Questions Further Pure Core (114 questions)

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OCR MEI Further Pure Core 2024 June Q7
7
  1. Explain why \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt [ 3 ] { x - 2 } } \mathrm {~d} x\) is an improper integral.
  2. In this question you must show detailed reasoning. Use an appropriate limit argument to evaluate this integral.
OCR MEI Further Pure Core 2024 June Q8
8
  1. Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda
    0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
    1. Find detM.
    2. Deduce two properties of the transformation T from the value of detM.
  3. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda
    0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
  4. Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.
OCR MEI Further Pure Core 2024 June Q9
9 A curve has polar equation \(r = \operatorname { asin } 3 \theta\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.
  1. Sketch the curve. Indicate the parts of the curve where \(r\) is negative by using a broken line.
  2. In this question you must show detailed reasoning. Determine the area of one of the loops of the curve.
OCR MEI Further Pure Core 2024 June Q10
10
  1. Write down the first three terms of the Maclaurin series for \(\ln \left( 1 + x ^ { 3 } \right)\).
  2. Use these three terms to show that \(\ln ( 1.125 ) \approx \frac { n } { 1536 }\), where \(n\) is an integer to be determined.
  3. Charlie uses the same first three terms of the series to approximate \(\ln 9\) and gets an answer of 147, correct to 3 significant figures. However, \(\ln 9 = 2.20\) correct to 3 significant figures. Explain Charlie's error.
OCR MEI Further Pure Core 2024 June Q11
11 The plane \(\Pi\) has equation \(2 x - y + 2 z = 4\). The point \(P\) has coordinates \(( 8,4,5 )\).
  1. Calculate the shortest distance from P to \(\Pi\). The line \(L\) has equation \(\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }\).
  2. Verify that P lies on L .
  3. Find the coordinates of the point of intersection of L and \(\Pi\).
  4. Determine the acute angle between L and \(\Pi\).
  5. Use the results of parts (b), (c) and (d) to verify your answer to part (a).
OCR MEI Further Pure Core 2024 June Q12
12 The diagram shows the curve with parametric equations \(x = 2 \cosh t + \sinh t , y = \cosh t - 2 \sinh t\).
\includegraphics[max width=\textwidth, alt={}, center]{83275e7c-7f5a-4f26-b81d-a041e67ac9a2-5_812_808_1283_246}
  1. The curve crosses the positive \(x\)-axis at A .
    1. Determine the value of the parameter \(t\) at A , giving your answer in logarithmic form.
    2. Find the \(x\)-coordinate of A , giving your answer correct to \(\mathbf { 3 }\) significant figures.
  2. The point B has parameter \(t = 0\). Determine the equation of the tangent to the curve at B .
OCR MEI Further Pure Core 2024 June Q13
13 The complex number \(z\) is defined as \(z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
On an Argand diagram, the point O represents the complex number 0 , and the points \(P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots\) represent the complex numbers \(z , z ^ { 2 } , z ^ { 3 } , \ldots\) respectively.
  1. Write down each of the following.
    1. The ratio of the lengths \(\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }\)
    2. The angle \(\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }\)
    1. Show that \(\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta\), where \(a\) and \(b\) are integers to be determined.
    2. By considering the sum to infinity of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots\), show that $$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$
OCR MEI Further Pure Core 2024 June Q14
14
  1. Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } - 2 y = 12 e ^ { - x }\). You are given that \(y\) tends to zero as \(x\) tends to infinity, and that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\) when \(x = 0\).
  2. Find the exact value of \(x\) for which \(y = 0\).
OCR MEI Further Pure Core 2024 June Q15
15 Three planes have equations $$\begin{aligned} x + k y + 3 z & = 1
3 x + 4 y + 2 z & = 3
x + 3 y - z & = - k \end{aligned}$$ where \(k\) is a constant.
  1. Show that the planes meet at a point except for one value of \(k\), which should be determined.
  2. Show that, when the planes do meet at a point, the \(y\)-coordinate of this point is independent of \(k\).
OCR MEI Further Pure Core 2024 June Q17
17 In an industrial process, a container initially contains 1000 litres of liquid. Liquid is drawn from the bottom of the container at a rate of 5 litres per minute. At the same time, salt is added to the top of the container at a constant rate of 10 grams per minute. After \(t\) minutes the mass of salt in the container is \(x\) grams, and you are given that \(x = 0\) when \(t = 0\). In modelling the situation, it is assumed that the salt dissolves instantly and uniformly in the liquid, and that adding the salt does not change the volume of the liquid.
    1. Show that the concentration of salt in the liquid after \(t\) minutes is \(\frac { \mathrm { X } } { 1000 - 5 \mathrm { t } }\) grams per litre.
    2. Hence show that the mass of salt in the container is given by the differential equation $$\frac { d x } { d t } + \frac { x } { 200 - t } = 10$$
  2. Show by integration that \(\mathrm { x } = 10 ( 200 - \mathrm { t } ) \ln \left( \frac { 200 } { 200 - \mathrm { t } } \right)\).
    1. Hence determine the mass of salt in the container when half the liquid is drawn off.
    2. Determine also the time at which the mass of salt in the container is greatest.
  4. When the process is run, it is found that the concentration of salt over time is higher than predicted by the model. Suggest a reason for this.
OCR MEI Further Pure Core 2020 November Q1
1 Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \(( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots\),
where \(n\) is a positive integer. Give your answer in fully factorised form.
