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OCR FP3 2014 June Q1
6 marks Standard +0.3
1
  1. Find a vector equation of the line of intersection of the planes \(2 x + y - z = 4\) and \(3 x + 5 y + 2 z = 13\).
  2. Find the exact distance of the point \(( 2,5 , - 2 )\) from the plane \(2 x + y - z = 4\).
OCR FP3 2014 June Q2
8 marks Challenging +1.2
2 Use the substitution \(u = y ^ { 2 }\) to find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = \frac { \mathrm { e } ^ { x } } { y }$$ for \(y\) in terms of \(x\).
OCR FP3 2014 June Q3
10 marks Standard +0.3
3
  1. Solve the equation \(z ^ { 6 } = 1\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), and sketch an Argand diagram showing the positions of the roots.
  2. Show that \(( 1 + \mathrm { i } ) ^ { 6 } = - 8 \mathrm { i }\).
  3. Hence, or otherwise, solve the equation \(z ^ { 6 } + 8 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
OCR FP3 2014 June Q4
11 marks Challenging +1.3
4 The group \(G\) consists of the set \(\{ 1,3,7,9,11,13,17,19 \}\) combined under multiplication modulo 20.
  1. Find the inverse of each element.
  2. Show that \(G\) is not cyclic.
  3. Find two isomorphic subgroups of order 4 and state an isomorphism between them.
OCR FP3 2014 June Q5
10 marks Standard +0.8
5 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = \mathrm { e } ^ { - x }$$ subject to the conditions \(y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
OCR FP3 2014 June Q6
8 marks Standard +0.8
6 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 7 } { 5 }\). The plane \(\Pi\) has equation \(4 x - y - z = 8\).
  1. Show that \(l\) is parallel to \(\Pi\) but does not lie in \(\Pi\).
  2. The point \(A ( 1 , - 2,7 )\) is on \(l\). Write down a vector equation of the line through \(A\) which is perpendicular to \(\Pi\). Hence find the position vector of the point on \(\Pi\) which is closest to \(A\).
  3. Hence write down a vector equation of the line in \(\Pi\) which is parallel to \(l\) and closest to it.
OCR FP3 2014 June Q7
8 marks Challenging +1.2
7
  1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta ) .$$
  2. Hence solve the equation $$\sin 5 \theta + 4 \sin \theta = 5 \sin 3 \theta$$ for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). 8 consists of the set of matrices of the form \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\), where \(a\) and \(b\) are real and \(a ^ { 2 } + b ^ { 2 } \neq 0\), combined under the operation of matrix multiplication.
  1. Prove that \(G\) is a group. You may assume that matrix multiplication is associative.
  2. Determine whether \(G\) is commutative.
  3. Find the order of \(\left( \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
OCR FP3 2015 June Q1
8 marks Standard +0.8
1 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = \sin x$$
OCR FP3 2015 June Q2
8 marks Standard +0.8
2 The elements of a group \(G\) are polynomials of the form \(a + b x + c x ^ { 2 }\), where \(a , b , c \in \{ 0,1,2,3,4 \}\). The group operation is addition, where the coefficients are added modulo 5 .
  1. State the identity element.
  2. State the inverse of \(3 + 2 x + x ^ { 2 }\).
  3. State the order of \(G\). The proper subgroup \(H\) contains \(2 + x\) and \(1 + x\).
  4. Find the order of \(H\), justifying your answer.
OCR FP3 2015 June Q3
11 marks Standard +0.3
3 The plane \(\Pi\) passes through the points \(( 1,2,1 ) , ( 2,3,6 )\) and \(( 4 , - 1,2 )\).
  1. Find a cartesian equation of the plane \(\Pi\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\).
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
  3. Find the acute angle between \(\Pi\) and \(l\).
OCR FP3 2015 June Q4
9 marks Standard +0.8
4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.
  1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
  2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).
OCR FP3 2015 June Q5
8 marks Standard +0.3
5 Find the particular solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } + x$$ for which \(y = 1\) when \(x = 1\), giving \(y\) in terms of \(x\).
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 C1 2009 January Q1
3 marks Easy -1.2
1 Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
OCR C1 2009 January Q2
4 marks Easy -1.3
2 Simplify
  1. \(( \sqrt [ 3 ] { x } ) ^ { 6 }\),
  2. \(\frac { 3 y ^ { 4 } \times ( 10 y ) ^ { 3 } } { 2 y ^ { 5 } }\).
OCR C1 2009 January Q3
5 marks Standard +0.3
3 Solve the equation \(3 x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } - 2 = 0\).
OCR C1 2009 January Q4
6 marks Moderate -0.8
4
  1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
  2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
  3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).
OCR C1 2009 January Q5
9 marks Easy -1.3
5 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = 10 x ^ { - 5 }\),
  2. \(y = \sqrt [ 4 ] { x }\),
  3. \(y = x ( x + 3 ) ( 1 - 5 x )\).
OCR C1 2009 January Q6
8 marks Moderate -0.8
6
  1. Express \(5 x ^ { 2 } + 20 x - 8\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the equation of the line of symmetry of the curve \(y = 5 x ^ { 2 } + 20 x - 8\).
  3. Calculate the discriminant of \(5 x ^ { 2 } + 20 x - 8\).
  4. State the number of real roots of the equation \(5 x ^ { 2 } + 20 x - 8 = 0\).
OCR C1 2009 January Q7
8 marks Moderate -0.8
7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).
  1. Find the value of \(k\).
  2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  3. Write down the coordinates of the centre and the radius of the circle.
  4. Verify that \(A B\) is a diameter of the circle.
OCR C1 2009 January Q8
10 marks Moderate -0.3
8
  1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
  2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
  3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.
OCR C1 2009 January Q9
7 marks Moderate -0.3
9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.
OCR C1 2009 January Q10
12 marks Standard +0.3
10 A curve has equation \(y = x ^ { 2 } + x\).
  1. Find the gradient of the curve at the point for which \(x = 2\).
  2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.