Questions — OCR FP2 (173 questions)

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OCR FP2 2009 January Q5
8 marks Challenging +1.2
\includegraphics{figure_5} The diagram shows the curve with equation \(y = f(x)\), where $$f(x) = 2x^3 - 9x^2 + 12x - 4.36.$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha\), \(x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).
  1. The Newton-Raphson method is to be used to find the roots of the equation \(f(x) = 0\), with \(x_1 = k\).
    1. To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\)? [2]
    2. What happens if \(1 < k < 2\)? [2]
  2. Sketch the curve with equation \(y^2 = f(x)\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points. [4]
OCR FP2 2009 January Q6
8 marks Standard +0.3
  1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$1 + 2\sinh^2 x = \cosh 2x.$$ [3]
  2. Solve the equation $$\cosh 2x - 5\sinh x = 4,$$ giving your answers in logarithmic form. [5]
OCR FP2 2009 January Q7
8 marks Challenging +1.3
\includegraphics{figure_7} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2\cos \theta, \quad \text{for } 0 \leq \theta < 2\pi.$$ The points \(P\), \(Q\), \(R\) and \(S\) on the curve are such that the straight lines \(POR\) and \(QOS\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates \((r, \alpha)\).
  1. Show that \(OP + OQ + OR + OS = k\), where \(k\) is a constant to be found. [3]
  2. Given that \(\alpha = \frac{1}{4}\pi\), find the exact area bounded by the curve and the lines \(OP\) and \(OQ\) (shaded in the diagram). [5]
OCR FP2 2009 January Q8
11 marks Standard +0.8
\includegraphics{figure_8} The diagram shows the curve with equation \(y = \frac{1}{x+1}\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.
  1. By considering the areas of these rectangles, explain why $$\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} < \ln(n+1).$$ [5]
  2. By considering the areas of another set of rectangles, show that $$1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > \ln(n+1).$$ [2]
  3. Hence show that $$\ln(n+1) + \frac{1}{n+1} < \sum_{r=1}^{n+1} \frac{1}{r} < \ln(n+1) + 1.$$ [2]
  4. State, with a reason, whether \(\sum_{r=1}^{\infty} \frac{1}{r}\) is convergent. [2]
OCR FP2 2009 January Q9
12 marks Standard +0.8
A curve has equation $$y = \frac{4x - 3a}{2(x^2 + a^2)},$$ where \(a\) is a positive constant.
  1. Explain why the curve has no asymptotes parallel to the \(y\)-axis. [2]
  2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve. [5]
  3. Find the exact value of \(\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx\), showing that it is independent of \(a\). [5]
OCR FP2 2010 January Q1
5 marks Standard +0.3
It is given that \(f(x) = x^2 - \sin x\).
  1. The iteration \(x_{n+1} = \sqrt{\sin x_n}\), with \(x_1 = 0.875\), is to be used to find a real root, \(\alpha\), of the equation \(f(x) = 0\). Find \(x_2, x_3\) and \(x_4\), giving the answers correct to 6 decimal places. [2]
  2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 0.876726\), correct to 6 decimal places, find \(e_3\) and \(e_4\). Given that \(g(x) = \sqrt{\sin x}\), use \(e_3\) and \(e_4\) to estimate \(g'(\alpha)\). [3]
OCR FP2 2010 January Q2
6 marks Standard +0.3
It is given that \(f(x) = \tan^{-1}(1 + x)\).
  1. Find \(f(0)\) and \(f'(0)\), and show that \(f''(0) = -\frac{1}{2}\). [4]
  2. Hence find the Maclaurin series for \(f(x)\) up to and including the term in \(x^2\). [2]
OCR FP2 2010 January Q3
7 marks Moderate -0.3
\includegraphics{figure_3} A curve with no stationary points has equation \(y = f(x)\). The equation \(f(x) = 0\) has one real root \(\alpha\), and the Newton-Raphson method is to be used to find \(\alpha\). The tangent to the curve at the point \((x_1, f(x_1))\) meets the \(x\)-axis where \(x = x_2\) (see diagram).
  1. Show that \(x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\). [3]
  2. Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation \(x = x_1\), gives a sequence of approximations approaching \(\alpha\). [2]
  3. Use the Newton-Raphson method, with a first approximation of 1, to find a second approximation to the root of \(x^2 - 2\sinh x + 2 = 0\). [2]
OCR FP2 2010 January Q4
7 marks Standard +0.8
The equation of a curve, in polar coordinates, is $$r = e^{-2\theta}, \quad \text{for } 0 \leq \theta \leq \pi.$$
  1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value. [2]
  2. The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) respectively, lie on the curve. Given that \(\theta_2 > \theta_1\), show that the area of the region enclosed by the curve and the lines \(OP\) and \(OQ\) can be expressed as \(k(r_1^2 - r_2^2)\), where \(k\) is a constant to be found. [5]
OCR FP2 2010 January Q5
8 marks Standard +0.3
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$\cosh^2 x - \sinh^2 x \equiv 1.$$ Deduce that \(1 - \tanh^2 x \equiv \operatorname{sech}^2 x\). [4]
  2. Solve the equation \(2\tanh^2 x - \operatorname{sech} x = 1\), giving your answer(s) in logarithmic form. [4]
OCR FP2 2010 January Q6
9 marks Standard +0.8
  1. Express \(\frac{4}{(1-x)(1+x)(1+x^2)}\) in partial fractions. [5]
  2. Show that \(\int_0^{\frac{\sqrt{3}}{3}} \frac{4}{1-x^4} dx = \ln\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) + \frac{1}{3}\pi\). [4]
OCR FP2 2010 January Q7
8 marks Standard +0.8
\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sqrt{x}\), together with a set of \(n\) rectangles of unit width.
