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OCR Further Pure Core 1 2020 November Q6
5 marks Standard +0.8
6 The equations of two non-intersecting lines, \(l _ { 1 }\) and \(l _ { 2 }\), are \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { c } 1 \\ - 1 \\ 4 \end{array} \right)\).
Find the shortest distance between lines \(l _ { 1 }\) and \(l _ { 2 }\).
OCR Further Pure Core 1 2020 November Q7
5 marks Standard +0.3
7 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .
OCR Further Pure Core 1 2020 November Q8
10 marks Challenging +1.2
8
  1. Using exponentials, show that \(\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1\).
  2. By differentiating both sides of the identity in part (a) with respect to \(u\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\).
  3. Use the substitution \(\mathrm { x } = \sinh ^ { 2 } \mathrm { u }\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form asinh \(^ { - 1 } \mathrm {~b} \sqrt { \mathrm { x } } + \mathrm { f } ( \mathrm { x } )\) where \(a\) and \(b\) are integers and \(\mathrm { f } ( x )\) is a function to be determined.
  4. Hence determine the exact area of the region between the curve \(\mathrm { y } = \sqrt { \frac { \mathrm { x } } { \mathrm { x } + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(\mathrm { p } + \mathrm { q } \mid \mathrm { nr }\) where \(p , q\) and \(r\) are numbers to be determined.
OCR Further Pure Core 1 2020 November Q9
9 marks Challenging +1.2
9 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + \mathrm { i } \sqrt { 2 }\).
  1. Write down a second complex root, \(\beta\).
  2. Determine the third root, \(\gamma\).
  3. Find the value of \(p\) and the value of \(q\).
  4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).
OCR Further Pure Core 1 2020 November Q10
13 marks Standard +0.8
10 A particle of mass 0.5 kg is initially at point \(O\). It moves from rest along the \(x\)-axis under the influence of two forces \(F _ { 1 } \mathrm {~N}\) and \(F _ { 2 } \mathrm {~N}\) which act parallel to the \(x\)-axis. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\). \(F _ { 1 }\) is acting in the direction of motion of the particle and \(F _ { 2 }\) is resisting motion.
In an initial model
  • \(F _ { 1 }\) is proportional to \(t\) with constant of proportionality \(\lambda > 0\),
  • \(F _ { 2 }\) is proportional to \(v\) with constant of proportionality \(\mu > 0\).
    1. Show that the motion of the particle can be modelled by the following differential equation.
$$\frac { 1 } { 2 } \frac { d v } { d t } = \lambda t - \mu v$$
  • Solve the differential equation in part (a), giving the particular solution for \(v\) in terms of \(t\), \(\lambda\) and \(\mu\). You are now given that \(\lambda = 2\) and \(\mu = 1\).
  • Find a formula for an approximation for \(v\) in terms of \(t\) when \(t\) is large. In a refined model
    • \(F _ { 1 }\) is constant, acting in the direction of motion with magnitude 2 N ,
    • \(F _ { 2 }\) is as before with \(\mu = 1\).
    • Write down a differential equation for the refined model.
    • Without solving the differential equation in part (d), write down what will happen to the velocity in the long term according to this refined model.
    •
    OCR Further Pure Core 1 2020 November Q11
    8 marks Standard +0.8
    11 A curve has cartesian equation \(x ^ { 3 } + y ^ { 3 } = 2 x y\). \(C\) is the portion of the curve for which \(x \geqslant 0\) and \(y \geqslant 0\). The equation of \(C\) in polar form is given by \(r = \mathrm { f } ( \theta )\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
    1. Find \(f ( \theta )\).
    2. Find an expression for \(\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)\), giving your answer in terms of \(\sin \theta\) and \(\cos \theta\).
    3. Hence find the line of symmetry of \(C\).
    4. Find the value of \(r\) when \(\theta = \frac { 1 } { 4 } \pi\).
    5. By finding values of \(\theta\) when \(r = 0\), show that \(C\) has a loop.
    OCR Further Pure Core 1 2020 November Q12
    6 marks Challenging +1.8
    12 Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } d x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
    OCR Further Pure Core 1 2021 November Q1
    6 marks Moderate -0.3
    1
    1. Sketch on a single Argand diagram the loci given by
      1. \(\quad | z - 1 + 2 i | = 3\),
      2. \(\quad | z + 1 | = | z - 2 |\).
    2. Indicate, by shading, the region of the Argand diagram for which \(| z - 1 + 2 i | \leqslant 3\) and \(| z + 1 | \leqslant | z - 2 |\).
