Questions Further Pure Core (115 questions)

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OCR MEI Further Pure Core 2022 June Q13
17 marks Standard +0.8
13 The points A and B have coordinates \(( 4,0 , - 1 )\) and \(( 10,4 , - 3 )\) respectively. The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(x - 2 y = 5\) and \(2 x + 3 y - z = - 4\) respectively.
  1. Find the acute angle between the line AB and the plane \(\Pi _ { 1 }\).
  2. Show that the line AB meets \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) at the same point, whose coordinates should be specified.
    1. Find \(( \mathbf { i } - 2 \mathbf { j } ) \times ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )\).
    2. Hence find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
    3. Find the shortest distance between the point A and the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
OCR MEI Further Pure Core 2022 June Q14
8 marks Challenging +1.2
14
  1. Find \(\left( 3 - \mathrm { e } ^ { 2 \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right)\) in terms of \(\cos 2 \theta\).
  2. Hence show that the sum of the infinite series \(\sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \frac { 1 } { 27 } \sin 7 \theta + \ldots\) can be expressed as \(\frac { 6 \sin \theta } { 5 - 3 \cos 2 \theta }\).
OCR MEI Further Pure Core 2022 June Q15
23 marks Challenging +1.2
15 In an oscillating system, a particle of mass \(m \mathrm {~kg}\) moves in a horizontal line. Its displacement from its equilibrium position O at time \(t\) seconds is \(x\) metres, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), and it is acted on by a force \(2 m x\) newtons acting towards O as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-6_212_914_408_242} Initially, the particle is projected away from O with speed \(1 \mathrm {~ms} ^ { - 1 }\) from a point 2 m from O in the positive direction.
    1. Show that the motion is modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 x = 0\).
    2. State the type of motion.
    3. Write down the period of the motion.
    4. Find \(x\) in terms of \(t\).
    5. Find the amplitude of the motion.
  2. The motion is now damped by a force \(2 m v\) newtons.
    1. Show that \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 0\).
    2. State, giving a reason, whether the system is under-damped, critically damped or over-damped.
    3. Determine the general solution of this differential equation.
  3. Finally, a variable force \(2 m \cos 2 t\) newtons is added, so that the motion is now modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 2 \cos 2 t\).
    1. Find \(x\) in terms of \(t\). In the long term, the particle is seen to perform simple harmonic motion with a period of just over 3 seconds.
    2. Verify that this behaviour is consistent with the answer to part (c)(i).
OCR MEI Further Pure Core 2023 June Q1
7 marks Moderate -0.8
1
  1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
    1. Write down \(z ^ { * }\).
    2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
  2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
    1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
    2. Convert \(w\) to modulus-argument form.
OCR MEI Further Pure Core 2023 June Q2
5 marks Moderate -0.3
2 In this question you must show detailed reasoning.
Find the angle between the vector \(3 i + 2 j + \mathbf { k }\) and the plane \(- x + 3 y + 2 z = 8\).
OCR MEI Further Pure Core 2023 June Q3
6 marks Standard +0.8
3
  1. Using partial fractions and the method of differences, show that $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots + \frac { 1 } { \mathrm { n } ( \mathrm { n } + 2 ) } = \frac { 3 } { 4 } - \frac { \mathrm { an } + \mathrm { b } } { 2 ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) }$$ where \(a\) and \(b\) are integers to be determined.
  2. Deduce the sum to infinity of the series. $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$
OCR MEI Further Pure Core 2023 June Q4
6 marks Standard +0.3
4
    1. Given that \(\mathrm { f } ( x ) = \sqrt { 1 + 2 x }\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Hence, find the first three terms of the Maclaurin series for \(\sqrt { 1 + 2 x }\).
  2. Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).
OCR MEI Further Pure Core 2023 June Q5
7 marks Moderate -0.3
5
  1. In this question you must show detailed reasoning.
    Determine the sixth roots of - 64 , expressed in \(r \mathrm { e } ^ { \mathrm { i } \theta }\) form.
