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OCR FP1 2010 June Q5
6 marks Easy -1.2
  1. Write down the matrix that represents a reflection in the line \(y = x\). [2]
  2. Describe fully the geometrical transformation represented by each of the following matrices:
    1. \(\begin{pmatrix} 5 & 0 \\ 0 & 1 \end{pmatrix}\), [2]
    2. \(\begin{pmatrix} \frac{1}{2} & \frac{1}{2}\sqrt{3} \\ -\frac{1}{2}\sqrt{3} & \frac{1}{2} \end{pmatrix}\). [2]
OCR FP1 2010 June Q6
6 marks Moderate -0.3
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z - 3 + 4\text{i}| = 5\), [2]
    2. \(|z| = |z - 6|\). [2]
  2. Indicate, by shading, the region of the Argand diagram for which $$|z - 3 + 4\text{i}| \leq 5 \quad \text{and} \quad |z| \geq |z - 6|.$$ [2]
OCR FP1 2010 June Q7
7 marks Standard +0.8
The quadratic equation \(x^2 + 2kx + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac{\alpha + \beta}{\alpha}\) and \(\frac{\alpha + \beta}{\beta}\). [7]
OCR FP1 2010 June Q8
9 marks Standard +0.3
  1. Show that \(\frac{1}{\sqrt{r + 2} + \sqrt{r}} = \frac{\sqrt{r + 2} - \sqrt{r}}{2}\). [2]
  2. Hence find an expression, in terms of \(n\), for $$\sum_{r=1}^{n} \frac{1}{\sqrt{r + 2} + \sqrt{r}}.$$ [6]
  3. State, giving a brief reason, whether the series \(\sum_{r=1}^{\infty} \frac{1}{\sqrt{r + 2} + \sqrt{r}}\) converges. [1]
OCR FP1 2010 June Q9
9 marks Standard +0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & a & -1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{pmatrix}\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf{A}\). [3]
  2. Three simultaneous equations are shown below. \begin{align} ax + ay - z &= -1
    ay + 2z &= 2a
    x + 2y + z &= 1 \end{align} For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
    1. \(a = 0\)
    2. \(a = 1\)
    3. \(a = 2\) [6]
OCR FP1 2010 June Q10
11 marks Standard +0.8
The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).
  1. Use an algebraic method to find \(z\). [5]
  2. Show that \(z^3 = 2 + 11\text{i}\). [1]
The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).
  1. Find \(w\). [5]
OCR MEI FP1 2006 June Q1
4 marks Easy -1.3
  1. State the transformation represented by the matrix \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). [1]
  2. Write down the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin. [1]
  3. Find the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin, followed by reflection in the \(x\)-axis. [2]
OCR MEI FP1 2006 June Q2
5 marks Easy -1.2
Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]
OCR MEI FP1 2006 June Q3
6 marks Moderate -0.3
The cubic equation \(z^3 + 4z^2 - 3z + 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma\), \(\alpha\beta + \beta\gamma + \gamma\alpha\) and \(\alpha\beta\gamma\). [3]
  2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 22\). [3]
OCR MEI FP1 2006 June Q4
8 marks Moderate -0.8
Indicate, on separate Argand diagrams,
  1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
  2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
  3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]
OCR MEI FP1 2006 June Q5
6 marks Moderate -0.3
  1. The matrix \(\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}\) represents a transformation.
    1. Show that the point \((1, 1)\) is invariant under this transformation. [1]
    2. Calculate \(\mathbf{S}^{-1}\). [2]
    3. Verify that \((1, 1)\) is also invariant under the transformation represented by \(\mathbf{S}^{-1}\). [1]
  2. Part (i) may be generalised as follows. If \((x, y)\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf{T}\), it is also invariant under the transformation represented by \(\mathbf{T}^{-1}\). Starting with \(\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), or otherwise, prove this result. [2]
OCR MEI FP1 2006 June Q6
7 marks Standard +0.3
Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2^{n-1} = 3(2^n - 1)\) for all positive integers \(n\). [7]
OCR MEI FP1 2006 June Q7
13 marks Standard +0.3
A curve has equation \(y = \frac{x^2}{(x-2)(x+1)}\).