  1. By considering the areas of these rectangles, explain why $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} > \int_0^n \sqrt{x} dx.$$ [2]
  2. By drawing another set of rectangles and considering their areas, show that $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} < \int_1^{n+1} \sqrt{x} dx.$$ [3]
  3. Hence find an approximation to \(\sum_{n=1}^{100} \sqrt{n}\), giving your answer correct to 2 significant figures. [3]
OCR FP2 2010 January Q8
10 marks Standard +0.3
The equation of a curve is $$y = \frac{kx}{(x-1)^2},$$ where \(k\) is a positive constant.
  1. Write down the equations of the asymptotes of the curve. [2]
  2. Show that \(y \geq -\frac{1}{4}k\). [4]
  3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve. [4]
OCR FP2 2010 January Q9
12 marks Standard +0.8
  1. Given that \(y = \tanh^{-1} x\), for \(-1 < x < 1\), prove that \(y = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\). [3]
  2. It is given that \(f(x) = a\cosh x - b\sinh x\), where \(a\) and \(b\) are positive constants.
    1. Given that \(b \geq a\), show that the curve with equation \(y = f(x)\) has no stationary points. [3]
    2. In the case where \(a > 1\) and \(b = 1\), show that \(f(x)\) has a minimum value of \(\sqrt{a^2 - 1}\). [6]
OCR FP2 2012 January Q1
4 marks Standard +0.3
Given that \(f(x) = \ln(\cos 3x)\), find \(f'(0)\) and \(f''(0)\). Hence show that the first term in the Maclaurin series for \(f(x)\) is \(ax^2\), where the value of \(a\) is to be found. [4]
OCR FP2 2012 January Q2
5 marks Easy -2.5
By first completing the square in the denominator, find the exact value of $$\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{4x^2 - 4x + 5} dx.$$ [5]
OCR FP2 2012 January Q3
7 marks Standard +0.3
Express \(\frac{2x^3 + x + 12}{(2x - 1)(x^2 + 4)}\) in partial fractions. [7]
OCR FP2 2012 January Q4
9 marks Standard +0.8
\includegraphics{figure_4} The diagram shows the curve \(y = e^{-\frac{1}{x}}\) for \(0 < x \leq 1\). A set of \((n - 1)\) rectangles is drawn under the curve as shown.
  1. Explain why a lower bound for \(\int_0^1 e^{-\frac{1}{x}} dx\) can be expressed as $$\frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}}\right).$$ [2]
  2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int_0^1 e^{-\frac{1}{x}} dx\). [2]
  3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures. [2]
  4. When \(n > N\), the difference between the upper and lower bounds is less than 0.001. By expressing this difference in terms of \(n\), find the least possible value of \(N\). [3]
OCR FP2 2012 January Q5
11 marks Standard +0.8
It is given that \(f(x) = x^3 - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(f(x) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x_1, x_2, \ldots, x_n, \ldots\).
  1. Show that \(x_{n+1} = \frac{2x_n^3 + k}{3x_n^2}\). [2]
  2. Sketch the graph of \(y = f(x)\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(|x_2 - x_1|\) to be greater than \(|x_1|\). [3]
It is now given that \(k = 100\) and \(x_1 = 5\).
  1. Write down the exact value of \(\alpha\) and find \(x_2\) and \(x_3\) correct to 5 decimal places. [3]
  2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). By finding \(e_1\), \(e_2\) and \(e_3\), verify that \(e_3 \approx \frac{e_2^2}{e_1}\). [3]
OCR FP2 2012 January Q6
8 marks Standard +0.8
  1. Prove that the derivative of \(\cos^{-1} x\) is \(-\frac{1}{\sqrt{1 - x^2}}\). [3]
A curve has equation \(y = \cos^{-1}(1 - x^2)\), for \(0 < x < \sqrt{2}\).
  1. Find and simplify \(\frac{dy}{dx}\), and hence show that $$(2 - x^2)\frac{d^2y}{dx^2} = x\frac{dy}{dx}.$$ [5]
OCR FP2 2012 January Q7
8 marks Standard +0.8
  1. Given that \(y = \sinh^{-1} x\), prove that \(y = \ln\left(x + \sqrt{x^2 + 1}\right)\). [3]
  2. It is given that \(x\) satisfies the equation \(\sinh^{-1} x - \cosh^{-1} x = \ln 2\). Use the logarithmic forms for \(\sinh^{-1} x\) and \(\cosh^{-1} x\) to show that $$\sqrt{x^2 + 1} - 2\sqrt{x^2 - 1} = x.$$ Hence, by squaring this equation, find the exact value of \(x\). [5]
OCR FP2 2012 January Q8
9 marks Challenging +1.3
\includegraphics{figure_8} The diagram shows two curves, \(C_1\) and \(C_2\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C_1\) is \(r = \sqrt{2}\cos\theta\) and the polar equation of \(C_2\) is \(r = \sqrt{2}\sin 2\theta\). For both curves, \(0 \leq \theta \leq \frac{1}{2}\pi\). The value of \(\theta\) at \(P\) is \(\alpha\).
  1. Show that \(\tan\alpha = \frac{1}{2}\). [2]
  2. Show that the area of the region common to \(C_1\) and \(C_2\), shaded in the diagram, is \(\frac{1}{4}\pi - \frac{1}{2}\alpha\). [7]
OCR FP2 2012 January Q9
11 marks Challenging +1.3
  1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]
It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).
  1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
  2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
  3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]