    OCR Further Pure Core 1 2021 November Q2
    8 marks Standard +0.3
    2 You are given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )\).
      1. Find the value of \(f ( 0 )\).
      2. Determine the value of \(f ^ { \prime } ( 0 )\).
      3. Show that \(f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }\).
    2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).
    OCR Further Pure Core 1 2021 November Q3
    8 marks Challenging +1.2
    3 A function \(\mathrm { f } ( \mathrm { z } )\) is defined on all complex numbers z by \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 3 \mathrm { z } ^ { 2 } + \mathrm { k } \mathrm { z } - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta\) and \(\gamma\). You are given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5\).
    1. Explain why \(f ( z ) = 0\) has only one real root.
    2. Find the value of \(k\).
    3. Find a cubic equation with integer coefficients that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
    OCR Further Pure Core 1 2021 November Q4
    11 marks Standard +0.8
    4 Points \(A , B\) and \(C\) have coordinates ( \(4,2,0\) ), ( \(1,5,3\) ) and ( \(1,4 , - 2\) ) respectively. The line \(l\) passes through \(A\) and \(B\).
    1. Find a cartesian equation for \(l\). \(M\) is the point on \(l\) that is closest to \(C\).
    2. Find the coordinates of \(M\).
    3. Find the exact area of the triangle \(A B C\).
    OCR Further Pure Core 1 2021 November Q5
    4 marks Standard +0.8
    5 Use de Moivre's theorem to find the constants \(A , B\) and \(C\) in the identity \(\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta\). \(6 O\) is the origin of a coordinate system whose units are cm .
    The points \(A , B , C\) and \(D\) have coordinates ( 1,0 ), ( 1,4 ), ( 6,9 ) and ( 0,9 ) respectively.
    The arc \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
    The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
    A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_641_1131_808_242} Find the volume of the funnel according to the model.
    OCR Further Pure Core 1 2021 November Q7
    9 marks Standard +0.8
    7 The diagram below shows the curve with polar equation \(r = \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_385_807_1834_260}
    1. Find the values of \(\theta\) at the pole.
    2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value.
    3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve.
    4. Given that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\), find a cartesian equation for the curve.
    OCR Further Pure Core 1 2021 November Q8
    8 marks Standard +0.3
    8 You are given that \(\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x\).
    1. Show that the curve \(y = f ( x )\) has no turning points.
    2. Determine the exact solution of the equation \(\mathrm { f } ( x ) = 5\).
    OCR Further Pure Core 1 2021 November Q9
    5 marks Standard +0.3
    9 You are given that the matrix \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation T .
    1. You are given that the line with equation \(\mathrm { y } = \mathrm { kx }\) is invariant under T . Determine the value of \(k\).
    2. Determine whether the line with equation \(\mathrm { y } = \mathrm { kx }\) in part (a) is a line of invariant points under T .
    OCR Further Pure Core 1 2021 November Q10
    8 marks Standard +0.8
    10 Using an algebraic method, determine the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } \geqslant 0.49\).
    OCR Further Pure Core 1 2021 November Q11
    5 marks Standard +0.8
    11 The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 3 \theta = 0\) where \(\lambda\) is a constant and \(\lambda \geqslant 0\).
      1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic?
      2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic.
    2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position.
    3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic.
    OCR Further Pure Core 1 Specimen Q1
    2 marks Easy -1.8
    1 Show that \(\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }\).
    OCR Further Pure Core 1 Specimen Q2
    5 marks Standard +0.3
    2 In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).
    1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
    2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).
    OCR Further Pure Core 1 Specimen Q3
    6 marks Challenging +1.2
    3 In this question you must show detailed reasoning. The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined. Calculate the exact area enclosed by the curve.
    OCR Further Pure Core 1 Specimen Q4
    3 marks Standard +0.3
    4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).
    OCR Further Pure Core 1 Specimen Q5
    5 marks Standard +0.8
    5
    1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }\).
    2. Find \(\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x\).
    OCR Further Pure Core 1 Specimen Q6
    5 marks Standard +0.8
    6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
    The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
    Find the values of \(p , q\) and \(r\).
    OCR Further Pure Core 1 Specimen Q7
    7 marks Challenging +1.2
    7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }\) and \(\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }\).
    1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
    OCR Further Pure Core 1 Specimen Q8
    8 marks Standard +0.3
    8
    1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
    2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r l } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$