Questions — OCR FP3 (182 questions)

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OCR FP3 Q5
9 marks Standard +0.8
The variables \(x\) and \(y\) are related by the differential equation $$x^3 \frac{dy}{dx} = xy + x + 1. \qquad (A)$$
  1. Use the substitution \(y = u - \frac{1}{x}\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x^2 \frac{du}{dx} = u.$$ [4]
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = f(x)\). [5]
OCR FP3 Q6
13 marks Standard +0.3
\includegraphics{figure_6} The cuboid \(OABCDEFG\) shown in the diagram has \(\overrightarrow{OA} = 4\mathbf{i}, \overrightarrow{OC} = 2\mathbf{j}, \overrightarrow{OD} = 3\mathbf{k}\), and \(M\) is the mid-point of \(GF\).
  1. Find the equation of the plane \(ACGE\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
  2. The plane \(OEFC\) has equation \(\mathbf{r} \cdot (3\mathbf{i} - 4\mathbf{k}) = 0\). Find the acute angle between the planes \(OEFC\) and \(ACGE\). [4]
  3. The line \(AM\) meets the plane \(OEFC\) at the point \(W\). Find the ratio \(AW : WM\). [5]
OCR FP3 Q7
13 marks Standard +0.8
  1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
    1. Prove that the set of real numbers, together with the operation \(*\), forms a group. [6]
    2. State, with a reason, whether the group is commutative. [1]
    3. Prove that there are no elements of order 2. [2]
  2. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied. [4]
OCR FP3 Q8
12 marks Standard +0.8
  1. By expressing \(\sin \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\sin^6 \theta \equiv \frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10).$$ [5]
  2. Replace \(\theta\) by \(\left(\frac{1}{2}\pi - \theta\right)\) in the identity in part (i) to obtain a similar identity for \(\cos^6 \theta\). [3]
  3. Hence find the exact value of \(\int_0^{2\pi} \left(\sin^6 \theta - \cos^6 \theta\right) d\theta\). [4]
OCR FP3 Q1
4 marks Standard +0.8
Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]
OCR FP3 Q2
5 marks Standard +0.3
It is given that the set of complex numbers of the form \(re^{i\theta}\) for \(-\pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.
  1. Write down the inverse of \(5e^{3\pi i}\). [1]
  2. Prove the closure property for the group. [2]
  3. \(Z\) denotes the element \(e^{i\gamma}\), where \(\frac{1}{2}\pi < \gamma < \pi\). Express \(Z^2\) in the form \(e^{i\theta}\), where \(-\pi < \theta \leqslant 0\). [2]
OCR FP3 Q3
8 marks Standard +0.8
A line \(l\) has equation \(\frac{x - 6}{-4} = \frac{y + 7}{8} = \frac{z + 10}{7}\) and a plane \(p\) has equation \(3x - 4y - 2z = 8\).
  1. Find the point of intersection of \(l\) and \(p\). [3]
  2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]
OCR FP3 Q4
8 marks Standard +0.8
The differential equation $$\frac{dy}{dx} + \frac{1}{1 - x^2} y = (1 - x)^{\frac{1}{2}}, \quad \text{where } |x| < 1,$$ can be solved by the integrating factor method.
  1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left(\frac{1 + x}{1 - x}\right)^{\frac{1}{2}}\). [2]
  2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = f(x)\). [6]
OCR FP3 Q5
9 marks Challenging +1.2
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = e^{3x}.$$
  1. Find the complementary function. [3]
  2. Explain briefly why there is no particular integral of either of the forms \(y = ke^{3x}\) or \(y = kxe^{3x}\). [1]
  3. Given that there is a particular integral of the form \(y = kx^2e^{3x}\), find the value of \(k\). [5]
OCR FP3 Q6
9 marks Standard +0.3
The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -5 \\ -2 \end{pmatrix}\).
  1. Express the equation of \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4] The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 7 \\ 1 \\ -3 \end{pmatrix} = 21\).
