Questions Further Pure Core 2 (116 questions)

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OCR Further Pure Core 2 2024 June Q5
5 Vectors, \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), are given by \(\mathbf { a } = \mathbf { i } + ( 1 - p ) \mathbf { j } + ( p + 2 ) \mathbf { k } , \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { c } = \mathbf { i } + 14 \mathbf { j } + ( p - 3 ) \mathbf { k }\) where \(p\) is a constant. You are given that \(\mathbf { a } \times \mathbf { b }\) is perpendicular to \(\mathbf { c }\). Determine the possible values of \(p\).
OCR Further Pure Core 2 2024 June Q6
6 In polar coordinates, the equation of a curve, \(C\), is \(r = 6 \sin ( 2 \theta ) \sinh \left( \frac { 1 } { 3 } \theta \right)\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.
  1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. The incomplete table below shows values of \(r\) for various values of \(\theta\).
    \(\theta\)0\(\frac { 1 } { 12 } \pi\)\(\frac { 1 } { 6 } \pi\)\(\frac { 1 } { 4 } \pi\)\(\frac { 1 } { 3 } \pi\)\(\frac { 5 } { 12 } \pi\)\(\frac { 1 } { 2 } \pi\)
    \(r\)00.2621.851
  2. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
  3. Show that \(\phi\) satisfies the equation \(\phi = \frac { 3 } { 2 } \ln \left( \frac { 6 - \tan 2 \phi } { 6 + \tan 2 \phi } \right)\)
  4. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to \(\mathbf { 3 }\) significant figures.
OCR Further Pure Core 2 2024 June Q7
7
  1. Express \(17 \cosh x - 15 \sinh x\) in the form \(\mathrm { e } ^ { - \mathrm { x } } \left( \mathrm { ae } ^ { \mathrm { bx } } + \mathrm { c } \right)\) where \(a , b\) and \(c\) are integers to be determined. A function is defined by \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 17 \cosh x - 15 \sinh x } }\). The region bounded by the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\). \section*{(b) In this question you must show detailed reasoning.} Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k \pi \tan ^ { - 1 } q\) where \(k\) and \(q\) are rational numbers to be determined.
OCR Further Pure Core 2 2024 June Q9
9 In this question, the argument of a complex number is defined as being in the range \([ 0,2 \pi )\).
You are given that \(\omega _ { k }\), where \(k = 0,1,2 , \ldots , n - 1\), are the \(n n ^ { \text {th } }\) roots of unity for some integer \(n , n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega _ { 0 } = 1\) ).
  1. With the help of a diagram explain why \(\omega _ { k } = \left( \omega _ { 1 } \right) ^ { k }\) for \(k = 2 , \ldots , n - 1\).
  2. Using the identity given in part (a), show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \omega _ { \mathrm { k } } = 0\).
  3. Show that if \(z\) is a complex number then \(z + z ^ { * } = 2 \operatorname { Re } ( z )\).
  4. Using the results from parts (b) and (c) show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = 0\).
  5. With the help of a diagram explain why \(\operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = \operatorname { Re } \left( \omega _ { \mathrm { n } - \mathrm { k } } \right)\) for \(k = 1,2 , \ldots , n - 1\). You should now consider the case where \(n = 5\).
    1. Use parts (d) and (e) to deduce that \(\cos \frac { 4 \pi } { 5 } = \mathrm { a } + \mathrm { b } \cos \frac { 2 \pi } { 5 }\), for some rational constants \(a\) and \(b\).
    2. Hence determine the exact value of \(\cos \frac { 2 \pi } { 5 }\).
OCR Further Pure Core 2 2020 November Q1
1 In this question you must show detailed reasoning.
Solve the equation \(4 z ^ { 2 } - 20 z + 169 = 0\). Give your answers in modulus-argument form.
OCR Further Pure Core 2 2020 November Q2
2 In this question you must show detailed reasoning.
The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
  2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).
OCR Further Pure Core 2 2020 November Q3
3 In this question you must show detailed reasoning.
  1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
  2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).
OCR Further Pure Core 2 2020 November Q4
4 The equations of two intersecting lines \(l _ { 1 }\) and \(l _ { 2 }\) are
\(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1
0
a \end{array} \right) + \lambda \left( \begin{array} { r } 2
1
- 3 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 7
9
- 2 \end{array} \right) + \mu \left( \begin{array} { r } - 1
1
2 \end{array} \right)\)
where \(a\) is a constant.
The equation of the plane \(\Pi\) is
r. \(\left( \begin{array} { l } 1
5
3 \end{array} \right) = - 14\).
\(l _ { 1 }\) and \(\Pi\) intersect at \(Q\).
\(l _ { 2 }\) and \(\Pi\) intersect at \(R\).
  1. Verify that the coordinates of \(R\) are (13, 3, -14).
  2. Determine the exact value of the length of \(Q R\).
OCR Further Pure Core 2 2020 November Q5
5 A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geqslant 0 , Q\) can be modelled by the differential equation \(\frac { d ^ { 2 } Q } { d t ^ { 2 } } - 2 \frac { d Q } { d t } - 15 Q = 0\). Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.
