Questions — OCR (4907 questions)

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OCR FP3 Q2
7 marks Standard +0.8
Find the acute angle between the line with equation \(\mathbf{r} = 2\mathbf{i} + 3\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} - \mathbf{k})\) and the plane with equation \(\mathbf{r} = 2\mathbf{i} + 3\mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}) + \mu(\mathbf{i} + 2\mathbf{j} - \mathbf{k})\). [7]
OCR FP3 Q3
7 marks Standard +0.8
  1. Use the substitution \(z = x + y\) to show that the differential equation $$\frac{dy}{dx} = \frac{x + y + 3}{x + y - 1} \qquad (A)$$ may be written in the form \(\frac{dz}{dx} = \frac{2(z + 1)}{z - 1}\). [3]
  2. Hence find the general solution of the differential equation (A). [4]
OCR FP3 Q4
9 marks Standard +0.8
  1. By expressing \(\cos \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\cos^5 \theta \equiv \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta).$$ [5]
  2. Hence solve the equation \(\cos 5\theta + 5\cos 3\theta + 9\cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\). [4]
OCR FP3 Q5
10 marks Standard +0.8
Two lines have equations $$\frac{x-k}{2} = \frac{y+1}{-5} = \frac{z-1}{-3} \quad \text{and} \quad \frac{x-k}{1} = \frac{y+4}{-4} = \frac{z}{-2},$$ where \(k\) is a constant.
  1. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\). [6]
  2. For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(ax + by + cz = d\). [4]
OCR FP3 Q6
10 marks Standard +0.8
The operation \(\circ\) on real numbers is defined by \(a \circ b = a|b|\).
  1. Show that \(\circ\) is not commutative. [2]
  2. Prove that \(\circ\) is associative. [4]
  3. Determine whether the set of real numbers, under the operation \(\circ\), forms a group. [4]
OCR FP3 Q7
11 marks Standard +0.3
The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).
  1. Sketch an Argand diagram to show these roots. [1]
  2. Show that \(1 + \omega + \omega^2 = 0\). [2]
  3. Hence evaluate
    1. \((2 + \omega)(2 + \omega^2)\), [2]
    2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
  4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]
OCR FP3 Q8
13 marks Challenging +1.8
  1. Find the complementary function of the differential equation $$\frac{d^2y}{dx^2} + y = \cosec x.$$ [2]
  2. It is given that \(y = p(\ln \sin x) \sin x + qx \cos x\), where \(p\) and \(q\) are constants, is a particular integral of this differential equation.
    1. Show that \(p - 2(p + q) \sin^2 x \equiv 1\). [6]
    2. Deduce the values of \(p\) and \(q\). [2]
  3. Write down the general solution of the differential equation. State the set of values of \(x\), in the interval \(0 \leqslant x \leqslant 2\pi\), for which the solution is valid, justifying your answer. [3]
OCR FP3 Q1
5 marks Challenging +1.2
In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\):
    1. if \(G\) is cyclic, [1]
    2. if \(G\) has a proper subgroup of order 3, [1]
    3. if \(G\) has at least two elements of order 2. [1]
  2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\). [2]
OCR FP3 Q2
5 marks Standard +0.3
  1. Express \(\frac{\sqrt{3} + i}{\sqrt{3} - i}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(0 \leqslant \theta < 2\pi\). [3]
  2. Hence find the smallest positive value of \(n\) for which \(\left(\frac{\sqrt{3} + i}{\sqrt{3} - i}\right)^n\) is real and positive. [2]
OCR FP3 Q3
6 marks Standard +0.8
Two skew lines have equations $$\frac{x}{2} = \frac{y + 3}{1} = \frac{z - 6}{3} \quad \text{and} \quad \frac{x - 5}{3} = \frac{y + 1}{1} = \frac{z - 7}{5}.$$
  1. Find the direction of the common perpendicular to the lines. [2]
  2. Find the shortest distance between the lines. [4]
OCR FP3 Q4
9 marks Standard +0.8
Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x.$$ [9]
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]