Questions — OCR FP3 (182 questions)

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OCR FP3 2015 June Q6
7 marks Standard +0.8
6 Find the shortest distance between the lines with equations $$\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 5 } { - 1 } \quad \text { and } \quad \frac { x - 3 } { 4 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 3 } .$$
OCR FP3 2015 June Q7
9 marks Challenging +1.2
7
  1. Use de Moivre's theorem to show that \(\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }\).
  2. Hence find the exact roots of \(t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0\).
OCR FP3 2015 June Q8
12 marks Challenging +1.8
8 Let \(G\) be any multiplicative group. \(H\) is a subset of \(G\). \(H\) consists of all elements \(h\) such that \(h g = g h\) for every element \(g\) in \(G\).
  1. Prove that \(H\) is a subgroup of \(G\). Now consider the case where \(G\) is given by the following table:
    \(e\)\(p\)\(q\)\(r\)\(s\)\(t\)
    \(e\)\(e\)\(p\)\(q\)\(r\)\(s\)\(t\)
    \(p\)\(p\)\(q\)\(e\)\(s\)\(t\)\(r\)
    \(q\)\(q\)\(e\)\(p\)\(t\)\(r\)\(s\)
    \(r\)\(r\)\(t\)\(s\)\(e\)\(q\)\(p\)
    \(s\)\(s\)\(r\)\(t\)\(p\)\(e\)\(q\)
    \(t\)\(t\)\(s\)\(r\)\(q\)\(p\)\(e\)
  2. Show that \(H\) consists of just the identity element.
OCR FP3 2009 June Q1
4 marks Standard +0.3
1 Find the cube roots of \(\frac { 1 } { 2 } \sqrt { 3 } + \frac { 1 } { 2 } \mathrm { i }\), giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(0 \leqslant \theta < 2 \pi\).
OCR FP3 2009 June Q2
5 marks Standard +0.8
2 It is given that the set of complex numbers of the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) for \(- \pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.
  1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
  2. Prove the closure property for the group.
  3. \(Z\) denotes the element \(\mathrm { e } ^ { \mathrm { i } \gamma }\), where \(\frac { 1 } { 2 } \pi < \gamma < \pi\). Express \(Z ^ { 2 }\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta < 0\).
OCR FP3 2009 June Q3
8 marks Standard +0.8
3 A line \(l\) has equation \(\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }\) and a plane \(p\) has equation \(3 x - 4 y - 2 z = 8\).
  1. Find the point of intersection of \(l\) and \(p\).
  2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).
OCR FP3 2009 June Q4
8 marks Challenging +1.2
4 The differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { 1 - x ^ { 2 } } y = ( 1 - x ) ^ { \frac { 1 } { 2 } } , \quad \text { where } | x | < 1$$ 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. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2009 June Q5
9 marks Challenging +1.2
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$
  1. Find the complementary function.
  2. Explain briefly why there is no particular integral of either of the forms \(y = k \mathrm { e } ^ { 3 x }\) or \(y = k x \mathrm { e } ^ { 3 x }\).
  3. Given that there is a particular integral of the form \(y = k x ^ { 2 } \mathrm { e } ^ { 3 x }\), find the value of \(k\).
OCR FP3 2009 June Q6
9 marks Standard +0.8
6 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 5 \\ - 2 \end{array} \right)\).
  1. Express the equation of \(\Pi _ { 1 }\) in the form r.n \(= p\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . \left( \begin{array} { r } 7 \\ 17 \\ - 3 \end{array} \right) = 21\).
  2. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
OCR FP3 2009 June Q7
14 marks Challenging +1.3
7
  1. Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
  2. (a) By putting \(\theta = \frac { 1 } { 12 } \pi\) in the identity in part (i), show that \(\tan \frac { 1 } { 12 } \pi\) is a solution of the equation $$t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0 .$$ (b) Hence show that \(\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }\).
  3. Use the substitution \(t = \tan \theta\) to show that $$\int _ { 0 } ^ { 2 - \sqrt { 3 } } \frac { t \left( 3 - t ^ { 2 } \right) } { \left( 1 - 3 t ^ { 2 } \right) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t = a \ln b$$ where \(a\) and \(b\) are positive constants to be determined.
