Questions FP3 (539 questions)

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OCR FP3 2010 June Q3
9 marks Standard +0.8
In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).
  1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
  2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
  3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]
OCR FP3 2010 June Q4
8 marks Challenging +1.2
  1. Use the substitution \(y = xz\) to find the general solution of the differential equation $$x \frac{dy}{dx} - y = x \cos \left(\frac{y}{x}\right),$$ giving your answer in a form without logarithms. (You may quote an appropriate result given in the List of Formulae (MF1).) [6]
  2. Find the solution of the differential equation for which \(y = \pi\) when \(x = 4\). [2]
OCR FP3 2010 June Q5
8 marks Challenging +1.2
Convergent infinite series \(C\) and \(S\) are defined by \begin{align} C &= 1 + \frac{1}{4} \cos \theta + \frac{1}{4} \cos 2\theta + \frac{1}{8} \cos 3\theta + \ldots,
S &= \frac{1}{2} \sin \theta + \frac{1}{4} \sin 2\theta + \frac{1}{8} \sin 3\theta + \ldots. \end{align}
  1. Show that \(C + iS = \frac{2}{2 - e^{i\theta}}\). [4]
  2. Hence show that \(C = \frac{4 - 2\cos \theta}{5 - 4\cos \theta}\) and find a similar expression for \(S\). [4]
OCR FP3 2010 June Q6
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 17y = 17x + 36.$$ [7]
  2. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation. [2]
OCR FP3 2010 June Q7
12 marks Challenging +1.2
A line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} -7 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\). A plane \(\Pi\) passes through the points \((1, 3, 5)\) and \((5, 2, 5)\), and is parallel to \(l\).
  1. Find an equation of \(\Pi\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
  2. Find the distance between \(l\) and \(\Pi\). [4]
  3. Find an equation of the line which is the reflection of \(l\) in \(\Pi\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [4]
OCR FP3 2010 June Q8
13 marks Challenging +1.2
A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.
  1. Write down the elements of a subgroup of order 2. [1]
  2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
  3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
  4. Find the inverses of \(B\) and \(E\). [3]
  5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]
OCR FP3 2011 June Q1
6 marks Standard +0.3
A line \(l\) has equation \(\frac{x-1}{5} = \frac{y-6}{6} = \frac{z+3}{-7}\) and a plane \(p\) has equation \(x + 2y - z = 40\).
  1. Find the acute angle between \(l\) and \(p\). [4]
  2. Find the perpendicular distance from the point \((1, 6, -3)\) to \(p\). [2]
OCR FP3 2011 June Q2
6 marks Standard +0.8
It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).
  1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
  2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]
OCR FP3 2011 June Q3
11 marks Standard +0.8
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} + 4y = 5 \cos 3x.$$
  1. Find the complementary function. [2]
  2. Hence, or otherwise, find the general solution. [7]
  3. Find the approximate range of values of \(y\) when \(x\) is large and positive. [2]
OCR FP3 2011 June Q4
9 marks Challenging +1.3
A group \(G\), of order 8, is generated by the elements \(a\), \(b\), \(c\). \(G\) has the properties $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb, \quad ca = ac,$$ where \(e\) is the identity.
  1. Using these properties and basic group properties as necessary, prove that \(abc = cba\). [2]
The operation table for \(G\) is shown below.
