Questions Further Pure Core 2 (129 questions)

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OCR Further Pure Core 2 2021 June Q5
11 marks Challenging +1.2
Two thin poles, \(OA\) and \(BC\), 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{figure_5} It is required to find the height of pole \(BC\) by modelling the shape of the curve that the chain forms. Jofra models the curve using the equation \(y = k \cosh(ax - b) - 1\) where \(k\), \(a\) and \(b\) are positive constants.
  1. Determine the value of \(k\). [2]
  2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. [5]
Holly models the curve using the equation \(y = \frac{1}{4}x^2 - 3x + 3\).
  1. Write down the coordinates of the point, \((u, v)\) where \(u\) and \(v\) are both non-zero, at which the two models will agree. [1]
  2. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(BC\) by \(3.32\)m to 3 significant figures. [3]
OCR Further Pure Core 2 2018 March Q1
8 marks Standard +0.3
Plane \(\Pi\) has equation \(3x - y + 2z = 33\). Line \(l\) has the following vector equation. $$l: \quad \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$$
  1. Find the acute angle between \(\Pi\) and \(l\). [3]
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\). [3]
  3. \(S\) is the point \((4, 5, -5)\). Find the shortest distance from \(S\) to \(\Pi\). [2]
OCR Further Pure Core 2 2018 March Q2
5 marks Moderate -0.8
The complex number \(2 + i\) is denoted by \(z\).
  1. Show that \(z^2 = 3 + 4i\). [2]
  2. Plot the following on the Argand diagram in the Printed Answer Booklet.
    [1]
  3. State the relationship between \(|z^2|\) and \(|z|\). [1]
  4. State the relationship between \(\arg(z^2)\) and \(\arg(z)\). [1]
OCR Further Pure Core 2 2018 March Q3
3 marks Standard +0.3
In this question you must show detailed reasoning. Use the formula \(\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1)\) to evaluate \(121^2 + 122^2 + 123^2 + \ldots + 300^2\). [3]
OCR Further Pure Core 2 2018 March Q4
4 marks Standard +0.8
You are given that the cubic equation \(2x^3 - 3x^2 + x + 4 = 0\) has three roots, \(\alpha\), \(\beta\) and \(\gamma\). By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
OCR Further Pure Core 2 2018 March Q5
5 marks Standard +0.8
In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [5]
OCR Further Pure Core 2 2018 March Q6
12 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
  2. By finding the second derivative, classify the stationary points found in part (i). [3]
OCR Further Pure Core 2 2018 March Q7
12 marks Challenging +1.2
In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}
  1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
  2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
    1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
    2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
    3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]
OCR Further Pure Core 2 2018 March Q8
12 marks Challenging +1.8
In this question you must show detailed reasoning. Show that \(\int_0^2 \frac{2x^2 + 3x - 1}{x^3 - 3x^2 + 4x - 12} dx = \frac{3}{8}\pi - \ln 9\). [12]
OCR Further Pure Core 2 2018 March Q9
14 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Show that \(e^{i\theta} - e^{-i\theta} = 2i\sin\theta\). [1]
  2. Hence, show that \(\frac{2}{e^{2i\theta} - 1} = -(1 + i\cot\theta)\). [3]
  3. Two series, \(C\) and \(S\), are defined as follows. $$C = 2 + 2\cos\frac{\pi}{10} + 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{10} + 2\cos\frac{2\pi}{5}$$ $$S = 2\sin\frac{\pi}{10} + 2\sin\frac{\pi}{5} + 2\sin\frac{3\pi}{10} + 2\sin\frac{2\pi}{5}$$ By considering \(C + iS\), find a simplified expression for \(C\) in terms of only integers and \(\cot\frac{\pi}{10}\). [8]
  4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. [2]
OCR Further Pure Core 2 2018 September Q1
8 marks Standard +0.3
Line \(l_1\) has Cartesian equation $$l_1: \frac{-x}{2} = \frac{y-5}{2} = \frac{-z-6}{7}.$$
  1. Find a vector equation for \(l_1\). [2]
Line \(l_2\) has vector equation $$l_2: \mathbf{r} = \begin{pmatrix} 2 \\ 7 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}.$$
  1. Find the point of intersection of \(l_1\) and \(l_2\). [3]
  2. Find the acute angle between \(l_1\) and \(l_2\). [3]
OCR Further Pure Core 2 2018 September Q2
5 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find \(\int_{-\frac{3\pi}{4}}^{\frac{3\pi}{4}} 2\tan x \, dx\) giving your answer in the form \(\ln p\). [3]
  2. Show that \(\int_0^{\frac{3\pi}{4}} 2\tan x \, dx\) is undefined explaining your reasoning. [2]
OCR Further Pure Core 2 2018 September Q3
6 marks Standard +0.3
The equation of a plane, \(\Pi\), is $$\Pi: \mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}.$$
  1. Find a vector which is perpendicular to \(\Pi\). [2]
  2. Hence find an equation for \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
  3. Find in the form \(\sqrt{q}\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. [2]
OCR Further Pure Core 2 2018 September Q4
10 marks Challenging +1.2
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}\) where \(a\) is a constant. It is given that if \(\mathbf{A}\) is not singular then $$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$
  1. Use \(\mathbf{A}^{-1}\) to solve the simultaneous equations below, giving your answer in terms of \(k\). \begin{align} x + 2y + 3z &= 6
    4x + 4y + 6z &= 8
    -2x + 2y + 9z &= k \end{align} [3]
  2. Consider the equations below where \(a\) takes the value which makes \(\mathbf{A}\) singular. \begin{align} ax + 2y + 3z &= b
    4x + 4y + 6z &= 10
    -2x + 2y + 9z &= -13 \end{align} \(b\) takes the value for which the equations have an infinite number of solutions.
