Questions — OCR Further Pure Core 1 (136 questions)

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OCR Further Pure Core 1 2021 June Q5
16 marks Challenging +1.2
5 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. \(\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x\)
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
    1. Find the general solution for \(x\).
    2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
  3. Find the particular solutions for \(x\) and \(y\).
  4. Determine whether the model predicts that the predators will die out before the prey.
OCR Further Pure Core 1 2021 June Q1
4 marks Standard +0.3
1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\). Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
OCR Further Pure Core 1 2021 June Q3
6 marks Standard +0.8
3 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\). You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
  2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
  3. For each of the values of \(a\) found in part (b) determine whether the equations have
    A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t\) where \(x\) is the displacement of the particle below the equilibrium position at time \(t\).
    When \(t = 0\) the particle is stationary and its displacement is 2 .
    1. Find the particular solution of the differential equation.
    2. Write down an approximate equation for the displacement when \(t\) is large.
OCR Further Pure Core 1 2021 June Q1
3 marks Standard +0.3
1 Indicate by shading on an Argand diagram the region $$\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \} .$$
OCR Further Pure Core 1 2021 June Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.
OCR Further Pure Core 1 2021 June Q3
7 marks Standard +0.3
3 The diagram shows part of the curve \(y = 5 \cosh x + 3 \sinh x\). \includegraphics[max width=\textwidth, alt={}, center]{ef967953-70b5-4dd1-a342-ad488b5fa79f-02_426_661_906_260}
  1. Solve the equation \(5 \cosh x + 3 \sinh x = 4\) giving your solution in exact form.
  2. In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x\) giving your answer in the form \(a \mathrm { e } + \frac { b } { \mathrm { e } }\) where \(a\) and \(b\) are integers to be determined.
OCR Further Pure Core 1 2021 June Q4
6 marks Challenging +1.2
4 You are given that \(y = \tan ^ { - 1 } \sqrt { 2 x }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 2 } } \frac { \sqrt { x } } { \left( x + 2 x ^ { 2 } \right) } \mathrm { d } x = k \pi\) where \(k\) is a number to be determined in exact form.
OCR Further Pure Core 1 2021 June Q5
6 marks Standard +0.8
5 The function \(\operatorname { sech } x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).
  1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
  2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).
OCR Further Pure Core 1 2021 June Q6
12 marks Challenging +1.2
6 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).
  1. Show that \(\omega\) is a root of the equation.
  2. Write down the other four roots of the equation.
  3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
  4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
  5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found. Total Marks for Question Set 4: 38
OCR Further Pure Core 1 2021 June Q2
4 marks Moderate -0.8
2 In this question you must show detailed reasoning.
  1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
  2. Illustrate the number 25 i and its square roots on an Argand diagram.
OCR Further Pure Core 1 2021 June Q3
5 marks Challenging +1.2
3 By expanding \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).
OCR Further Pure Core 1 2021 June Q4
5 marks Standard +0.8
4 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 2021 June Q5
5 marks Standard +0.3
5 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .
OCR Further Pure Core 1 2021 June Q6
9 marks Challenging +1.2
6 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 \(\alpha = 1 + i \sqrt { 2 }\).
  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 2021 June Q7
8 marks Standard +0.8
7 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 2021 June Q1
3 marks Standard +0.8
1 Find an expression for \(1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }\) in terms of \(n\). Give your answer in fully factorised form.
OCR Further Pure Core 1 2021 June Q2
5 marks Moderate -0.3
2
You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { 4 }\).
  2. Describe the transformation that A represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
  3. Write down the matrix \(B\). The point \(P\) has coordinates \(( 2,3,4 )\). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
  4. Find the coordinates of \(P ^ { \prime }\).
OCR Further Pure Core 1 2021 June Q3
10 marks Challenging +1.2
3
  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 \(x = \sinh ^ { 2 } u\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form \(a \sinh ^ { - 1 } b \sqrt { x } + \mathrm { f } ( x )\) 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 \(y = \sqrt { \frac { x } { x + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(p + q \ln r\) where \(p , q\) and \(r\) are numbers to be determined.
OCR Further Pure Core 1 2021 June Q4
13 marks Standard +0.3
4 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.
$$\frac { 1 \mathrm {~d} v } { 2 \mathrm {~d} t } = \lambda t - \mu v$$
  • 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
    • OCR Further Pure Core 1 2021 June Q5
    6 marks Challenging +1.2
    5
    Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
    OCR Further Pure Core 1 2021 November Q1
    6 marks Moderate -0.8
    1. Sketch on a single Argand diagram the loci given by
      1. \(|z - 1 + 2\mathrm{i}| = 3\), [2]
      2. \(|z + 1| = |z - 2|\). [2]
    2. Indicate, by shading, the region of the Argand diagram for which \(|z - 1 + 2\mathrm{i}| \leqslant 3\) and \(|z + 1| \leqslant |z - 2|\). [2]
    OCR Further Pure Core 1 2021 November Q2
    8 marks Standard +0.3
    You are given that \(\mathrm{f}(x) = \tan^{-1}(1 + x)\).
      1. Find the value of \(\mathrm{f}(0)\). [1]
      2. Determine the value of \(\mathrm{f}'(0)\). [2]
      3. Show that \(\mathrm{f}''(0) = -\frac{1}{2}\). [3]
    2. Hence find the Maclaurin series for \(\mathrm{f}(x)\) up to and including the term in \(x^2\). [2]
    OCR Further Pure Core 1 2021 November Q3
    8 marks Standard +0.8
    A function \(\mathrm{f}(z)\) is defined on all complex numbers \(z\) by \(\mathrm{f}(z) = z^3 - 3z^2 + kz - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm{f}(z) = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha^2 + \beta^2 + \gamma^2 = -5\).
    1. Explain why \(\mathrm{f}(z) = 0\) has only one real root. [3]
    2. Find the value of \(k\). [3]
    3. Find a cubic equation with integer coefficients that has roots \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\). [2]
    OCR Further Pure Core 1 2021 November Q4
    11 marks Standard +0.3
    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 \(l\). [3]
    \(M\) is the point on \(l\) that is closest to \(C\).
    1. Find the coordinates of \(M\). [4]
    2. Find the exact area of the triangle \(ABC\). [4]
    OCR Further Pure Core 1 2021 November Q5
    4 marks Standard +0.8
    Use de Moivre's theorem to find the constants \(A\), \(B\) and \(C\) in the identity \(\sin^3 \theta \equiv A \sin \theta + B \sin 3\theta + C \sin 5\theta\). [4]