OCR MEI Further Pure Core (Further Pure Core) Specimen

1 Find the acute angle between the lines with vector equations \(\mathbf { r } = \left( \begin{array} { c } 3
0
- 2 \end{array} \right) + \lambda \left( \begin{array} { c } 1
2
- 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 1
5
3 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right)\).

2

1. On an Argand diagram draw the locus of points which satisfy \(\arg ( z - 4 \mathrm { i } ) = \frac { \pi } { 4 }\).
2. Give, in complex form, the equation of the circle which has centre at \(6 + 4 \mathrm { i }\) and touches the locus in part (i).

3 Transformation M is represented by matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 3
1 & 4 \end{array} \right)\).

1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M .
2. (A) Show that there is a constant \(k\) such that \(\mathbf { M } \binom { x } { k x } = 5 \binom { x } { k x }\) for all \(x\).
   (B) Hence find the equation of an invariant line under M .
   (C) Draw the invariant line from part (ii) (B) on your diagram for part (i).

4 You are given that \(z = 1 + 2 \mathrm { i }\) is a root of the equation \(z ^ { 3 } - 5 z ^ { 2 } + q z - 15 = 0\), where \(q \in \mathbb { R }\). Find

• the other roots,
• the value of \(q\).

5

1. Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }\), expressing your answer as a single fraction.

6

1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t , y = \cosh t - \sinh t\) where \(t \in \mathbb { R }\). Show that the cartesian equation of the curve is \(x y = 1\). Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-3_931_1050_598_479} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer.

1. Use the Maclaurin series for \(\ln ( 1 + x )\) up to the term in \(x ^ { 3 }\) to obtain an approximation to \(\ln 1.5\).
2. (A) Find the error in the approximation in part (i).
   (B) Explain why the Maclaurin series in part (i), with \(x = 2\), should not be used to find an approximation to \(\ln 3\).
3. Find a cubic approximation to \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
4. (A) Use the approximation in part (iii) to find approximations to
   • ln 1.5 and
   • \(\quad \ln 3\).
     (B) Comment on your answers to part (iv) (A).

8 Find the cartesian equation of the plane which contains the three points \(( 1,0 , - 1 ) , ( 2,2,1 )\) and \(( 1,1,2 )\).

9 A curve has polar equation \(r = a \sin 3 \theta\) for \(- \frac { 1 } { 3 } \pi \leq \theta \leq \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.

1. Sketch the curve.
2. In this question you must show detailed reasoning. Find, in terms of \(a\) and \(\pi\), the area enclosed by one of the loops of the curve.
3. Obtain the solution to the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { 1 } { x } , \text { where } x > 0 ,$$ given that \(y = 1\) when \(x = 1\).
4. Deduce that \(y\) decreases as \(x\) increases.

11

1. It is conjectured that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = a - \frac { b } { n ! } ,$$ where \(a\) and \(b\) are constants, and \(n\) is an integer such that \(n \geq 2\). By considering particular cases, show that if the conjecture is correct then \(a = b = 1\).
2. Use induction to prove that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = 1 - \frac { 1 } { n ! } \text { for } n \geq 2 .$$

12 In this question you must show detailed reasoning.

1. Given that \(y = \arctan x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\). Fig. 12 shows the curve \(y = \frac { 1 } { 1 + x ^ { 2 } }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-5_444_1435_1371_332} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure}
2. Find, in exact form, the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 1 + x ^ { 2 } }\) for \(- 1 \leq x \leq 1\).
3. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = - 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated.

13 Matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c c } k & 1 & - 5
2 & 3 & - 3
- 1 & 2 & 2 \end{array} \right)\), where \(k\) is a constant.

1. Show that \(\operatorname { det } \mathbf { M } = 12 ( k - 3 )\).
2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$\begin{aligned} 4 x ^ { 2 } + y ^ { 2 } - 5 z ^ { 2 } & = 6
   2 x ^ { 2 } + 3 y ^ { 2 } - 3 z ^ { 2 } & = 6
   - x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } & = - 6 \end{aligned}$$
3. (A) Verify that the point ( \(2,0,1\) ) lies on each of the following three planes. $$\begin{aligned} 3 x + y - 5 z & = 1
   2 x + 3 y - 3 z & = 1
   - x + 2 y + 2 z & = 0 \end{aligned}$$ (B) Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer.
4. Find the values of \(k\) for which the transformation represented by \(\mathbf { M }\) has a volume scale factor of 6 .

1. Starting with the result $$\mathrm { e } ^ { \mathrm { i } \theta } = \cos \theta + \mathrm { i } \sin \theta$$ show that
   (A) \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta\)
   (B) \(\cos \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right)\).
2. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity
3. Using the result in part (i) (B), obtain the values of the constants \(P , Q , R\) and \(S\) in the identity
4. Show that \(\cos \frac { \pi } { 12 } = \left( \frac { 26 + 15 \sqrt { 3 } } { 64 } \right) ^ { \frac { 1 } { 6 } }\).
5. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity $$\cos 6 \theta \equiv a \cos ^ { 6 } \theta + b \cos ^ { 4 } \theta + c \cos ^ { 2 } \theta + d$$ $$\cos ^ { 6 } \theta \equiv P \cos 6 \theta + Q \cos 4 \theta + R \cos 2 \theta + S$$

15 In this question you must show detailed reasoning. Show that $$\int _ { 0 } ^ { \frac { 2 } { 3 } } \operatorname { arsinh } 2 x \mathrm {~d} x = \frac { 2 } { 3 } \ln 3 - \frac { 1 } { 3 }$$

16 A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time \(t\) seconds is denoted by \(x \mathrm {~cm}\). Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which \(x = 0\).

1. (A) Write down a differential equation to model this motion.
   (B) Give the general solution of the differential equation in part (i) (A). Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( k ^ { 2 } + 9 \right) x = 0$$ where \(k\) is a positive constant.
2. (A) Obtain the general solution of (*).
   (B) State two ways in which the motion given by this model differs from that in part (i). The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.
3. Find the value of \(k\). At the start of the object's motion, \(x = 0\) and the velocity is \(12 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction.
4. Find an equation for \(x\) as a function of \(t\).
5. Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm . \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
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