Questions Further Pure Core 2 (129 questions)

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OCR Further Pure Core 2 Specimen Q1
4 marks Standard +0.3
Find \(\sum_{r=1}^{n}(r+1)(r+5)\). Give your answer in a fully factorised form. [4]
OCR Further Pure Core 2 Specimen Q2
4 marks Standard +0.8
In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac{8}{\sqrt{16+x^3}}\), the \(x\)-axis and the lines \(x=0\) and \(x=4\). Region \(R\) is rotated through \(360°\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
OCR Further Pure Core 2 Specimen Q3
4 marks Standard +0.3
\begin{enumerate}[label=(\roman*)] \item Find \(\sum_{r=1}^{n}\left(\frac{1}{r}-\frac{1}{r+2}\right)\). [3] \item What does the sum in part (i) tend to as \(n \to \infty\)? Justify your answer. [1]
OCR Further Pure Core 2 Specimen Q4
5 marks Challenging +1.2
It is given that \(\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}\) where \(k\), \(A\), \(B\) and \(C\) are positive integers. Determine the set of possible values of \(k\). [5]
OCR Further Pure Core 2 Specimen Q5
4 marks Standard +0.8
In this question you must show detailed reasoning. Evaluate \(\int_0^{\infty} 2xe^{-x} dx\). [You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]
OCR Further Pure Core 2 Specimen Q6
8 marks Standard +0.3
The equation of a plane \(\Pi\) is \(x-2y-z=30\). \begin{enumerate}[label=(\roman*)] \item Find the acute angle between the line \(\mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}\) and \(\Pi\). [4] \item Determine the geometrical relationship between the line \(\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}\) and \(\Pi\). [4]
OCR Further Pure Core 2 Specimen Q7
7 marks Challenging +1.8
\begin{enumerate}[label=(\roman*)] \item Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2x \sin 4x\) up to and including the term in \(x^3\). [4] \item Hence find, in exact surd form, an approximation to the least positive root of the equation \(2\sin x \sin 2x \sin 4x = x\). [3]
OCR Further Pure Core 2 Specimen Q8
8 marks Challenging +1.2
The equation of a curve is \(y = \cosh^2 x - 3\sinh x\). Show that \(\left(\ln\left(\frac{3+\sqrt{13}}{2}\right), -\frac{5}{4}\right)\) is the only stationary point on the curve. [8]
OCR Further Pure Core 2 Specimen Q9
6 marks Standard +0.8
A curve has equation \(x^4 + y^4 = x^2 + y^2\), where \(x\) and \(y\) are not both zero. \begin{enumerate}[label=(\roman*)] \item Show that the equation of the curve in polar coordinates is \(r^2 = \frac{2}{2-\sin^2 2\theta}\). [4] \item Deduce that no point on the curve \(x^4 + y^4 = x^2 + y^2\) is further than \(\sqrt{2}\) from the origin. [2]
OCR Further Pure Core 2 Specimen Q10
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]
OCR Further Pure Core 2 Specimen Q11
17 marks Challenging +1.2
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x + 0.2y \text{ and } \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
    1. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). [2]
    2. Comment on this result in the context of the industrial process. [2]
  3. Express \(y\) in terms of \(t\). [5]
  4. Determine the maximum amount of substance \(Y\) present during the process. [3]
  5. How long does it take to produce 9 kg of substance \(Z\)? [2]
OCR Further Pure Core 2 2021 June Q1
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 2021 June Q2
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 2021 June Q3
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
OCR Further Pure Core 2 2021 June Q4
7 marks Challenging +1.8
\includegraphics{figure_4} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the x-axis is rotated by \(2\pi\) radians around the x-axis to form a solid of revolution, S. Determine the exact volume of S. [7]
OCR Further Pure Core 2 2021 June Q5
7 marks Challenging +1.8
\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]
OCR Further Pure Core 2 2021 June Q1
8 marks Standard +0.3
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]
OCR Further Pure Core 2 2021 June Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]
OCR Further Pure Core 2 2021 June Q3
6 marks Standard +0.3
\(A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after \(0.2\) seconds at point \(B\) whose displacement is \(0.2\) m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x\) m.
  1. Sketch a graph of \(x\) against \(t\) for \(0 \leq t \leq 0.4\). [4]
  2. Find the displacement of \(P\) from \(M\) at \(0.75\) seconds after release. [2]
OCR Further Pure Core 2 2021 June Q4
7 marks Standard +0.8
In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square \(S\).
  1. Find the area of \(S\). [3]
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
OCR Further Pure Core 2 2021 June Q5
11 marks Challenging +1.3
In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin\theta}e^{\cos\theta}\) for \(0 \leq \theta < \pi\). \includegraphics{figure_5}
  1. Find the exact area enclosed by the curve. [4]
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}}e^{\frac{1}{2}}\). [7]
OCR Further Pure Core 2 2021 June Q1
5 marks Moderate -0.5
In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]
OCR Further Pure Core 2 2021 June Q2
9 marks Standard +0.3
The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).
  1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
  2. Determine the exact value of the length of \(QR\). [7]
OCR Further Pure Core 2 2021 June Q3
7 marks Standard +0.3
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 \(t \geq 0\), \(Q\) can be modelled by the differential equation $$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$ 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 \(t = 0.5\). [6]
  2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. [1]
OCR Further Pure Core 2 2021 June Q4
6 marks Standard +0.8
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).
  1. Find \(\det \mathbf{A}\). [1]
The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
  1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
  2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
  3. Find the angle of rotation. [2]