OCR Further Pure Core 1 (Further Pure Core 1) 2018 March

1 In this question you must show detailed reasoning.
Find the square roots of \(24 + 10 \mathrm { i }\), giving your answers in the form \(a + b \mathrm { i }\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)\).

1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).

3 Prove by mathematical induction that, for all integers \(n \geqslant 1 , n ^ { 5 } - n\) is divisible by 5 .

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).

5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$

6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$

1. Write down the general solution to this differential equation.
2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
3. State any limitations on the model.

7

1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$ \section*{(ii) In this question you must show detailed reasoning.} By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3 .$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.

8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).

1. Find the value of \(a\) and the value of \(b\).
2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.

9 In an experiment, at time \(t\) minutes there is \(Q\) grams of substance present.
It is known that the substance decays at a rate that is proportional to \(1 + Q ^ { 2 }\). Initially there are 100 grams of the substance present and after 100 minutes there are 50 grams present. Find the amount of the substance present after 400 minutes.

10

1. (a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
   (b) Explain why the curve is symmetrical in the \(x\)-axis.
   (c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
2. The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}