  2. Represent the roots on an Argand diagram.
OCR MEI Further Pure Core 2023 June Q6
9 marks Standard +0.3
6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.
  1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
  2. Specify the transformation of the plane associated with each of the following matrices.
    1. M
    2. N
  3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
  4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.
OCR MEI Further Pure Core 2023 June Q7
6 marks Standard +0.8
7 The diagram below shows the curve with polar equation \(r = a ( 1 - 2 \sin \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{76631941-3cd5-4b3e-a7e4-27b8f991975a-4_634_865_486_239} The curve crosses the initial line at A , and the points B and C are the lowest points on the two loops.
  1. Find the values of \(r\) and \(\theta\) at the points A , B and C .
  2. Find the set of values of \(\theta\) for the points on the inner loop (shown in the diagram with a broken line).
OCR MEI Further Pure Core 2023 June Q8
5 marks Moderate -0.3
8 Prove by mathematical induction that \(8 ^ { n } - 3 ^ { n }\) is divisible by 5 for all positive integers \(n\).
OCR MEI Further Pure Core 2023 June Q9
6 marks Standard +0.3
9 In an electrical circuit, the alternating current \(I\) amps is given by \(\mathbf { I } =\) asinnt, where \(t\) is the time in seconds and \(a\) and \(n\) are positive constants. The RMS value of the current, in amps, is defined to be the square root of the mean value of \(I ^ { 2 }\) over one complete period of \(\frac { 2 \pi } { n }\) seconds. Show that the RMS value of the current is \(\frac { a } { \sqrt { 2 } }\) amps.
OCR MEI Further Pure Core 2023 June Q10
7 marks Standard +0.8
10 The equation \(\mathrm { x } ^ { 3 } - 4 \mathrm { x } ^ { 2 } + 7 \mathrm { x } + \mathrm { c } = 0\), where \(c\) is a constant, has roots \(\alpha , \beta\) and \(\alpha + \beta\).
  1. Determine the roots of the equation.
  2. Find c.
OCR MEI Further Pure Core 2023 June Q11
7 marks Standard +0.3
11 Solve the differential equation \(\cosh x \frac { d y } { d x } - 2 y \sinh x = \cosh x\), given that \(y = 1\) when \(x = 0\).
OCR MEI Further Pure Core 2023 June Q12
7 marks Challenging +1.2
12 Show that \(\sin ^ { 5 } \theta = \operatorname { asin } 5 \theta + \mathrm { b } \sin 3 \theta + \mathrm { csin } \theta\), where \(a , b\) and \(c\) are constants to be determined.
OCR MEI Further Pure Core 2023 June Q13
14 marks Challenging +1.2
13
  1. On separate Argand diagrams, show the set of points representing each of the following inequalities.
    1. \(| z | \leqslant \sqrt { 5 }\)
    2. \(\quad | z + 2 - 4 i | \geqslant | z - 2 - 6 i |\)
  2. Show that there is a unique value of \(z\), which should be determined, for which both \(| z | \leqslant \sqrt { 5 }\) and \(| z + 2 - 4 i | \geqslant | z - 2 - 6 i |\).
OCR MEI Further Pure Core 2023 June Q14
13 marks Challenging +1.2
14 Three planes have equations $$\begin{aligned} k x - z & = 2 \\ - x + k y + 2 z & = 1 \\ 2 k x + 2 y + 3 z & = 0 \end{aligned}$$ where \(k\) is a constant.
  1. By considering a suitable determinant, show that the three planes meet at a point for all values of \(k\).
  2. Using a matrix method, find, in terms of \(k\), the coordinates of the point of intersection of the planes.
OCR MEI Further Pure Core 2023 June Q16
10 marks Challenging +1.8
16 The point \(P ( 4,1,0 )\) is equidistant from the plane \(2 x + y + 2 z = 0\) and the line \(\frac { x - 3 } { 2 } = \frac { y - 1 } { b } = \frac { z + 5 } { 3 }\), where \(b > 0\). Determine the value of \(b\).
OCR MEI Further Pure Core 2023 June Q17
24 marks Challenging +1.2
17 Two similar species, X and Y , of a small mammal compete for food and habitat. A model of this competition assumes, in a particular area, the following.
