Questions — OCR FP1 (210 questions)

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OCR FP1 Q1
6 marks Moderate -0.5
Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]
OCR FP1 Q2
6 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
  1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
  2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]
OCR FP1 Q3
7 marks Moderate -0.8
The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 Q4
6 marks Standard +0.3
Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]
OCR FP1 Q5
7 marks Standard +0.3
  1. Show that $$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$ [2]
  2. Hence find an expression, in terms of \(n\), for $$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$ [4]
  3. Hence write down the value of \(\sum_{r=1}^\infty \frac{1}{(r+1)(r+2)}\). [1]
OCR FP1 Q6
7 marks Standard +0.3
The loci \(C_1\) and \(C_2\) are given by $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
  2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]
OCR FP1 Q7
10 marks Standard +0.3
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).
  1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
  2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
  3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
    2x + y - z &= 4,
    y + 2z &= -1. \end{align} [3]
OCR FP1 Q8
11 marks Moderate -0.3
  1. The quadratic equation \(x^2 - 2x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha\beta\). [2]
    2. Show that \(\alpha^2 + \beta^2 = -4\). [2]
    3. Hence find a quadratic equation which has roots \(\alpha^2\) and \(\beta^2\). [3]
  2. The cubic equation \(x^3 - 12x^2 + ax - 48 = 0\) has roots \(p\), \(2p\) and \(3p\).
    1. Find the value of \(p\). [2]
    2. Hence find the value of \(a\). [2]
OCR FP1 Q9
12 marks Standard +0.3
  1. Write down the matrix \(\mathbf{C}\) which represents a stretch, scale factor \(2\), in the \(x\)-direction. [2]
  2. The matrix \(\mathbf{D}\) is given by \(\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\). Describe fully the geometrical transformation represented by \(\mathbf{D}\). [2]
  3. The matrix \(\mathbf{M}\) represents the combined effect of the transformation represented by \(\mathbf{C}\) followed by the transformation represented by \(\mathbf{D}\). Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
  4. Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]
OCR FP1 2013 January Q1
5 marks Easy -1.2
The matrix A is given by \(A = \begin{pmatrix} a & 1 \\ 1 & a \end{pmatrix}\), where \(a \neq \frac{1}{2}\), and I denotes the \(2 \times 2\) identity matrix. Find
  1. \(2A - 3I\), [3]
  2. \(A^{-1}\). [2]
OCR FP1 2013 January Q2
6 marks Moderate -0.3
Find \(\sum_{r=1}^{n} (r-1)(r+1)\), giving your answer in a fully factorised form. [6]
OCR FP1 2013 January Q3
7 marks Moderate -0.3
The complex number \(2 - i\) is denoted by \(z\).
  1. Find \(|z|\) and \(\arg z\). [2]
  2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]
OCR FP1 2013 January Q4
4 marks Moderate -0.3
The quadratic equation \(x^2 + x + k = 0\) has roots \(\alpha\) and \(\beta\).
  1. Use the substitution \(x = 2u + 1\) to obtain a quadratic equation in \(u\). [2]
  2. Hence, or otherwise, find the value of \(\left(\frac{\alpha - 1}{2}\right)\left(\frac{\beta - 1}{2}\right)\) in terms of \(k\). [2]
OCR FP1 2013 January Q5
6 marks Standard +0.8
By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations \begin{align} 3x + 2y + 4z &= 5,
\lambda y + z &= 1,
x + \lambda y + \lambda z &= 4, \end{align} do not have a unique solution for \(x\), \(y\) and \(z\). [6]
OCR FP1 2013 January Q6
10 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).
  1. Write down the matrix, X, for this transformation. [2]
  2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
  3. Find the matrix that represents transformation Q followed by transformation P. [2]
OCR FP1 2013 January Q7
7 marks Moderate -0.3
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z| = 2\), [2]
    2. \(\arg(z - 3 - i) = \pi\). [3]
  2. Indicate, by shading, the region of the Argand diagram for which $$|z| < 2 \text{ and } 0 < \arg(z - 3 - i) < \pi.$$ [2]
OCR FP1 2013 January Q8
9 marks Standard +0.8
  1. Show that \(\frac{1}{r} - \frac{3}{r+1} + \frac{2}{r+2} = \frac{2-r}{r(r+1)(r+2)}\). [2]
  2. Hence show that \(\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = -\frac{n}{(n+1)(n+2)}\). [5]
  3. Find the value of \(\sum_{r=3}^{\infty} \frac{2-r}{r(r+1)(r+2)}\). [2]
OCR FP1 2013 January Q9
8 marks Standard +0.8
  1. Show that \((\alpha\beta + \beta\gamma + \gamma\alpha)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma)\). [3]
  2. It is given that \(\alpha\), \(\beta\) and \(\gamma\) are the roots of the cubic equation \(x^3 + px^2 - 4x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}\) in terms of \(p\). [5]
OCR FP1 2013 January Q10
10 marks Standard +0.3
The sequence \(u_1, u_2, u_3, \ldots\) is defined by \(u_1 = 2\) and \(u_{n+1} = \frac{u_n}{1 + u_n}\) for \(n \geq 1\).
  1. Find \(u_2\) and \(u_3\), and show that \(u_4 = \frac{2}{7}\). [3]
  2. Hence suggest an expression for \(u_n\). [2]
  3. Use induction to prove that your answer to part (ii) is correct. [5]
OCR FP1 2005 June Q3
7 marks Easy -1.2
The complex numbers \(2 + 3\text{i}\) and \(4 - \text{i}\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \text{i}y\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z^*w\), where \(z^*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 2005 June Q4
6 marks Standard +0.3
Use an algebraic method to find the square roots of the complex number \(21 - 20\text{i}\). [6]
OCR FP1 2005 June Q5
7 marks Standard +0.3
  1. Show that $$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$ [2]
  2. Hence find an expression, in terms of \(n\), for $$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$ [4]
  3. Hence write down the value of \(\sum_{r=1}^{\infty} \frac{1}{(r+1)(r+2)}\). [1]
OCR FP1 2005 June Q6
7 marks Moderate -0.3
The loci \(C_1\) and \(C_2\) are given by $$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
  2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]
OCR FP1 2005 June Q7
10 marks Standard +0.3
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).
  1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
  2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
  3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
    2x + y - z &= 4,
    y + 2z &= -1. \end{align} [3]
OCR FP1 2005 June Q9
12 marks Standard +0.3
  1. Write down the matrix \(\mathbf{C}\) which represents a stretch, scale factor 2, in the \(x\)-direction. [2]
  2. The matrix \(\mathbf{D}\) is given by \(\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\). Describe fully the geometrical transformation represented by \(\mathbf{D}\). [2]
  3. The matrix \(\mathbf{M}\) represents the combined effect of the transformation represented by \(\mathbf{C}\) followed by the transformation represented by \(\mathbf{D}\). Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
  4. Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]