Questions — OCR FP3 (182 questions)

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OCR FP3 2011 January Q1
6 marks Standard +0.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + xy = xe^{\frac{x^2}{2}},$$ giving your answer in the form \(y = f(x)\). [4]
  2. Find the particular solution for which \(y = 1\) when \(x = 0\). [2]
OCR FP3 2011 January Q2
6 marks Standard +0.8
Two intersecting lines, lying in a plane \(p\), have equations $$\frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{-3} \quad \text{and} \quad \frac{x-1}{-1} = \frac{y-3}{2} = \frac{z-4}{4}.$$
  1. Obtain the equation of \(p\) in the form \(2x - y + z = 3\). [3]
  2. Plane \(q\) has equation \(2x - y + z = 21\). Find the distance between \(p\) and \(q\). [3]
OCR FP3 2011 January Q3
8 marks Standard +0.3
  1. Express \(\sin \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\) and show that $$\sin^4 \theta \equiv \frac{1}{8}(\cos 4\theta - 4\cos 2\theta + 3).$$ [4]
  2. Hence find the exact value of \(\int_0^{\frac{\pi}{4}} \sin^4 \theta \, d\theta\). [4]
OCR FP3 2011 January Q4
8 marks Standard +0.3
The cube roots of 1 are denoted by \(1\), \(\omega\) and \(\omega^2\), where the imaginary part of \(\omega\) is positive.
  1. Show that \(1 + \omega + \omega^2 = 0\). [2]
\includegraphics{figure_1} In the diagram, \(ABC\) is an equilateral triangle, labelled anticlockwise. The points \(A\), \(B\) and \(C\) represent the complex numbers \(z_1\), \(z_2\) and \(z_3\) respectively.
  1. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z_1 - z_3 = \omega(z_3 - z_2)\). [4]
  2. Hence show that \(z_1 + \omega z_2 + \omega^2 z_3 = 0\). [2]
OCR FP3 2011 January Q5
13 marks Standard +0.3
  1. Find the general solution of the differential equation $$3\frac{d^2y}{dx^2} + 5\frac{dy}{dx} - 2y = -2x + 13.$$ [7]
  2. Find the particular solution for which \(y = -\frac{7}{2}\) and \(\frac{dy}{dx} = 0\) when \(x = 0\). [5]
  3. Write down the function to which \(y\) approximates when \(x\) is large and positive. [1]
OCR FP3 2011 January Q6
9 marks Challenging +1.8
\(Q\) is a multiplicative group of order 12.
  1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]
The table below shows the number of elements of \(Q\) with each possible order.
Order of element12346
Number of elements11262
\(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
  1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]
OCR FP3 2011 January Q7
10 marks Challenging +1.2
Three planes \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\) have equations $$\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 5, \quad \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 6, \quad \mathbf{r} \cdot (\mathbf{i} + 5\mathbf{j} - 12\mathbf{k}) = 12,$$ respectively. Planes \(\Pi_1\) and \(\Pi_2\) intersect in a line \(l\); planes \(\Pi_2\) and \(\Pi_3\) intersect in a line \(m\).
  1. Show that \(l\) and \(m\) are in the same direction. [5]
  2. Write down what you can deduce about the line of intersection of planes \(\Pi_1\) and \(\Pi_3\). [1]
  3. By considering the cartesian equations of \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\), or otherwise, determine whether or not the three planes have a common line of intersection. [4]
OCR FP3 2011 January Q8
12 marks Challenging +1.3
The operation \(*\) is defined on the elements \((x, y)\), where \(x, y \in \mathbb{R}\), by $$(a, b) * (c, d) = (ac, ad + b).$$ It is given that the identity element is \((1, 0)\).
  1. Prove that \(*\) is associative. [3]
  2. Find all the elements which commute with \((1, 1)\). [3]
  3. It is given that the particular element \((m, n)\) has an inverse denoted by \((p, q)\), where $$(m, n) * (p, q) = (p, q) * (m, n) = (1, 0).$$ Find \((p, q)\) in terms of \(m\) and \(n\). [2]
  4. Find all self-inverse elements. [3]
  5. Give a reason why the elements \((x, y)\), under the operation \(*\), do not form a group. [1]
OCR FP3 2006 June Q1
5 marks Moderate -0.8
  1. For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of \(1 + 2\mathrm{i}\), giving your answers in the form \(a + ib\). [3]
  2. For the group of matrices of the form \(\begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}\) under matrix addition, where \(a \in \mathbb{R}\), state the identity element and the inverse of \(\begin{pmatrix} 3 & 0 \\ 0 & 0 \end{pmatrix}\). [2]
OCR FP3 2006 June Q2
7 marks Moderate -0.8
  1. Given that \(z_1 = 2e^{\frac{5\pi i}{6}}\) and \(z_2 = 3e^{\frac{2\pi i}{3}}\), express \(z_1z_2\) and \(\frac{z_1}{z_2}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [4]
  2. Given that \(w = 2(\cos \frac{1}{3}\pi + i \sin \frac{1}{3}\pi)\), express \(w^{-5}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [3]
OCR FP3 2006 June Q3
6 marks Standard +0.3
Find the perpendicular distance from the point with position vector \(12\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}\) to the line with equation \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + t(8\mathbf{i} + 3\mathbf{j} - 6\mathbf{k})\). [6]
OCR FP3 2006 June Q4
8 marks Standard +0.8
Find the solution of the differential equation $$\frac{dy}{dx} - \frac{x^2y}{1 + x^3} = x^2$$ for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = f(x)\). [8]
OCR FP3 2006 June Q5
10 marks Challenging +1.2
A line \(l_1\) has equation \(\frac{x}{2} = \frac{y + 4}{3} = \frac{z + 9}{5}\).
