Questions — OCR FP3 (140 questions)

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OCR FP3 2012 June Q2
2
  1. Solve the equation \(z ^ { 4 } = 2 ( 1 + \mathrm { i } \sqrt { 3 } )\), giving the roots exactly in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Sketch an Argand diagram to show the lines from the origin to the point representing \(2 ( 1 + i \sqrt { 3 } )\) and from the origin to the points which represent the roots of the equation in part (i).
OCR FP3 2012 June Q3
3 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \cot x = 2 x$$ for which \(y = 2\) when \(x = \frac { 1 } { 6 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2012 June Q4
4 The elements \(a , b , c , d\) are combined according to the operation table below, to form a group \(G\) of order 4.
\(a\)\(b\)\(c\)\(d\)
\(a\)\(b\)\(a\)\(d\)\(c\)
\(b\)\(a\)\(b\)\(c\)\(d\)
\(c\)\(d\)\(c\)\(a\)\(b\)
\(d\)\(c\)\(d\)\(b\)\(a\)
Group \(G\) is isomorphic either to the multiplicative group \(H = \left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\) or to the multiplicative group \(K = \{ e , p , q , p q \}\). It is given that \(r ^ { 4 } = e\) in group \(H\) and that \(p ^ { 2 } = q ^ { 2 } = e\) in group \(K\), where \(e\) denotes the identity in each group.
  1. Write down the operation tables for \(H\) and \(K\).
  2. State the identity element of \(G\).
  3. Demonstrate the isomorphism between \(G\) and either \(H\) or \(K\) by listing how the elements of \(G\) correspond to the elements of the other group. If the correspondence can be shown in more than one way, list the alternative correspondence(s).
OCR FP3 2012 June Q5
5
  1. By expressing \(\sin \theta\) and \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), prove that $$\sin ^ { 3 } \theta \cos ^ { 2 } \theta \equiv - \frac { 1 } { 16 } ( \sin 5 \theta - \sin 3 \theta - 2 \sin \theta )$$
  2. Hence show that all the roots of the equation $$\sin 5 \theta = \sin 3 \theta + 2 \sin \theta$$ are of the form \(\theta = \frac { n \pi } { k }\), where \(n\) is any integer and \(k\) is to be determined.
OCR FP3 2012 June Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 12 \mathrm { e } ^ { 2 x }$$
  1. Find the general solution of the differential equation.
  2. It is given that the curve which represents a particular solution of the differential equation has gradient 6 when \(x = 0\), and approximates to \(y = \mathrm { e } ^ { 2 x }\) when \(x\) is large and positive. Find the equation of the curve.
OCR FP3 2012 June Q7
7 With respect to the origin \(O\), the position vectors of the points \(U , V\) and \(W\) are \(\mathbf { u } , \mathbf { v }\) and \(\mathbf { w }\) respectively. The mid-points of the sides \(V W , W U\) and \(U V\) of the triangle \(U V W\) are \(M , N\) and \(P\) respectively.
  1. Show that \(\overrightarrow { U M } = \frac { 1 } { 2 } ( \mathbf { v } + \mathbf { w } - 2 \mathbf { u } )\).
  2. Verify that the point \(G\) with position vector \(\frac { 1 } { 3 } ( \mathbf { u } + \mathbf { v } + \mathbf { w } )\) lies on \(U M\), and deduce that the lines \(U M , V N\) and \(W P\) intersect at \(G\).
  3. Write down, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line through \(G\) which is perpendicular to the plane \(U V W\). (It is not necessary to simplify the expression for \(\mathbf { b }\).)
  4. It is now given that \(\mathbf { u } = \left( \begin{array} { l } 1
    0
    0 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } 0
    1
    0 \end{array} \right)\) and \(\mathbf { w } = \left( \begin{array} { l } 0
    0
    1 \end{array} \right)\). Find the perpendicular distance from \(O\) to the plane \(U V W\).
OCR FP3 2012 June Q8
8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b
c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta
\sin \theta & \cos \theta \end{array} \right)\).
  1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
  2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR FP3 2013 June Q1
1 The plane \(\Pi\) passes through the points with coordinates \(( 1,6,2 ) , ( 5,2,1 )\) and \(( 1,0 , - 2 )\).
  1. Find a vector equation of \(\Pi\) in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
  2. Find a cartesian equation of \(\Pi\).
    \(2 G\) consists of the set \(\{ 1,3,5,7 \}\) with the operation of multiplication modulo 8 .
OCR FP3 2013 June Q3
3 The differential equation $$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y ^ { 3 } = \frac { \cos x } { x }$$ is to be solved for \(x > 0\). Use the substitution \(u = y ^ { 3 }\) to find the general solution for \(y\) in terms of \(x\).
OCR FP3 2013 June Q4
4 The complex numbers 0,3 and \(3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\) are represented in an Argand diagram by the points \(O , A\) and \(B\) respectively.
  1. Sketch the triangle \(O A B\) and show that it is equilateral.
  2. Hence express \(3 - 3 e ^ { \frac { 1 } { 3 } \pi i }\) in polar form.
  3. Hence find \(\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }\), giving your answer in the form \(a + b \sqrt { 3 } \mathrm { i }\) where \(a\) and \(b\) are rational numbers.
OCR FP3 2013 June Q5
5 Find the solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = \mathrm { e } ^ { - x }\) for which \(y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
OCR FP3 2013 June Q6
6 The plane \(\Pi\) has equation \(x + 2 y - 2 z = 5\). The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y + 1 } { 5 } = \frac { z - 2 } { 1 }\).
  1. Find the coordinates of the point of intersection of \(l\) with the plane \(\Pi\).
  2. Calculate the acute angle between \(l\) and \(\Pi\).
  3. Find the coordinates of the two points on the line \(l\) such that the distance of each point from the plane \(\Pi\) is 2 .
OCR FP3 2013 June Q7
7 A commutative group \(G\) has order 18. The elements \(a , b\) and \(c\) have orders 2, 3 and 9 respectively.
  1. Prove that \(a b\) has order 6 .
  2. Show that \(G\) is cyclic.
OCR FP3 2013 June Q8
8
  1. Use de Moivre's theorem to show that \(\cos 5 \theta \equiv 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta\).
  2. Hence find the roots of \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) in the form \(\cos \alpha\) where \(0 \leqslant \alpha \leqslant \pi\).
  3. Hence find the exact value of \(\cos \frac { 1 } { 10 } \pi\).
OCR FP3 2014 June Q1
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
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
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
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
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
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
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.
  3. Prove that \(G\) is a group. You may assume that matrix multiplication is associative.
  4. Determine whether \(G\) is commutative.
  5. Find the order of \(\left( \begin{array} { c c } 0 & - 1
    1 & 0 \end{array} \right)\).
OCR FP3 2015 June Q1
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
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
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
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 \(a + b \mathrm { i }\).