OCR FP3 2009 June Q8
15 marks Challenging +1.8
8 A multiplicative group \(Q\) of order 8 has elements \(\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}\), where \(e\) is the identity. The elements have the properties \(p ^ { 4 } = e\) and \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).
  1. Prove that \(a = p a p\) and that \(p = a p a\).
  2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
  3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
  4. Determine whether \(Q\) is a commutative group.
OCR FP3 2016 June Q1
6 marks Standard +0.3
1 In this question, give all non-real numbers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 < \theta < 2 \pi\).
  1. Solve \(z ^ { 5 } = 1\).
  2. Hence, or otherwise, solve \(z ^ { 5 } + 32 = 0\). Sketch an Argand diagram showing the roots.
OCR FP3 2016 June Q2
4 marks Standard +0.8
2 Find the shortest distance between the lines \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { c } - 1 \\ 1 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\).
OCR FP3 2016 June Q3
10 marks Challenging +1.2
3 The differential equation $$\frac { 2 } { y } - \frac { x } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } }$$ is to be solved subject to the condition \(y = 1\) when \(x = 1\).
  1. Show that \(y = \frac { 1 } { u }\) transforms the differential equation into $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } + 2 u = \frac { 1 } { x ^ { 2 } } .$$
  2. Find \(y\) in terms of \(x\).
OCR FP3 2016 June Q4
5 marks Standard +0.3
4 Let \(A\) be the set of non-zero integers.
  1. Show that \(A\) does not form a group under multiplication.
  2. State the largest subset of \(A\) which forms a group under multiplication. Show that this is a group.
OCR FP3 2016 June Q5
8 marks Standard +0.8
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 85 \cos x .$$
OCR FP3 2016 June Q6
10 marks Standard +0.3
6 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations $$\mathbf { r } \cdot \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) = 3 \text { and } \mathbf { r } \cdot \left( \begin{array} { l } 2 \\ 1 \\ 4 \end{array} \right) = 5$$ respectively. They intersect in the line \(l\).
  1. Find cartesian equations of \(l\). The plane \(\Pi _ { 3 }\) has equation \(\mathbf { r } . \left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right) = 1\).
  2. Show that \(\Pi _ { 3 }\) is parallel to \(l\) but does not contain it.
  3. Verify that \(( 2,0,1 )\) lies on planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\). Hence write down a vector equation of the line of intersection of these planes.
OCR FP3 2016 June Q8
17 marks Challenging +1.8
8 A non-commutative multiplicative group \(G\) of order eight has the elements $$\left\{ e , a , a ^ { 2 } , a ^ { 3 } , b , a b , a ^ { 2 } b , a ^ { 3 } b \right\}$$ where \(e\) is the identity and \(a ^ { 4 } = b ^ { 2 } = e\).
  1. Show that \(b a \neq a ^ { n }\) for any integer \(n\).
  2. Prove, by contradiction, that \(b a \neq a ^ { 2 } b\) and also that \(b a \neq a b\). Deduce that \(b a = a ^ { 3 } b\).
  3. Prove that \(b a ^ { 2 } = a ^ { 2 } b\).
  4. Construct group tables for the three subgroups of \(G\) of order four. \section*{END OF QUESTION PAPER}
OCR FP3 2007 June Q7
10 marks Standard +0.3
  1. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
  2. Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
  3. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors.
OCR FP3 2013 June Q2
9 marks Challenging +1.2
  1. Write down the operation table and, assuming associativity, show that \(G\) is a group.
  2. State the order of each element.
  3. Find all the proper subgroups of \(G\). The group \(H\) consists of the set \(\{ 1,3,7,9 \}\) with the operation of multiplication modulo 10 .
  4. Explaining your reasoning, determine whether \(H\) is isomorphic to \(G\).
OCR FP3 2016 June Q7
12 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
  2. Hence show that, for \(\sin 2 \theta \neq 0\), $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$
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]