  • In the absence of the other species, each species would increase at a rate proportional to the number present with the same constant of proportionality in each case.
  • The competition reduces the rate of increase of each species by an amount proportional to the number of the other species present.
So if the numbers of species X and Y present at time \(t\) years are \(x\) and \(y\) respectively, the model gives the differential equations \(\frac { d x } { d t } = k x - a y\) and \(\frac { d y } { d t } = k y - b x\),
where \(k , a\) and \(b\) are positive constants.
    1. Show that the general solution for \(x\) is \(x = A e ^ { ( k + n ) t } + B e ^ { ( k - n ) t }\), where \(n = \sqrt { a b }\) and \(A\) and \(B\) are arbitrary constants.
    2. Hence find the general solution for \(y\) in terms of \(A , B , k , n , a\) and \(t\). Observations suggest that suitable values for the model are \(k = 0.015 , a = 0.04\) and \(b = 0.01\). You should use these values in the rest of this question.
  2. When \(t = 0\), the numbers present of species X and Y in this area are \(x _ { 0 }\) and \(y _ { 0 }\) respectively.
    1. Show that \(\mathrm { x } = \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } - 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { 0.035 \mathrm { t } } + \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } + 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { - 0.005 \mathrm { t } }\).
    2. Hence show that \(y = \frac { 1 } { 4 } \left( x _ { 0 } + 2 y _ { 0 } \right) e ^ { - 0.005 t } - \frac { 1 } { 4 } \left( x _ { 0 } - 2 y _ { 0 } \right) e ^ { 0.035 t }\).
  3. Use initial values \(x _ { 0 } = 500\) and \(y _ { 0 } = 300\) with the results in part (b) to determine what the model predicts for each of the following questions.
    1. What numbers of each species will be present after 25 years?
    2. In this question you must show detailed reasoning. When will the numbers of the two species be equal?
    3. Does either species ever disappear from the area? Justify your answer.
  4. Different initial values will apply in other areas where the two species compete, but previous studies indicate that one species or the other will eventually dominate in any given area.
    1. Identify a relationship between \(x _ { 0 }\) and \(y _ { 0 }\) where the model does not predict this outcome.
    2. Explain what the model predicts in the long term for this exceptional case.
OCR MEI Further Pure Core 2024 June Q1
4 marks Moderate -0.3
1 By expressing \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 }\) as a single fraction, find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\) in terms of \(n\).
OCR MEI Further Pure Core 2024 June Q2
7 marks Moderate -0.8
2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).
    1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
    2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
  2. Express \(u\) in exact modulus-argument form.
OCR MEI Further Pure Core 2024 June Q3
4 marks Standard +0.3
3 The equation \(2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
OCR MEI Further Pure Core 2024 June Q4
4 marks Standard +0.8
4 The equation of a curve is \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm {~K} ^ { 2 } + \mathrm { x } ^ { 2 } } }\), where \(k\) is a positive constant. The region between the \(x\)-axis, the \(y\)-axis and the line \(x = k\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Given that the volume of the solid of revolution formed is 1 unit \({ } ^ { 3 }\), find the exact value of \(k\).
OCR MEI Further Pure Core 2024 June Q5
6 marks Standard +0.3
5
  1. Given that \(\mathbf { u } = \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } a \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { u } \times \mathbf { v } = \left( \begin{array} { l } 1 \\ b \\ 3 \end{array} \right)\), find \(a\) and \(b\).
  2. Using \(\mathbf { u } \times \mathbf { v }\), determine the angle between the vectors \(\mathbf { u }\) and \(\mathbf { v }\), given that this angle is acute.
OCR MEI Further Pure Core 2024 June Q6
6 marks Moderate -0.8
6 On separate Argand diagrams, sketch the set of points represented by each of the following.
  1. \(| z - 1 - 2 i | \leqslant 4\).
  2. \(\quad \arg ( z + \mathrm { i } ) = \frac { 1 } { 3 } \pi\).