Questions Further Pure Core 1 (136 questions)

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OCR Further Pure Core 1 2021 November Q6
3 marks Standard +0.8
\(O\) 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 \(BC\) is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape \(OABCD\) is formed, in turn, from the line segments \(OA\) and \(AB\), the arc \(BC\) and the line segments \(CD\) and \(DO\) (see diagram). A funnel can be modelled by rotating \(OABCD\) by \(2\pi\) radians about the \(y\)-axis. \includegraphics{figure_6} Find the volume of the funnel according to the model. [3]
OCR Further Pure Core 1 2021 November Q7
9 marks Standard +0.8
The diagram below shows the curve with polar equation \(r = \sin 3\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{3}\pi\). \includegraphics{figure_7}
  1. Find the values of \(\theta\) at the pole. [1]
  2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value. [2]
  3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve. [4]
  4. Given that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\), find a cartesian equation for the curve. [2]
OCR Further Pure Core 1 2021 November Q8
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = 4 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm{f}(x)\) has no turning points. [3]
  2. Determine the exact solution of the equation \(\mathrm{f}(x) = 5\). [5]
OCR Further Pure Core 1 2021 November Q9
5 marks Standard +0.3
You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.
  1. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of \(k\). [4]
  2. Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]
OCR Further Pure Core 1 2021 November Q10
8 marks Challenging +1.2
Using an algebraic method, determine the least value of \(n\) for which \(\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49\). [8]
OCR Further Pure Core 1 2021 November Q11
5 marks Standard +0.3
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? [1]
    2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic. [1]
  2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position. [2]
  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. [1]
OCR Further Pure Core 1 2021 June Q1
9 marks Standard +0.3
The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_1}
  1. The greatest value of \(r\) on the curve occurs at the point \(P\).
    1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
    2. Find the value of \(r\) at the point \(P\). [1]
    3. Mark the point \(P\) on a copy of the graph. [1]
  2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]
OCR Further Pure Core 1 2021 June Q2
7 marks Standard +0.3
You are given that \(f(x) = \ln(2 + x)\).
  1. Determine the exact value of \(f'(0)\). [2]
  2. Show that \(f''(0) = -\frac{1}{4}\). [2]
  3. Hence write down the first three terms of the Maclaurin series for \(f(x)\). [3]
OCR Further Pure Core 1 2021 June Q3
6 marks Standard +0.3
You are given that \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}\).
  1. Find \(\mathbf{AB}\). [1]
  2. Hence write down \(\mathbf{A}^{-1}\). [1]
  3. You are given three simultaneous equations $$x + 2y + z = 0$$ $$2x + 5y + 2z = 1$$ $$3x - 2y - z = 4$$
    1. Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
    2. Find this unique solution. [2]
OCR Further Pure Core 1 2021 June Q4
9 marks Standard +0.3
  1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
  2. Solve the equation \(4\tanh^2 x + \tanh x - 3 = 0\), giving the solution in the form \(a\ln b\) where \(a\) and \(b\) are rational numbers to be determined. [4]
  3. Explain why the equation in part (b) has only one root. [1]
OCR Further Pure Core 1 2021 June Q5
7 marks Challenging +1.2
In this question you must show detailed reasoning. Find \(\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx\) giving your answer in the form \(\ln(p + q\sqrt{2})\) where \(p\) and \(q\) are integers to be determined. [7]