Questions FP3 (539 questions)

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OCR FP3 2010 January Q2
6 marks Challenging +1.2
\(H\) denotes the set of numbers of the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.
  1. Show that the product of any two members of \(H\) is a member of \(H\). [2] It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
  2. State the identity element of the group. [1]
  3. Find the inverse of \(a + b\sqrt{5}\). [2]
  4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse. [1]
OCR FP3 2010 January Q3
6 marks Moderate -0.3
Use the integrating factor method to find the solution of the differential equation $$\frac{\text{d}y}{\text{d}x} + 2y = \text{e}^{-3x}$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \text{f}(x)\). [6]
OCR FP3 2010 January Q4
7 marks Challenging +1.2
  1. Write down, in cartesian form, the roots of the equation \(z^4 = 16\). [2]
  2. Hence solve the equation \(w^4 = 16(1-w)^4\), giving your answers in cartesian form. [5]
OCR FP3 2010 January Q5
11 marks Challenging +1.3
A regular tetrahedron has vertices at the points $$A\left(0, 0, \frac{2}{\sqrt{3}}\sqrt{6}\right), \quad B\left(\frac{2}{\sqrt{3}}\sqrt{3}, 0, 0\right), \quad C\left(-\frac{1}{3}\sqrt{3}, 1, 0\right), \quad D\left(-\frac{1}{3}\sqrt{3}, -1, 0\right).$$
  1. Obtain the equation of the face \(ABC\) in the form $$x + \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [5] (Answers which only verify the given equation will not receive full credit.)
  2. Give a geometrical reason why the equation of the face \(ABD\) can be expressed as $$x - \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [2]
  3. Hence find the cosine of the angle between two faces of the tetrahedron. [4]
OCR FP3 2010 January Q6
12 marks Challenging +1.2
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{\text{d}^2y}{\text{d}x^2} + 16y = 8\cos 4x.$$
  1. Find the complementary function of the differential equation. [2]
  2. Given that there is a particular integral of the form \(y = px\sin 4x\), where \(p\) is a constant, find the general solution of the equation. [6]
  3. Find the solution of the equation for which \(y = 2\) and \(\frac{\text{d}y}{\text{d}x} = 0\) when \(x = 0\). [4]
OCR FP3 2010 January Q7
13 marks Challenging +1.3
  1. Solve the equation \(\cos 6\theta = 0\), for \(0 < \theta < \pi\). [3]
  2. By using de Moivre's theorem, show that $$\cos 6\theta \equiv (2\cos^2\theta - 1)(16\cos^4\theta - 16\cos^2\theta + 1).$$ [5]
  3. Hence find the exact value of $$\cos\left(\frac{1}{12}\pi\right)\cos\left(\frac{5}{12}\pi\right)\cos\left(\frac{7}{12}\pi\right)\cos\left(\frac{11}{12}\pi\right),$$ justifying your answer. [5]
OCR FP3 2010 January Q8
12 marks Challenging +1.2
The function f is defined by \(\text{f} : x \mapsto \frac{1}{2-2x}\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\). The function g is defined by \(\text{g}(x) = \text{ff}(x)\).
  1. Show that \(\text{g}(x) = \frac{1-x}{1-2x}\) and that \(\text{gg}(x) = x\). [4]
It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where \(\text{e} : x \mapsto x\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\).
  1. State the orders of the elements f and g. [2]
  2. The inverse of the element f is denoted by h. Find \(\text{h}(x)\). [2]
  3. Construct the operation table for the elements e, f, g, h of the group \(K\). [4]
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]