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OCR MEI FP2 2011 January Q1
19 marks Standard +0.3
  1. A curve has polar equation \(r = 2(\cos \theta + \sin \theta)\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{3}{4}\pi\).
    1. Show that a cartesian equation of the curve is \(x^2 + y^2 = 2x + 2y\). Hence or otherwise sketch the curve. [5]
    2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{2}\pi\). Give your answer in terms of \(\pi\). [7]
    1. Given that \(f(x) = \arctan(\frac{1}{2}x)\), find \(f'(x)\). [2]
    2. Expand \(f'(x)\) in ascending powers of \(x\) as far as the term in \(x^4\). Hence obtain an expression for \(f(x)\) in ascending powers of \(x\) as far as the term in \(x^5\). [5]
OCR MEI FP2 2011 January Q2
19 marks Standard +0.3
    1. Given that \(z = \cos \theta + j \sin \theta\), express \(z^n + z^{-n}\) and \(z^n - z^{-n}\) in simplified trigonometrical form. [2]
    2. By considering \((z + z^{-1})^6\), show that $$\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6 \cos 4\theta + 15 \cos 2\theta + 10).$$ [3]
    3. Obtain an expression for \(\cos^6 \theta - \sin^6 \theta\) in terms of \(\cos 2\theta\) and \(\cos 6\theta\). [5]
  2. The complex number \(w\) is \(8e^{i\pi/3}\). You are given that \(z_1\) is a square root of \(w\) and that \(z_2\) is a cube root of \(w\). The points representing \(z_1\) and \(z_2\) in the Argand diagram both lie in the third quadrant.
    1. Find \(z_1\) and \(z_2\) in the form \(re^{i\theta}\). Draw an Argand diagram showing \(w\), \(z_1\) and \(z_2\). [6]
    2. Find the product \(z_1z_2\), and determine the quadrant of the Argand diagram in which it lies. [3]
OCR MEI FP2 2011 January Q3
16 marks Standard +0.3
  1. Show that the characteristic equation of the matrix $$\mathbf{M} = \begin{pmatrix} 1 & -4 & 5 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \end{pmatrix}$$ is \(\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0\). [4]
  2. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf{M}\), and determine whether or not \(\mathbf{M}\) has any other real eigenvalues. [4]
  3. Find an eigenvector, \(\mathbf{v}\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf{M}^n\mathbf{v}\), where \(n\) is an integer. [5]
  4. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [3]
OCR MEI FP2 2011 January Q4
18 marks Standard +0.8
  1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]
A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).
  1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
  2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]
OCR MEI FP2 2011 January Q5
18 marks Challenging +1.2
A curve has parametric equations $$x = t + a \sin t, \quad y = 1 - a \cos t,$$ where \(a\) is a positive constant.
  1. Draw, on separate diagrams, sketches of the curve for \(-2\pi < t < 2\pi\) in the cases \(a = 1\), \(a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
    1. loops,
    2. cusps. [7]
  2. Suppose that the point P\((x, y)\) lies on the curve. Show that the point P\('(-x, y)\) also lies on the curve. What does this indicate about the symmetry of the curve? [3]
  3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(-2\pi < t < 2\pi\) and \(a \neq 1\). [5]
  4. In the case \(a = \frac{1}{2}\pi\), show that \(t = \frac{1}{3}\pi\) and \(t = \frac{5}{3}\pi\) give the same point. Find the angle at which the curve crosses itself at this point. [3]
OCR MEI FP2 2009 June Q1
16 marks Standard +0.3
    1. Use the Maclaurin series for \(\ln(1 + x)\) and \(\ln(1 - x)\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln\left(\frac{1 + x}{1 - x}\right)\). State the range of validity of this series. [4]
    2. Find the value of \(x\) for which \(\frac{1 + x}{1 - x} = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places. [4]
  2. A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
    1. By plotting suitable points, or otherwise, sketch the curve. [3]
    2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]
OCR MEI FP2 2009 June Q2
19 marks Standard +0.3
  1. Obtain the characteristic equation for the matrix \(\mathbf{M}\) where $$\mathbf{M} = \begin{pmatrix} 3 & 1 & -2 \\ 6 & -1 & 0 \\ 2 & 0 & 1 \end{pmatrix}.$$ Hence or otherwise obtain the value of \(\det(\mathbf{M})\). [3]
  2. Show that \(-1\) is an eigenvalue of \(\mathbf{M}\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue \(-1\). Hence or otherwise write down the solution to the following system of equations. [9] $$3x + y - 2z = -0.1$$ $$-y = 0.6$$ $$2x + z = 0.1$$
  3. State the Cayley-Hamilton theorem and use it to show that $$\mathbf{M}^3 = 3\mathbf{M}^2 - 3\mathbf{M} - 7\mathbf{I}.$$ Obtain an expression for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [4]
  4. Find the numerical values of the elements of \(\mathbf{M}^{-1}\), showing your working. [3]
OCR MEI FP2 2009 June Q3
19 marks Standard +0.8
    1. Sketch the graph of \(y = \arcsin x\) for \(-1 \leq x \leq 1\). [1] Find \(\frac{dy}{dx}\), justifying the sign of your answer by reference to your sketch. [4]
    2. Find the exact value of the integral \(\int_0^1 \frac{1}{\sqrt{2 - x^2}} dx\). [3]
