^ { 3 } + 3 x ^ { 2 } + 6 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine a cubic equation with integer coefficients that has roots \(\alpha ^ { 2 } \beta \gamma , \alpha \beta ^ { 2 } \gamma\) and \(\alpha \beta \gamma ^ { 2 }\).

5 Express \(\frac { 12 x ^ { 3 } } { ( 2 x + 1 ) \left( 2 x ^ { 2 } + 1 \right) }\) using partial fractions.

6 In this question you must show detailed reasoning.
Determine the exact value of \(\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x\).

7

1. By using the definitions of \(\cosh u\) and \(\sinh u\) in terms of \(\mathrm { e } ^ { u }\) and \(\mathrm { e } ^ { - u }\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\). The equation of a curve, \(C\), is \(\mathrm { y } = 16 \cosh \mathrm { x } - \sinh 2 \mathrm { x }\).
2. Show that there is only one solution to the equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 0\) You are now given that \(C\) has exactly one point of inflection.
3. Use your answer to part (b) to determine the exact coordinates of this point of inflection. Give your answer in a logarithmic form where appropriate.

8 Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).

9

1. Find the Maclaurin series of \(( \ln ( 1 + x ) ) ^ { 2 }\) up to and including the term in \(x ^ { 4 }\). The diagram below shows parts of the graphs of the curves with equations \(y = ( \ln ( 1 + x ) ) ^ { 2 }\) and \(y = 2 x ^ { 3 }\). The curves intersect at the origin, \(O\), and at the point \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-4_663_906_831_248} \section*{(b) In this question you must show detailed reasoning.} Use your answer to part (a) to determine an approximation for the value of the \(x\)-coordinate of \(A\). Give your answer to \(\mathbf { 2 }\) decimal places.

10 A particle \(B\), of mass 3 kg , moves in a straight line and has velocity \(v \mathrm {~ms} ^ { - 1 }\).
At time \(t\) seconds, where \(0 \leqslant t < \frac { 1 } { 4 } \pi\), a variable force of \(- ( 15 \sin 4 \mathrm { t } + 6 \mathrm { v } \tan 2 \mathrm { t } )\) Newtons is applied to \(B\). There are no other forces acting on \(B\). Initially, when \(t = 0 , B\) has velocity \(4.5 \mathrm {~ms} ^ { - 1 }\). The motion of \(B\) can be modelled by the differential equation \(\frac { d v } { d t } + P ( t ) v = Q ( t )\) where \(P ( t )\) and \(\mathrm { Q } ( \mathrm { t } )\) are functions of \(t\).

1. Find the functions \(\mathrm { P } ( \mathrm { t } )\) and \(\mathrm { Q } ( \mathrm { t } )\).
2. Using an integrating factor, determine the first time at which \(B\) is stationary according to the model.

11 A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles \(P _ { 1 } , P _ { 2 } , P _ { 3 }\) and \(P _ { 4 }\). The floor of the construction site is modelled as lying in the \(x - y\) plane and the poles are modelled as vertical line segments. One end of each pole lies on the floor of the construction site, and the other end of each pole is modelled by the points \(( 0,0,18 ) , ( 12,14,20 ) , ( 0,11,7 )\) and \(( 18,2,16 )\) respectively. A wire, \(S\), runs from the top of \(P _ { 1 }\) to the top of \(P _ { 2 }\). A second wire, \(T\), runs from the top of \(P _ { 3 }\) to the top of \(P _ { 4 }\). The wires are modelled by straight lines segments. The layout of the construction site is illustrated on the diagram below which is not drawn to scale.
\includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-5_707_871_696_242} A vector equation of the line segment that represents the wire \(S\) is given by
\(\mathbf { r } = \left( \begin{array} { c } 0
0
18 \end{array} \right) + \lambda \left( \begin{array} { l } 6
7
1 \end{array} \right) , 0 \leqslant \lambda \leqslant 2\).

1. Find, in the same form, a vector equation of the line segment that represents the wire \(T\). The components of the direction vector should be integers whose only positive common factor is 1 . For the construction site to be considered safe, it must pass two tests.
   Test 1: The wires \(S\) and \(T\) need to be at least 5 metres apart at all positions on \(S\) and \(T\).
2. By using an appropriate formula, determine whether the construction site passes Test 1. A security camera is placed at a point \(Q\) on wire \(S\). Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of \(P _ { 3 }\) must be at least 19 m .
3. Determine whether it is possible to find point \(Q\) on \(S\) such that the construction site passes Test 2.

12 For any positive parameter \(k\), the curve \(C _ { k }\) is defined by the polar equation
\(\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi\).
For each value of \(k\) the curve is a single, closed loop with no self-intersections. The diagram shows \(C _ { 10.5 }\) for the purpose of illustration.
\includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, \(C _ { k }\), encloses a certain area, \(A _ { k }\).
You are given that there is a single minimum value of \(A _ { k }\).
Determine, in an exact form, the value of \(k\) for which \(C _ { k }\) encloses this minimum area.