  3. For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]
OCR Further Pure Core 2 2018 September Q5
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p+x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form
OCR Further Pure Core 2 2018 September Q6
8 marks Challenging +1.2
  1. By considering \(\sum_{r=1}^n ((r+1)^5 - r^5)\) show that \(\sum_{r=1}^n r^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1)\). [6]
  2. Use the formula given in part (i) to find \(50^4 + 51^4 + \ldots + 80^4\). [2]
OCR Further Pure Core 2 2018 September Q7
9 marks Challenging +1.2
The roots of the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).
  1. Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha\beta\). [4]
  2. Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots. [2]
  3. Show that the discriminant of the equation found in part (i) is always positive. [3]
OCR Further Pure Core 2 2018 September Q8
6 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Express \((6+5i)(7+5i)\) in the form \(a+bi\). [2]
  2. You are given that \(17^2 + 65^2 = 4514\). Using the result in part (i) and by considering \((6-5i)(7-5i)\) express \(4514\) as a product of its prime factors. [4]
OCR Further Pure Core 2 2018 September Q9
15 marks Challenging +1.2
The quantity of grass on an island at time \(t\) years is \(x\), in appropriate units. At time \(t = 0\) some rabbits are introduced to the island. The population of rabbits on the island at time \(t\) years is \(y\), in units of \(100\)s of rabbits. An ecologist who is studying the island suggests that the following pair of simultaneous first order differential equations can be used to model the population of rabbits and quantity of grass for \(t \geq 0\). $$\frac{dx}{dt} = 3x - 2y,$$ $$\frac{dy}{dt} = y + 5x$$
    1. Show that \(\frac{d^2x}{dt^2} = a\frac{dx}{dt} + bx\) where \(a\) and \(b\) are constants which should be found. [2]
    2. Find the general solution for \(x\) in real form. [3]
  2. Find the corresponding general solution for \(y\). [3]
At time \(t = 0\) the quantity of grass on the island was \(4\) units. The number of rabbits introduced at this time was \(500\).
  1. Find the particular solutions for \(x\) and \(y\). [5]
  2. The ecologist finds that the model predicts that there will be no grass at time \(T\), when there are still rabbits on the island. Find the value of \(T\). [1]
  3. State one way in which the model is not appropriate for modelling the quantity of grass and the population of rabbits for \(0 \leq t \leq T\). [1]
OCR Further Pure Core 2 2018 December Q1
6 marks Easy -1.2
  1. The Argand diagram below shows the two points which represent two complex numbers, \(z_1\) and \(z_2\). \includegraphics{figure_1} On the copy of the diagram in the Printed Answer Booklet
    • draw an appropriate shape to illustrate the geometrical effect of adding \(z_1\) and \(z_2\),
    • indicate with a cross (\(\times\)) the location of the point representing the complex number \(z_1 + z_2\).
    [2]
  2. You are given that \(\arg z_3 = \frac{1}{4}\pi\) and \(\arg z_4 = \frac{3}{8}\pi\). In each part, sketch and label the points representing the numbers \(z_3\), \(z_4\) and \(z_3z_4\) on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
    1. \(|z_3| = 1.5\) and \(|z_4| = 1.2\) [2]
    2. \(|z_3| = 0.7\) and \(|z_4| = 0.5\) [2]
OCR Further Pure Core 2 2018 December Q2
8 marks Moderate -0.8
In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the \(x\)-axis. A is the matrix which represents S.
  1. Write down A. [1]
  2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]
Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.
  1. Describe T. [1]
  2. Determine the matrix which represents the transformation S followed by T. [2]
  3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]
OCR Further Pure Core 2 2018 December Q3
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the equation \(2\cosh^2 x + 5\sinh x - 5 = 0\) giving each answer in the form \(\ln(p + q\sqrt{r})\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. [6]
OCR Further Pure Core 2 2018 December Q4
6 marks Standard +0.3
You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.
  1. State the plane of reflection of R. [1]
  2. Determine the value of \(a\). [3]
  3. With reference to R explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3\times 3\) identity matrix. [2]
OCR Further Pure Core 2 2018 December Q5
7 marks Standard +0.8
  1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]
Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}.$$
  1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
  2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]
OCR Further Pure Core 2 2018 December Q6
9 marks Standard +0.3
Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.
  1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
  2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]
For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)
  1. Find
    [3]