  2. The infinite series \(C\) and \(S\) are defined as follows. $$C = \cos \theta + \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta + \ldots$$ $$S = \sin \theta + \frac{1}{3}\sin 3\theta + \frac{1}{5}\sin 5\theta + \ldots$$ By considering \(C + jS\), show that $$C = \frac{3\cos \theta}{5 - 3\cos 2\theta},$$ and find a similar expression for \(S\). [11]
OCR MEI FP2 2009 June Q4
18 marks Standard +0.8
  1. Prove, from definitions involving exponentials, that $$\cosh 2u = 2\cosh^2 u - 1.$$ [3]
  2. Prove that \(\arsinh y = \ln\left(y + \sqrt{y^2 + 1}\right)\). [4]
  3. Use the substitution \(x = 2\sinh u\) to show that $$\int \sqrt{x^2 + 4} dx = 2\arsinh \frac{x}{2} + \frac{x}{2}\sqrt{x^2 + 4} + c,$$ where \(c\) is an arbitrary constant. [6]
  4. By first expressing \(t^2 + 2t + 5\) in completed square form, show that $$\int_{-1}^1 \sqrt{t^2 + 2t + 5} dt = 2\left(\ln(1 + \sqrt{2}) + \sqrt{2}\right).$$ [5]
OCR MEI FP2 2009 June Q5
18 marks Challenging +1.3
Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}
  1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
  2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
  3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
  4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]
OCR FP3 Q1
3 marks Easy -1.2
  1. By writing \(z\) in the form \(re^{i\theta}\), show that \(zz^* = |z|^2\). [1]
  2. Given that \(zz^* = 9\), describe the locus of \(z\). [2]
OCR FP3 Q2
5 marks Standard +0.3
A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]
OCR FP3 Q3
6 marks Standard +0.8
Find the general solution of the differential equation $$\frac{d^2y}{dx^2} - c\frac{dy}{dx} + 8y = e^{3x}.$$ [6]
OCR FP3 Q4
8 marks Standard +0.8
Elements of the set \(\{p, q, r, s, t\}\) are combined according to the operation table shown below.
\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(t\)\(s\)\(p\)\(r\)\(q\)
\(q\)\(s\)\(p\)\(q\)\(t\)\(r\)
\(r\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(s\)\(r\)\(t\)\(s\)\(q\)\(p\)
\(t\)\(q\)\(r\)\(t\)\(p\)\(s\)
  1. Verify that \(q(st) = (qs)t\). [2]
  2. Assuming that the associative property holds for all elements, prove that the set \(\{p, q, r, s, t\}\), with the operation table shown, forms a group \(G\). [4]
  3. A multiplicative group \(H\) is isomorphic to the group \(G\). The identity element of \(H\) is \(e\) and another element is \(d\). Write down the elements of \(H\) in terms of \(e\) and \(d\). [2]
OCR FP3 Q5
8 marks Standard +0.8
  1. Use de Moivre's theorem to prove that $$\cos 6\theta = 32\cos^6 \theta - 48\cos^4 \theta + 18\cos^2 \theta - 1.$$ [4]
  2. Hence find the largest positive root of the equation $$64x^6 - 96x^4 + 36x^2 - 3 = 0,$$ giving your answer in trigonometrical form. [4]
OCR FP3 Q6
10 marks Standard +0.8
Lines \(l_1\) and \(l_2\) have equations $$\frac{x-3}{2} = \frac{y-4}{-1} = \frac{z+1}{1} \quad \text{and} \quad \frac{x-5}{4} = \frac{y-1}{3} = \frac{z-1}{2}$$ respectively.
  1. Find the equation of the plane \(\Pi_1\) which contains \(l_1\) and is parallel to \(l_2\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [5]
  2. Find the equation of the plane \(\Pi_2\) which contains \(l_2\) and is parallel to \(l_1\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
  3. Find the distance between the planes \(\Pi_1\) and \(\Pi_2\). [2]
  4. State the relationship between the answer to part (iii) and the lines \(l_1\) and \(l_2\). [1]
OCR FP3 Q7
10 marks Standard +0.8
  1. Show that \((z - e^{i\theta})(z - e^{-i\theta}) \equiv z^2 - (2\cos \theta)z + 1\). [1]
  2. Write down the seven roots of the equation \(z^7 = 1\) in the form \(e^{i\theta}\) and show their positions in an Argand diagram. [4]
  3. Hence express \(z^7 - 1\) as the product of one real linear factor and three real quadratic factors. [5]
OCR FP3 Q8
10 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + y\tan x = \cos^3 x,$$ expressing \(y\) in terms of \(x\) in your answer. [8]
  2. Find the particular solution for which \(y = 2\) when \(x = \pi\). [2]
OCR FP3 Q9
12 marks Challenging +1.2
The set \(S\) consists of the numbers \(3^n\), where \(n \in \mathbb{Z}\). (\(\mathbb{Z}\) denotes the set of integers \(\{0, \pm 1, \pm 2, \ldots\}\).)
  1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.) [6]
  2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
    1. The numbers \(3^{2n}\), where \(n \in \mathbb{Z}\). [2]
    2. The numbers \(3^n\), where \(n \in \mathbb{Z}\) and \(n \geqslant 0\). [2]
    3. The numbers \(3^{(±n^2)}\), where \(n \in \mathbb{Z}\). [2]
OCR FP3 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 Q2
7 marks Standard +0.3
Find the general solution of the differential equation $$\frac{d^2y}{dx^2} - 8\frac{dy}{dx} + 16y = 4x.$$ [7]
OCR FP3 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 \(0 \leqslant \lambda \leqslant 1\). [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 Q4
8 marks Challenging +1.2
The integrals \(C\) and \(S\) are defined by $$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = \frac{1}{13}(2 + 3e^\pi),$$ 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 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 \(x = \frac{1}{4}\pi\).
  2. Find the solution of the differential equation in this case. [2]
  3. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]
OCR FP3 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]