  1. Write down the equations of the three asymptotes. [3]
  2. Determine whether the curve approaches the horizontal asymptote from above or from below for
    1. large positive values of \(x\),
    2. large negative values of \(x\). [3]
  3. Sketch the curve. [4]
  4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]
OCR MEI FP1 2006 June Q8
10 marks Moderate -0.3
  1. Verify that \(2 + \mathrm{j}\) is a root of the equation \(2x^3 - 11x^2 + 22x - 15 = 0\). [5]
  2. Write down the other complex root. [1]
  3. Find the third root of the equation. [4]
OCR MEI FP1 2006 June Q9
13 marks Standard +0.3
  1. Show that \(r(r+1)(r+2) - (r-1)r(r+1) \equiv 3r(r+1)\). [2]
  2. Hence use the method of differences to find an expression for \(\sum_{r=1}^{n} r(r+1)\). [6]
  3. Show that you can obtain the same expression for \(\sum_{r=1}^{n} r(r+1)\) using the standard formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\). [5]
OCR MEI FP1 2007 June Q1
3 marks Moderate -0.8
You are given the matrix \(\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}\).
  1. Find the inverse of \(\mathbf{M}\). [2]
  2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf{M}\). Find the area of the image of the triangle following this transformation. [1]
OCR MEI FP1 2007 June Q2
3 marks Easy -1.2
Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}
OCR MEI FP1 2007 June Q3
5 marks Easy -1.2
Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]
OCR MEI FP1 2007 June Q4
7 marks Moderate -0.8
Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2\mathrm{j}\) and \(\beta = -2 - \mathrm{j}\).
  1. Represent \(\beta\) and its complex conjugate \(\beta^*\) on an Argand diagram. [2]
  2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
  3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]
OCR MEI FP1 2007 June Q5
6 marks Standard +0.3
The roots of the cubic equation \(x^3 + 3x^2 - 7x + 1 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). Find the cubic equation whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\), expressing your answer in a form with integer coefficients. [6]
OCR MEI FP1 2007 June Q6
6 marks Moderate -0.8
  1. Show that \(\frac{1}{r+2} - \frac{1}{r+3} = \frac{1}{(r+2)(r+3)}\). [2]
  2. Hence use the method of differences to find \(\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \ldots + \frac{1}{52 \times 53}\). [4]
OCR MEI FP1 2007 June Q7
6 marks Moderate -0.3
Prove by induction that \(\sum_{r=1}^{n} 3^{r-1} = \frac{3^n - 1}{2}\). [6]
OCR MEI FP1 2007 June Q8
14 marks Standard +0.3
A curve has equation \(y = \frac{x^2 - 4}{(x-3)(x+1)(x-1)}\).
  1. Write down the coordinates of the points where the curve crosses the axes. [3]
  2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
  3. Determine whether the curve approaches the horizontal asymptote from above or below for
    1. large positive values of \(x\),
    2. large negative values of \(x\). [3]
  4. Sketch the curve. [4]
OCR MEI FP1 2007 June Q9
11 marks Standard +0.8
The cubic equation \(x^3 + Ax^2 + Bx + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2\mathrm{j}\).
  1. Write down the other complex root. [1]
  2. Explain why the equation must have a real root. [1]
  3. Find the value of the real root and the values of \(A\) and \(B\). [9]
OCR MEI FP1 2007 June Q10
11 marks Standard +0.8
You are given that \(\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}\) and that \(\mathbf{AB}\) is of the form \(\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}\).
  1. Find the value of \(n\). [2]
  2. Write down the inverse matrix \(\mathbf{A}^{-1}\) and state the condition on \(k\) for this inverse to exist. [4]
  3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
    2x + y + 2z &= 12
    3x + 2y - z &= 3 \end{align} [5]