Questions Further Pure Core (114 questions)

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OCR MEI Further Pure Core 2021 November Q2
2 In this question you must show detailed reasoning.
Find the gradient of the curve \(y = 6 \arcsin ( 2 x )\) at the point with \(x\)-coordinate \(\frac { 1 } { 4 }\). Express the result in the form \(\mathrm { m } \sqrt { \mathrm { n } }\), where \(m\) and \(n\) are integers.
OCR MEI Further Pure Core 2021 November Q3
3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = - 2 + 2 i\) and \(z _ { 2 } = 2 \left( \cos \frac { 1 } { 6 } \pi + i \sin \frac { 1 } { 6 } \pi \right)\).
  1. Find the modulus and argument of \(z _ { 1 }\).
  2. Hence express \(\frac { z _ { 1 } } { z _ { 2 } }\) in exact modulus-argument form.
OCR MEI Further Pure Core 2021 November Q4
4 In this question you must show detailed reasoning.
Determine the mean value of \(\frac { 1 } { 1 + 4 x ^ { 2 } }\) between \(x = - 1\) and \(x = 1\). Give your answer to 3 significant
figures. figures.
OCR MEI Further Pure Core 2021 November Q5
5
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.
OCR MEI Further Pure Core 2021 November Q6
6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2
2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.
OCR MEI Further Pure Core 2021 November Q7
7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).
OCR MEI Further Pure Core 2021 November Q8
8 The equation \(4 \mathrm { x } ^ { 4 } - 4 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , - \alpha , \beta\) and \(\frac { 1 } { \beta }\).
  1. Determine the exact roots of the equation.
  2. Determine the values of \(p\) and \(q\).
OCR MEI Further Pure Core 2021 November Q9
9 The transformation Too the plane has associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } - 1 & 0
- 2 & 1 \end{array} \right)\).
  1. On the grid in the Printed Answer Booklet, plot the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) of the unit square OABC under the transformation T.
    1. Calculate the value of \(\operatorname { det } \mathbf { M }\).
    2. Explain the significance of the value of \(\operatorname { det } \mathbf { M }\) in relation to the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\).
  3. T is equivalent to a sequence of two transformations of the plane.
    1. Specify fully two transformations equivalent to T .
    2. Use matrices to verify your answer.
OCR MEI Further Pure Core 2021 November Q10
10
  1. Show on an Argand diagram the points representing the three cube roots of unity.
    1. Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
    2. The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\). State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
    3. The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\). By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.
    4. Hence show that \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).
OCR MEI Further Pure Core 2021 November Q11
11
  1. Given that \(\mathbf { u } = \lambda \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(\mathbf { v } = \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\), find the following, giving your answers in terms of \(\lambda\).
    1. u.v
    2. \(\mathbf { u } \times \mathbf { v }\)
  2. Hence determine
    1. the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x + 2 y - 2 z = 10\),
    2. the shortest distance between the lines \(\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z - 2 } { - 3 }\) and \(\frac { x } { 1 } = \frac { y - 4 } { 2 } = \frac { z + 2 } { - 2 }\), giving your answer as a multiple of \(\sqrt { 2 }\).
OCR MEI Further Pure Core 2021 November Q12
12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers \(z\) and \(w\) respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82808722-0abc-411a-9aa3-c0f368a4c95e-4_641_659_1201_242} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Prove that \(\arg ( z + w ) = \frac { 1 } { 2 } ( \arg z + \arg w )\).
[0pt] [A copy of Fig. 12 is provided in the Printed Answer Booklet.]
OCR MEI Further Pure Core 2021 November Q13
13 Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = 2 e ^ { x }\).
OCR MEI Further Pure Core 2021 November Q14
14 A curve has polar equation \(\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \pi\).
  1. Determine the polar coordinates of the point on the curve which is furthest from the pole.
    1. Show that the curve is a circle whose radius should be specified.
    2. Write down the polar coordinates of the centre of the circle.
OCR MEI Further Pure Core 2021 November Q15
15 The equations of three planes are $$\begin{aligned} - 4 x + k y + 7 z & = 4
x - 2 y + 5 z & = 1
2 x + 3 y + z & = 2 \end{aligned}$$ Given that the planes form a sheaf, determine the values of \(k\) and \(l\).