  1. Find the cartesian equation of the plane which is parallel to \(l_1\) and which contains the points \((2, 1, 5)\) and \((0, -1, 5)\). [5]
  2. Write down the position vector of a point on \(l_1\) with parameter \(t\). [1]
  3. Hence, or otherwise, find an equation of the line \(l_2\) which intersects \(l_1\) at right angles and which passes through the point \((-5, 3, 4)\). Give your answer in the form \(\frac{x - a}{p} = \frac{y - b}{q} = \frac{z - c}{r}\). [4]
OCR FP3 2006 June Q6
10 marks Standard +0.3
  1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 4y = \sin x.$$ [6]
  2. Find the solution of the differential equation for which \(y = 0\) and \(\frac{dy}{dx} = \frac{4}{3}\) when \(x = 0\). [4]
OCR FP3 2006 June Q7
12 marks Challenging +1.3
The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by \begin{align} C &= 1 + \cos \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta + \cos 5\theta,
S &= \sin \theta + \sin 2\theta + \sin 3\theta + \sin 4\theta + \sin 5\theta. \end{align}
  1. Show that \(C + iS = \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta} - e^{-i\theta}} \cdot e^{i\theta}\). [4]
  2. Deduce that \(C = \sin 3\theta \cos \frac{5}{2}\theta \operatorname{cosec} \frac{1}{2}\theta\) and write down the corresponding expression for \(S\). [4]
  3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\). [4]
OCR FP3 2006 June Q8
14 marks Challenging +1.2
A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}
  1. Give a reason why \(D\) is not commutative. [1]
  2. Write down the orders of any possible proper subgroups of \(D\). [2]
  3. List the elements of a proper subgroup which contains
    1. the element \(a\), [1]
    2. the element \(r\). [1]
  4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
  5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]
OCR FP3 2010 June Q1
7 marks Standard +0.8
The line \(l_1\) passes through the points \((0, 0, 10)\) and \((7, 0, 0)\) and the line \(l_2\) passes through the points \((4, 6, 0)\) and \((3, 3, 1)\). Find the shortest distance between \(l_1\) and \(l_2\). [7]
OCR FP3 2010 June Q2
6 marks Challenging +1.2
A multiplicative group with identity \(e\) contains distinct elements \(a\) and \(r\), with the properties \(r^6 = e\) and \(ar = r^2a\).
  1. Prove that \(rar = a\). [2]
  2. Prove, by induction or otherwise, that \(r^n ar^n = a\) for all positive integers \(n\). [4]
OCR FP3 2010 June Q3
9 marks Standard +0.8
In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).
  1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
  2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
  3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]
OCR FP3 2010 June Q4
8 marks Challenging +1.2
  1. Use the substitution \(y = xz\) to find the general solution of the differential equation $$x \frac{dy}{dx} - y = x \cos \left(\frac{y}{x}\right),$$ giving your answer in a form without logarithms. (You may quote an appropriate result given in the List of Formulae (MF1).) [6]
  2. Find the solution of the differential equation for which \(y = \pi\) when \(x = 4\). [2]
OCR FP3 2010 June Q5
8 marks Challenging +1.2
Convergent infinite series \(C\) and \(S\) are defined by \begin{align} C &= 1 + \frac{1}{4} \cos \theta + \frac{1}{4} \cos 2\theta + \frac{1}{8} \cos 3\theta + \ldots,
S &= \frac{1}{2} \sin \theta + \frac{1}{4} \sin 2\theta + \frac{1}{8} \sin 3\theta + \ldots. \end{align}
  1. Show that \(C + iS = \frac{2}{2 - e^{i\theta}}\). [4]
  2. Hence show that \(C = \frac{4 - 2\cos \theta}{5 - 4\cos \theta}\) and find a similar expression for \(S\). [4]
OCR FP3 2010 June Q6
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 17y = 17x + 36.$$ [7]
  2. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation. [2]
OCR FP3 2010 June Q7
12 marks Challenging +1.2
A line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} -7 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\). A plane \(\Pi\) passes through the points \((1, 3, 5)\) and \((5, 2, 5)\), and is parallel to \(l\).
  1. Find an equation of \(\Pi\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
  2. Find the distance between \(l\) and \(\Pi\). [4]
  3. Find an equation of the line which is the reflection of \(l\) in \(\Pi\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [4]
OCR FP3 2010 June Q8
13 marks Challenging +1.2
A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.
  1. Write down the elements of a subgroup of order 2. [1]
  2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
  3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
  4. Find the inverses of \(B\) and \(E\). [3]
  5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]
OCR FP3 2011 June Q1
6 marks Standard +0.3
A line \(l\) has equation \(\frac{x-1}{5} = \frac{y-6}{6} = \frac{z+3}{-7}\) and a plane \(p\) has equation \(x + 2y - z = 40\).
  1. Find the acute angle between \(l\) and \(p\). [4]
  2. Find the perpendicular distance from the point \((1, 6, -3)\) to \(p\). [2]