\(e\)\(a\)\(b\)\(c\)\(bc\)\(ca\)\(ab\)\(abc\)
\(e\)\(e\)\(a\)\(b\)\(c\)\(bc\)\(ca\)\(ab\)\(abc\)
\(a\)\(a\)\(e\)\(ab\)\(ca\)\(abc\)\(c\)\(b\)\(bc\)
\(b\)\(b\)\(ab\)\(e\)\(bc\)\(c\)\(abc\)\(a\)\(ca\)
\(c\)\(c\)\(ca\)\(bc\)\(e\)\(b\)\(a\)\(abc\)\(ab\)
\(bc\)\(bc\)\(abc\)\(c\)\(b\)\(e\)\(ab\)\(ca\)\(a\)
\(ca\)\(ca\)\(c\)\(abc\)\(a\)\(ab\)\(e\)\(bc\)\(b\)
\(ab\)\(ab\)\(b\)\(a\)\(abc\)\(ca\)\(bc\)\(e\)\(c\)
\(abc\)\(abc\)\(bc\)\(ca\)\(ab\)\(a\)\(b\)\(c\)\(e\)
  1. List all the subgroups of order 2. [2]
  2. List five subgroups of order 4. [3]
  3. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic. [2]
OCR FP3 2011 June Q5
9 marks Standard +0.8
The substitution \(y = u^k\), where \(k\) is an integer, is to be used to solve the differential equation $$x \frac{dy}{dx} + 3y = x^2 y^2 \qquad (A)$$ by changing it into an equation (B) in the variables \(u\) and \(x\).
  1. Show that equation (B) may be written in the form $$\frac{du}{dx} + \frac{3}{kx} u = \frac{1}{k} x u^{k+1}.$$ [4]
  2. Write down the value of \(k\) for which the integrating factor method may be used to solve equation (B). [1]
  3. Using this value of \(k\), solve equation (B) and hence find the general solution of equation (A), giving your answer in the form \(y = f(x)\). [4]
OCR FP3 2011 June Q6
10 marks Challenging +1.2
  1. The set of polynomials \(\{ax + b\}\), where \(a, b \in \mathbb{R}\), is denoted by \(P\). Assuming that the associativity property holds, prove that \(P\), under addition, is a group. [4]
  2. The set of polynomials \(\{ax + b\}\), where \(a, b \in \{0, 1, 2\}\), is denoted by \(Q\). It is given that \(Q\), under addition modulo 3, is a group, denoted by \((Q, +(\text{mod}3))\).
    1. State the order of the group. [1]
    2. Write down the inverse of the element \(2x + 1\). [1]
    3. \(q(x) = ax + b\) is any element of \(Q\) other than the identity. Find the order of \(q(x)\) and hence determine whether \((Q, +(\text{mod}3))\) is a cyclic group. [4]
OCR FP3 2011 June Q7
10 marks Challenging +1.2
(In this question, the notation \(\Delta ABC\) denotes the area of the triangle \(ABC\).) The points \(P\), \(Q\) and \(R\) have position vectors \(p\mathbf{i}\), \(q\mathbf{j}\) and \(r\mathbf{k}\) respectively, relative to the origin \(O\), where \(p\), \(q\) and \(r\) are positive. The points \(O\), \(P\), \(Q\) and \(R\) are joined to form a tetrahedron.
  1. Draw a sketch of the tetrahedron and write down the values of \(\Delta OPQ\), \(\Delta OQR\) and \(\Delta ORP\). [3]
  2. Use the definition of the vector product to show that \(\frac{1}{2}|\overrightarrow{RP} \times \overrightarrow{RQ}| = \Delta PQR\). [1]
  3. Show that \((\Delta OPQ)^2 + (\Delta OQR)^2 + (\Delta ORP)^2 = (\Delta PQR)^2\). [6]
OCR FP3 2011 June Q8
11 marks Challenging +1.2
  1. Use de Moivre's theorem to express \(\cos 4\theta\) as a polynomial in \(\cos \theta\). [4]
  2. Hence prove that \(\cos 4\theta \cos 2\theta \equiv 16 \cos^6 \theta - 24 \cos^4 \theta + 10 \cos^2 \theta - 1\). [1]
  3. Use part (ii) to show that the only roots of the equation \(\cos 4\theta \cos 2\theta = 1\) are \(\theta = n\pi\), where \(n\) is an integer. [3]
  4. Show that \(\cos 4\theta \cos 2\theta = -1\) only when \(\cos \theta = 0\). [3]