Questions — OCR MEI (4455 questions)

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OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.8
  1. Express \(\frac{1}{2r-1} - \frac{1}{2r+1}\) as a single fraction. [2]
  2. Find how many terms of the series $$\frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + \ldots + \frac{2}{(2r-1)(2r+1)} + \ldots$$ are needed for the sum to exceed \(0.999999\). [7]
OCR MEI Further Pure Core AS 2018 June Q8
6 marks Standard +0.3
Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]
OCR MEI Further Pure Core AS 2018 June Q9
9 marks Standard +0.3
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
OCR MEI Further Pure Core AS 2018 June Q10
8 marks Standard +0.3
Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    [6]
  2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]
OCR MEI Further Pure Core AS Specimen Q1
4 marks Moderate -0.8
The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).
  1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
  2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]
OCR MEI Further Pure Core AS Specimen Q2
4 marks Moderate -0.8
  1. Describe fully the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). [2]
  2. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). Explaining your reasoning, find the area of the image of the triangle following this transformation. [2]
OCR MEI Further Pure Core AS Specimen Q3
4 marks Moderate -0.3
  1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. [2] \includegraphics{figure_3}
  2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg(z - 2i) = \frac{\pi}{4}\). [2]
OCR MEI Further Pure Core AS Specimen Q4
6 marks Standard +0.3
  1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$x - 2y - z = 6$$ $$3x + y + 5z = -4$$ $$-4x + 2y - 3z = a$$ [4]
  2. Determine whether the intersection of the three planes could be on the \(z\)-axis. [2]
OCR MEI Further Pure Core AS Specimen Q5
7 marks Challenging +1.2
The cubic equation \(x^3 - 4x^2 + px + q = 0\) has roots \(\alpha\), \(\frac{2}{\alpha}\) and \(\alpha + \frac{2}{\alpha}\). Find
  • the values of the roots of the equation,
  • the value of \(p\).
[7]
OCR MEI Further Pure Core AS Specimen Q6
5 marks Standard +0.8
  1. Show that, when \(n = 5\), \(\sum_{r=n+1}^{2n} r^2 = 330\). [1]
  2. Find, in terms of \(n\), a fully factorised expression for \(\sum_{r=n+1}^{2n} r^2\). [4]
OCR MEI Further Pure Core AS Specimen Q7
7 marks Moderate -0.3
The plane \(\Pi\) has equation \(3x - 5y + z = 9\).
  1. Show that \(\Pi\) contains
    and
    [4]
  2. Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through \((4,1,2)\). [3]
OCR MEI Further Pure Core AS Specimen Q8
9 marks Challenging +1.8
In this question you must show detailed reasoning.
  1. Explain why all cubic equations with real coefficients have at least one real root. [2]
  2. Points representing the three roots of the equation \(z^3 + 9z^2 + 27z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices. [7]
OCR MEI Further Pure Core AS Specimen Q9
14 marks Challenging +1.2
You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).
  1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
  2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]
It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).
  1. Explain geometrically why this claim is true. [2]
  2. Verify algebraically that this claim is true. [3]
OCR MEI Further Pure Core Specimen Q1
3 marks Moderate -0.5
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]
OCR MEI Further Pure Core Specimen Q2
6 marks Standard +0.8
  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]
OCR MEI Further Pure Core Specimen Q3
6 marks Standard +0.3
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]
    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]
OCR MEI Further Pure Core Specimen Q4
5 marks Standard +0.3
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]
OCR MEI Further Pure Core Specimen Q5
7 marks Standard +0.8
  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]
OCR MEI Further Pure Core Specimen Q6
6 marks Standard +0.8
  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]
OCR MEI Further Pure Core Specimen Q7
11 marks Standard +0.3
  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]
    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]
    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]
OCR MEI Further Pure Core Specimen Q8
5 marks Standard +0.3
Find the cartesian equation of the plane which contains the three points \((1, 0, -1)\), \((2, 2, 1)\) and \((1, 1, 2)\). [5]
OCR MEI Further Pure Core Specimen Q9
7 marks Challenging +1.3
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]
OCR MEI Further Pure Core Specimen Q10
9 marks Standard +0.8
  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]
OCR MEI Further Pure Core Specimen Q11
9 marks Standard +0.8
  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]
OCR MEI Further Pure Core Specimen Q12
13 marks Standard +0.3
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]