OCR MEI Further Pure Core 2021 November Q16
16
  1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
  2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).
OCR MEI Further Pure Core 2021 November Q17
17 In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of \(a\) litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of \(b\) litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After \(t\) minutes, the mixture in the vessel contains \(x\) litres of the chemical.
    1. Show that the proportion of chemical present in the vessel after \(t\) minutes is $$\frac { x } { 1 + ( a - b ) t } .$$
    2. Hence show that \(\frac { d x } { d t } + \frac { b x } { 1 + ( a - b ) t } = a\).
  2. First, consider the case where \(\mathbf { b } = \mathbf { a }\).
    1. Solve the differential equation to find \(x\) in terms of \(a\) and \(t\).
    2. Given that after 1 minute the vessel contains equal amounts of water and chemical, find the rate of inflow of chemical.
  3. Now consider the case where \(\mathrm { b } = 2 \mathrm { a }\).
    1. Explain why the differential equation in part (a)(ii) is now invalid for \(\mathrm { t } \geqslant \frac { 1 } { \mathrm { a } }\).
    2. Find the maximum amount of chemical in the vessel.
OCR MEI Further Pure Core Specimen Q1
1 Find the acute angle between the lines with vector equations \(\mathbf { r } = \left( \begin{array} { c } 3
0
- 2 \end{array} \right) + \lambda \left( \begin{array} { c } 1
2
- 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 1
5
3 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right)\).
OCR MEI Further Pure Core Specimen Q2
2
  1. On an Argand diagram draw the locus of points which satisfy \(\arg ( z - 4 \mathrm { i } ) = \frac { \pi } { 4 }\).
  2. Give, in complex form, the equation of the circle which has centre at \(6 + 4 \mathrm { i }\) and touches the locus in part (i).
OCR MEI Further Pure Core Specimen Q3
3 Transformation M is represented by matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 3
1 & 4 \end{array} \right)\).
  1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M .
  2. (A) Show that there is a constant \(k\) such that \(\mathbf { M } \binom { x } { k x } = 5 \binom { x } { k x }\) for all \(x\).
    (B) Hence find the equation of an invariant line under M .
    (C) Draw the invariant line from part (ii) (B) on your diagram for part (i).
OCR MEI Further Pure Core Specimen Q4
4 You are given that \(z = 1 + 2 \mathrm { i }\) is a root of the equation \(z ^ { 3 } - 5 z ^ { 2 } + q z - 15 = 0\), where \(q \in \mathbb { R }\). Find
  • the other roots,
  • the value of \(q\).
OCR MEI Further Pure Core Specimen Q5
5
  1. Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }\), expressing your answer as a single fraction.
OCR MEI Further Pure Core Specimen Q6
6
  1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t , y = \cosh t - \sinh t\) where \(t \in \mathbb { R }\). Show that the cartesian equation of the curve is \(x y = 1\). Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-3_931_1050_598_479} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  2. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer.
OCR MEI Further Pure Core Specimen Q8
8 Find the cartesian equation of the plane which contains the three points \(( 1,0 , - 1 ) , ( 2,2,1 )\) and \(( 1,1,2 )\).
OCR MEI Further Pure Core Specimen Q9
9 A curve has polar equation \(r = a \sin 3 \theta\) for \(- \frac { 1 } { 3 } \pi \leq \theta \leq \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
  1. Sketch the curve.
  2. In this question you must show detailed reasoning. Find, in terms of \(a\) and \(\pi\), the area enclosed by one of the loops of the curve.
  3. Obtain the solution to the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { 1 } { x } , \text { where } x > 0 ,$$ given that \(y = 1\) when \(x = 1\).
  4. Deduce that \(y\) decreases as \(x\) increases.
OCR MEI Further Pure Core Specimen Q11
11
  1. It is conjectured that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = a - \frac { b } { n ! } ,$$ where \(a\) and \(b\) are constants, and \(n\) is an integer such that \(n \geq 2\). By considering particular cases, show that if the conjecture is correct then \(a = b = 1\).
  2. Use induction to prove that $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = 1 - \frac { 1 } { n ! } \text { for } n \geq 2 .$$