OCR MEI Further Pure Core 2020 November Q2
2
  1. The matrices \(\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a
    1 & b & 0 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { c c } b & - 5
    - 1 & c
    - 1 & 1 \end{array} \right)\) are such that \(\mathbf { M } \mathbf { N } = \mathbf { I }\).
    Find \(a , b\) and \(c\).
  2. State with a reason whether or not \(\mathbf { N }\) is the inverse of \(\mathbf { M }\).
OCR MEI Further Pure Core 2020 November Q3
3 In this question you must show detailed reasoning.
Find \(\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { 1 } { \sqrt { 4 - 9 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in terms of \(\pi\).
OCR MEI Further Pure Core 2020 November Q4
4 The roots of the equation \(2 x ^ { 3 } - 5 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
  2. Find an equation with integer coefficients whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).
OCR MEI Further Pure Core 2020 November Q5
5 Fig. 5 shows the curve with polar equation \(r = a ( 3 + 2 \cos \theta )\) for \(- \pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-3_607_718_351_244} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Write down the polar coordinates of the points A and B .
  2. Explain why the curve is symmetrical about the initial line.
  3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve.
OCR MEI Further Pure Core 2020 November Q6
6 The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).
Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Section B (108 marks)
Answer all the questions.
OCR MEI Further Pure Core 2020 November Q7
7 Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
OCR MEI Further Pure Core 2020 November Q8
8
  1. Given that the lines \(\mathbf { r } = \left( \begin{array} { l } 0
    2
    2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
    1
    3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1
    2
    k \end{array} \right) + \mu \left( \begin{array} { l } 2
    3
    4 \end{array} \right)\) meet, determine \(k\).
  2. In this question you must show detailed reasoning. Find the acute angle between the two lines.
OCR MEI Further Pure Core 2020 November Q9
9 A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2
\lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant. constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
OCR MEI Further Pure Core 2020 November Q12
12
  1. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
  2. By considering \(\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 }\), find constants \(A\) and \(B\) such that \(\sin ^ { 3 } \theta \cos ^ { 3 } \theta = A \sin 6 \theta + B \sin 2 \theta\).
OCR MEI Further Pure Core 2020 November Q13
13
  1. Using exponentials, prove that \(\sinh 2 x = 2 \cosh x \sinh x\).
  2. Hence show that if \(\mathrm { f } ( x ) = \sinh ^ { 2 } x\), then \(\mathrm { f } ^ { \prime \prime } ( x ) = 2 \cosh 2 x\).
  3. Explain why the coefficients of odd powers in the Maclaurin series for \(\sinh ^ { 2 } x\) are all zero.
  4. Find the coefficient of \(x ^ { n }\) in this series when \(n\) is a positive even number.
OCR MEI Further Pure Core 2020 November Q14
14 Solve the simultaneous differential equations
\(\frac { \mathrm { d } x } { \mathrm {~d} t } + 2 x = 4 y , \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 x = 5 y\),
given that when \(t = 0 , x = 0\) and \(y = 1\).
OCR MEI Further Pure Core 2020 November Q15
15
  1. Show that the three planes with equations $$\begin{aligned} x + \lambda y + 3 z & = - 12
    2 x + y + 5 z & = - 11
    x - 2 y + 2 z & = - 9 \end{aligned}$$ where \(\lambda\) is a constant, meet at a unique point except for one value of \(\lambda\) which is to be determined.
  2. In the case \(\lambda = - 2\), use matrices to find the point of intersection P of the planes, showing your method clearly. The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z + 2 } { - 2 }\).
  3. Find a vector equation of \(l\).
  4. Find the shortest distance between the point P and \(l\).
    1. Show that \(l\) is parallel to the plane \(x - 2 y + 2 z = - 9\).
    2. Find the distance between \(l\) and the plane \(x - 2 y + 2 z = - 9\).
OCR MEI Further Pure Core 2020 November Q16
16 The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is \(A\).
  1. One simple model is to assume that the rate of change of population density is directly proportional to \(A - P\).
    1. Formulate a differential equation for this model.
    2. Verify that \(P = A \left( 1 - \mathrm { e } ^ { - k t } \right)\), where \(k\) is a positive constant, satisfies
      • this differential equation,
  2. the initial condition,
  3. the long-term condition.
    4. An alternative model uses the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } - \frac { P } { t \left( 1 + t ^ { 2 } \right) } = \mathrm { Q } ( t )$$ where \(\mathrm { Q } ( t )\) is a function of \(t\).
  4. Find the integrating factor for this differential equation, showing that it can be written in the $$\text { form } \frac { \sqrt { 1 + t ^ { 2 } } } { t } \text {. }$$
  5. Suppose that \(\mathrm { Q } ( t ) = 0\). $$\text { (i) Show that } P = \frac { A t } { \sqrt { 1 + t ^ { 2 } } } \text {. }$$ (ii) Find the time predicted by this model for the population density to reach half its longterm value. Give your answer correct to the nearest minute.
  6. Now suppose that \(\mathrm { Q } ( t ) = \frac { t \mathrm { e } ^ { - t } } { \sqrt { 1 + t ^ { 2 } } }\). $$\text { Show that } \left. P = \frac { A t - t e ^ { - t } } { \sqrt { 1 + t ^ { 2 } } } \text {. [You may assume that } \lim _ { t \rightarrow \infty } t e ^ { - t } = 0 . \right]$$ It is found that the long-term value of \(P\) is 10, and \(P\) reaches half this value after 37 minutes.
  7. Determine which of the models proposed in parts (c) and (d) is more consistent with these data.
OCR MEI Further Pure Core 2021 November Q1
1
  1. Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.