  2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [5]
OCR FP3 2008 January Q1
6 marks Standard +0.3
  1. A group \(G\) of order 6 has the combination table shown below.
    \(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(e\)\(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(a\)\(a\)\(b\)\(e\)\(r\)\(p\)\(q\)
    \(b\)\(b\)\(e\)\(a\)\(q\)\(r\)\(p\)
    \(p\)\(p\)\(q\)\(r\)\(e\)\(a\)\(b\)
    \(q\)\(q\)\(r\)\(p\)\(b\)\(e\)\(a\)
    \(r\)\(r\)\(p\)\(q\)\(a\)\(b\)\(e\)
    1. State, with a reason, whether or not \(G\) is commutative. [1]
    2. State the number of subgroups of \(G\) which are of order 2. [1]
    3. List the elements of the subgroup of \(G\) which is of order 3. [1]
  2. A multiplicative group \(H\) of order 6 has elements \(e, c, c^2, c^3, c^4, c^5\), where \(e\) is the identity. Write down the order of each of the elements \(c^3, c^4\) and \(c^5\). [3]
OCR FP3 2008 January Q3
7 marks Standard +0.3
Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf{r}\).
  1. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} = \lambda\mathbf{a}\), where \(0 \leq \lambda \leq 1\). [2]
  2. Given that \(P\) is a point on the line \(AB\), use a property of the vector product to explain why \((\mathbf{r} - \mathbf{a}) \times (\mathbf{r} - \mathbf{b}) = \mathbf{0}\). [2]
  3. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} \times (\mathbf{a} - \mathbf{b}) = \mathbf{0}\). [3]
OCR FP3 2008 January Q4
8 marks Standard +0.8
The integrals \(C\) and \(S\) are defined by $$C = \int_0^{3\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{3\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = -\frac{1}{3}(2 + 3e^{\pi}),$$ and obtain a similar expression for \(S\). [8] (You may assume that the standard result for \(\int e^{kx} dx\) remains true when \(k\) is a complex constant, so that \(\int e^{(a+ib)x} dx = \frac{1}{a+ib}e^{(a+ib)x}\).)
OCR FP3 2008 January Q5
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{dy}{dx} - \frac{y}{x} = \sin 2x,$$ expressing \(y\) in terms of \(x\) in your answer. [6]
In a particular case, it is given that \(y = \frac{2}{\pi}\) when \(x = \frac{1}{4}\pi\).
  1. Find the solution of the differential equation in this case. [2]
  2. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]
OCR FP3 2008 January Q6
11 marks Challenging +1.2
A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). The coordinates of the points \(A\), \(C\) and \(D\) are \((-1, -7, 2)\), \((5, 0, 3)\) and \((-1, 3, 3)\) respectively, and the equation of the plane \(BCD\) is \(x + 2y - 2z = -1\).
  1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
  2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]
OCR FP3 2008 January Q7
11 marks Challenging +1.3
    1. Verify, without using a calculator, that \(\theta = \frac{1}{8}\pi\) is a solution of the equation \(\sin 6\theta = \sin 2\theta\). [1]
    2. By sketching the graphs of \(y = \sin 6\theta\) and \(y = \sin 2\theta\) for \(0 < \theta < \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \theta < \frac{1}{2}\pi\). [2]
  2. Use de Moivre's theorem to prove that $$\sin 6\theta = \sin 2\theta (16 \cos^4 \theta - 16 \cos^2 \theta + 3).$$ [5]
  3. Hence show that one of the solutions obtained in part (i) satisfies \(\cos^2 \theta = \frac{1}{4}(2 - \sqrt{2})\), and justify which solution it is. [3]
OCR FP3 2008 January Q8
13 marks Challenging +1.3
Groups \(A\), \(B\), \(C\) and \(D\) are defined as follows: \(A\): the set of numbers \(\{2, 4, 6, 8\}\) under multiplication modulo 10, \(B\): the set of numbers \(\{1, 5, 7, 11\}\) under multiplication modulo 12, \(C\): the set of numbers \(\{2^0, 2^1, 2^2, 2^3\}\) under multiplication modulo 15, \(D\): the set of numbers \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) under multiplication.
  1. Write down the identity element for each of groups \(A\), \(B\), \(C\) and \(D\). [2]
  2. Determine in each case whether the groups
    \(A\) and \(B\), \(B\) and \(C\), \(A\) and \(C\)
    are isomorphic or non-isomorphic. Give sufficient reasons for your answers. [5]
  3. Prove the closure property for group \(D\). [4]
  4. Elements of the set \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) [2]
OCR FP3 2010 January Q1
5 marks Standard +0.3
Determine whether the lines $$\frac{x-1}{-1} = \frac{y+2}{2} = \frac{z+4}{2} \quad \text{and} \quad \frac{x+3}{2} = \frac{y-1}{3} = \frac{z-5}{4}$$ intersect or are skew. [5]
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]