  1. Determine the charge on the capacitor when \(t = 0.5\).
  2. Determine the finite limit of \(Q\) as \(t\) tends to infinity.
OCR Further Pure Core 2 2020 November Q6
6 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.
  1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
  2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
  3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.
OCR Further Pure Core 2 2020 November Q7
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4
- 0.8 & 1.8 \end{array} \right)\).
  1. Find \(\operatorname { det } \mathbf { A }\). The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
  2. By considering the determinants of these transformations, determine the scale factor of the stretch.
  3. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
  4. Find the angle of rotation.
OCR Further Pure Core 2 2020 November Q9
9 Two thin poles, \(O A\) and \(B C\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \(( 0,3 ) , ( 5,0 )\) and \(( 2,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{c07ba83a-75fa-42dc-9bfd-6fc2f9226a23-5_805_1554_452_258} It is required to find the height of pole \(B C\) by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation \(\mathrm { y } = \mathrm { k } \cosh ( \mathrm { ax } - \mathrm { b } ) - 1\) where \(k , a\) and \(b\) are positive constants.
  1. Determine the value of \(k\).
  2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. Holly models the curve using the equation \(y = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 3\).
  3. Write down the coordinates of the point, \(( u , v )\) where \(u\) and \(v\) are both non-zero, at which the two models will agree.
  4. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(B C\) by 3.32 m to 3 significant figures.
OCR Further Pure Core 2 2020 November Q10
10 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
    1. Determine \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
    3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
  2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.
OCR Further Pure Core 2 2021 November Q1
1 Two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), are given by \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & - 1
2 & - 3 & 1
a & 1 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 6 & 3 & - 4
- 1 & 6 & - 4
8 & - 8 & - 1 \end{array} \right)\) where \(a\) is a constant. Find the value of \(a\) for which \(\mathbf { A B } = \mathbf { B A }\).
OCR Further Pure Core 2 2021 November Q3
3 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1
- 3
3 \end{array} \right) + \lambda \left( \begin{array} { r } 3
2
- 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2
- 5
- 3 \end{array} \right) = 4\).
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
  2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\).
    \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\).
    \(l _ { 2 }\) is the line with the following properties.
    • \(l _ { 2 }\) passes through \(A\)
    • \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
    • \(l _ { 2 }\) is parallel to \(\Pi\)
    • Find, in vector form, the equation of \(l _ { 2 }\).
OCR Further Pure Core 2 2021 November Q4
4 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 100 } ( 2 r + 3 ) ^ { 2 }\).
OCR Further Pure Core 2 2021 November Q5
5 In this question you must show detailed reasoning.
  1. Using the definition of \(\cosh x\) in terms of exponentials, show that \(\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1\).
  2. Solve the equation \(\cosh 2 x = 3 \cosh x + 1\), giving all your answers in exact logarithmic form.
OCR Further Pure Core 2 2021 November Q6
6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right)\).
  1. Define the transformation represented by \(\mathbf { A }\).
  2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2
    21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.
    • A shear which leaves the \(y\)-axis invariant and which transforms the point \(( 1,1 )\) to the point (1, 4).
    • The transformation represented by \(\mathbf { A }\).
    • A stretch of scale factor \(p\) which leaves the \(x\)-axis invariant.
    • Determine the value of \(p\).
OCR Further Pure Core 2 2021 November Q7
7 In this question you must show detailed reasoning.
  1. Find the values of \(A , B\) and \(C\) for which \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\).
  2. Hence express \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\) using partial fractions.
  3. Using your answer to part (b), determine \(\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x\) expressing your answer in the form \(a + \ln b + c \pi\) where \(a\) is an integer, and \(b\) and \(c\) are both rational.
OCR Further Pure Core 2 2021 November Q8
8 A particle \(P\) of mass 2 kg can only move along the straight line segment \(O A\), where \(O A\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(O A\) is 0.9 m . When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). \(P\) is subject to a force of magnitude \(4 \mathrm { e } ^ { - 2 t } \mathrm {~N}\) in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2 , v = 0.5\) and the resultant force on \(P\) is 0 N .
  1. Show that, according to the model, \(\frac { d v } { d t } + v = 2 e ^ { - 2 t }\).
  2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
  3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\) 's speed must reach a maximum value for some \(t > 0\).
  4. Determine the maximum speed considered in part (c).
  5. Determine the greatest value of \(t\) for which the model is valid.
OCR Further Pure Core 2 2021 November Q9
9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3
0 & 2 \end{array} \right)\).
  1. By considering \(\mathbf { A } , \mathbf { A } ^ { 2 } , \mathbf { A } ^ { 3 }\) and \(\mathbf { A } ^ { 4 }\) make a conjecture about the form of the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\) for \(n \geqslant 1\).
  2. Use induction to prove the conjecture made in part (a).
OCR Further Pure Core 2 2021 November Q10
10 In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
  2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
OCR Further Pure Core 2 Specimen Q1
1 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 )\). Give your answer in a fully factorised form.
OCR Further Pure Core 2 Specimen Q2
4 marks
2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
  1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
  2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.
OCR Further Pure Core 2 Specimen Q4
4 It is given that \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + k x } \equiv \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + k }\) where \(k , A , B\) and \(C\) are positive integers.
Determine the set of possible values of \(k\).