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OCR FP2 2012 January Q3
7 marks Standard +0.3
Express \(\frac{2x^3 + x + 12}{(2x - 1)(x^2 + 4)}\) in partial fractions. [7]
OCR FP2 2012 January Q4
9 marks Standard +0.8
\includegraphics{figure_4} The diagram shows the curve \(y = e^{-\frac{1}{x}}\) for \(0 < x \leq 1\). A set of \((n - 1)\) rectangles is drawn under the curve as shown.
  1. Explain why a lower bound for \(\int_0^1 e^{-\frac{1}{x}} dx\) can be expressed as $$\frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}}\right).$$ [2]
  2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int_0^1 e^{-\frac{1}{x}} dx\). [2]
  3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures. [2]
  4. When \(n > N\), the difference between the upper and lower bounds is less than 0.001. By expressing this difference in terms of \(n\), find the least possible value of \(N\). [3]
OCR FP2 2012 January Q5
11 marks Standard +0.8
It is given that \(f(x) = x^3 - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(f(x) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x_1, x_2, \ldots, x_n, \ldots\).
  1. Show that \(x_{n+1} = \frac{2x_n^3 + k}{3x_n^2}\). [2]
  2. Sketch the graph of \(y = f(x)\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(|x_2 - x_1|\) to be greater than \(|x_1|\). [3]
It is now given that \(k = 100\) and \(x_1 = 5\).
  1. Write down the exact value of \(\alpha\) and find \(x_2\) and \(x_3\) correct to 5 decimal places. [3]
  2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). By finding \(e_1\), \(e_2\) and \(e_3\), verify that \(e_3 \approx \frac{e_2^2}{e_1}\). [3]
OCR FP2 2012 January Q6
8 marks Standard +0.8
  1. Prove that the derivative of \(\cos^{-1} x\) is \(-\frac{1}{\sqrt{1 - x^2}}\). [3]
A curve has equation \(y = \cos^{-1}(1 - x^2)\), for \(0 < x < \sqrt{2}\).
  1. Find and simplify \(\frac{dy}{dx}\), and hence show that $$(2 - x^2)\frac{d^2y}{dx^2} = x\frac{dy}{dx}.$$ [5]
OCR FP2 2012 January Q7
8 marks Standard +0.8
  1. Given that \(y = \sinh^{-1} x\), prove that \(y = \ln\left(x + \sqrt{x^2 + 1}\right)\). [3]
  2. It is given that \(x\) satisfies the equation \(\sinh^{-1} x - \cosh^{-1} x = \ln 2\). Use the logarithmic forms for \(\sinh^{-1} x\) and \(\cosh^{-1} x\) to show that $$\sqrt{x^2 + 1} - 2\sqrt{x^2 - 1} = x.$$ Hence, by squaring this equation, find the exact value of \(x\). [5]
OCR FP2 2012 January Q8
9 marks Challenging +1.3
\includegraphics{figure_8} The diagram shows two curves, \(C_1\) and \(C_2\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C_1\) is \(r = \sqrt{2}\cos\theta\) and the polar equation of \(C_2\) is \(r = \sqrt{2}\sin 2\theta\). For both curves, \(0 \leq \theta \leq \frac{1}{2}\pi\). The value of \(\theta\) at \(P\) is \(\alpha\).
  1. Show that \(\tan\alpha = \frac{1}{2}\). [2]
  2. Show that the area of the region common to \(C_1\) and \(C_2\), shaded in the diagram, is \(\frac{1}{4}\pi - \frac{1}{2}\alpha\). [7]
OCR FP2 2012 January Q9
11 marks Challenging +1.3
  1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]
It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).
  1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
  2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
  3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]
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]
OCR FP3 Q7
11 marks Challenging +1.2
    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 \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \frac{1}{2}\pi\). [2]
  2. Use de Moivre's theorem to prove that $$\sin 6\theta \equiv \sin 2\theta(16\cos^4 \theta - 16\cos^2 \theta + 3).$$ [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 Q8
13 marks Challenging +1.3
Groups \(A, B, C\) and \(D\) are defined as follows: \begin{align} A: &\quad \text{the set of numbers } \{2, 4, 6, 8\} \text{ under multiplication modulo 10,}
B: &\quad \text{the set of numbers } \{1, 5, 7, 11\} \text{ under multiplication modulo 12,}
C: &\quad \text{the set of numbers } \{2^0, 2^1, 2^2, 2^3\} \text{ under multiplication modulo 15,}
D: &\quad \text{the set of numbers } \left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\} \text{ under multiplication.} \end{align}
  1. Write down the identity element for each of groups \(A, B, C\) and \(D\). [2]
  2. Determine in each case whether the groups \begin{align} &A \text{ and } B,
    &B \text{ and } C,
    &A \text{ and } C \end{align} 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 Q1
5 marks Standard +0.8
  1. A cyclic multiplicative group \(G\) has order 12. The identity element of \(G\) is \(e\) and another element is \(r\), with order 12.
    1. Write down, in terms of \(e\) and \(r\), the elements of the subgroup of \(G\) which is of order 4. [2]
    2. Explain briefly why there is no proper subgroup of \(G\) in which two of the elements are \(e\) and \(r\). [1]
  2. A group \(H\) has order \(mnp\), where \(m, n\) and \(p\) are prime. State the possible orders of proper subgroups of \(H\). [2]