OCR MEI Further Pure Core (Further Pure Core) Specimen

Find the acute angle between the lines with vector equations \(\mathbf{r} = \begin{pmatrix} 3 \\ 0 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}\). [3]

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

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

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

You are given that \(z = 1 + 2i\) is a root of the equation \(z^3 - 5z^2 + qz - 15 = 0\), 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]
2. Hence find \(\sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}\), expressing your answer as a single fraction. [5]

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 \(xy = 1\). [2]

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. \includegraphics{figure_6}

1. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]

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

Find the cartesian equation of the plane which contains the three points \((1, 0, -1)\), \((2, 2, 1)\) and \((1, 1, 2)\). [5]

A curve has polar equation \(r = a \sin 3\theta\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{1}{4}\pi\), where \(a\) is a positive constant.

1. Sketch the curve. [2]
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. [5]

1. Obtain the solution to the differential equation $$x \frac{dy}{dx} + 3y = \frac{1}{x}, \text{ where } x > 0,$$ given that \(y = 1\) when \(x = 1\). [7]
2. Deduce that \(y\) decreases as \(x\) increases. [2]

1. It is conjectured that $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + ... + \frac{n-1}{n!} = a - \frac{b}{n!},$$ 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]
2. Use induction to prove that $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + ... + \frac{n-1}{n!} = 1 - \frac{1}{n!} \text{ for } n \geq 2.$$ [7]

In this question you must show detailed reasoning.

1. Given that \(y = \arctan x\), show that \(\frac{dy}{dx} = \frac{1}{1+x^2}\). [3]

Fig. 12 shows the curve \(y = \frac{1}{1+x^2}\). \includegraphics{figure_12}

1. Find, in exact form, the mean value of the function \(f(x) = \frac{1}{1+x^2}\) for \(-1 \leq x \leq 1\). [3]
2. 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. [7]

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

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

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

In this question you must show detailed reasoning. Show that $$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]

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 \(x\) 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. 1. Write down a differential equation to model this motion. [3]
   2. Give the general solution of the differential equation in part (i) (A). [1]

Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac{d^2x}{dt^2} + 2k \frac{dx}{dt} + (k^2 + 9)x = 0 \qquad (*)$$ where \(k\) is a positive constant.

1. 1. Obtain the general solution of (*). [3]
   2. State two ways in which the motion given by this model differs from that in part (i). [2]

The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.

1. Find the value of \(k\). [3]

At the start of the object's motion, \(x = 0\) and the velocity is 12 cm s\(^{-1}\) in the positive \(x\) direction.

1. Find an equation for \(x\) as a function of \(t\). [4]
2. 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. [2]