Questions FP2 (1279 questions)

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AQA FP2 2011 June Q7
16 marks Challenging +1.3
    1. Use de Moivre's Theorem to show that $$\cos 5\theta = \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta$$ and find a similar expression for \(\sin 5\theta\). [5 marks]
    2. Deduce that $$\tan 5\theta = \frac{\tan \theta(5 - 10 \tan^2 \theta + \tan^4 \theta)}{1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$$ [3 marks]
  2. Explain why \(t = \tan \frac{\pi}{5}\) is a root of the equation $$t^4 - 10t^2 + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form. [3 marks]
  3. Deduce that $$\tan \frac{\pi}{5} \tan \frac{2\pi}{5} = \sqrt{5}$$ [5 marks]
AQA FP2 2016 June Q1
6 marks Standard +0.3
  1. Given that \(f(r) = \frac{1}{4r-1}\), show that $$f(r) - f(r+1) = \frac{A}{(4r-1)(4r+3)}$$ where \(A\) is an integer. [2 marks]
  2. Use the method of differences to find the value of \(\sum_{r=1}^{50} \frac{1}{(4r-1)(4r+3)}\), giving your answer as a fraction in its simplest form. [4 marks]
AQA FP2 2016 June Q2
8 marks Standard +0.3
The cubic equation \(3z^3 + pz^2 + 17z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2\mathrm{i}\).
    1. Write down the value of another non-real root, \(\beta\), of this equation. [1 mark]
    2. Hence find the value of \(\alpha\beta\). [1 mark]
  2. Find the value of the third root, \(\gamma\), of this equation. [3 marks]
  3. Find the values of \(p\) and \(q\). [3 marks]
AQA FP2 2016 June Q3
10 marks Challenging +1.3
The arc of the curve with equation \(y = 4 - \ln(1-x^2)\) from \(x = 0\) to \(x = \frac{3}{4}\) has length \(s\).
  1. Show that \(s = \int_0^{\frac{3}{4}} \frac{\sqrt{1+x^2}}{1-x^2} \, dx\). [4 marks]
  2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer. [6 marks]
AQA FP2 2016 June Q4
6 marks Standard +0.8
  1. Given that \(y = \tan^{-1} \sqrt{3x}\), find \(\frac{dy}{dx}\), giving your answer in terms of \(x\). [2 marks]
  2. Hence, or otherwise, show that \(\int_{\frac{1}{3}}^1 \frac{1}{(1+3x)\sqrt{x}} \, dx = \frac{\sqrt{3}\pi}{n}\), where \(n\) is an integer. [4 marks]
AQA FP2 2016 June Q5
12 marks Standard +0.3
  1. Find the modulus of the complex number \(-4\sqrt{3} + 4\mathrm{i}\), giving your answer as an integer. [2 marks]
  2. The locus of points, \(L\), satisfies the equation \(|z + 4\sqrt{3} - 4\mathrm{i}| = 4\).
    1. Sketch the locus \(L\) on the Argand diagram below. [3 marks]
    2. The complex number \(w\) lies on \(L\) so that \(-\pi < \arg w \leq \pi\). Find the least possible value of \(\arg w\), giving your answer in terms of \(\pi\). [2 marks]
  3. Solve the equation \(z^3 = -4\sqrt{3} + 4\mathrm{i}\), giving your answers in the form \(re^{\mathrm{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). [5 marks]
AQA FP2 2016 June Q6
14 marks Challenging +1.2
  1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that $$x = \ln(y + \sqrt{y^2 + 1}).$$ [4 marks]
  2. A curve has equation \(y = 6\cosh^2 x + 5\sinh x\).
    1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number. [5 marks]
    2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh^{-1} 2\) has area \(A\). Show that $$A = a\cosh^{-1} 2 + b\sqrt{3} + c$$ where \(a\), \(b\) and \(c\) are integers. [5 marks]
AQA FP2 2016 June Q7
6 marks Standard +0.3
Given that \(p \geq -1\), prove by induction that, for all integers \(n \geq 1\), $$(1 + p)^n \geq 1 + np$$ [6 marks]
AQA FP2 2016 June Q8
13 marks Challenging +1.8
  1. By applying de Moivre's theorem to \((\cos \theta + \mathrm{i} \sin \theta)^4\), where \(\cos \theta \neq 0\), show that $$(1 + \mathrm{i} \tan \theta)^4 + (1 - \mathrm{i} \tan \theta)^4 = \frac{2\cos 4\theta}{\cos^4 \theta}$$ [3 marks]
  2. Hence show that \(z = \mathrm{i} \tan \frac{\pi}{8}\) satisfies the equation \((1 + z)^4 + (1 - z)^4 = 0\), and express the three other roots of this equation in the form \(\mathrm{i} \tan \phi\), where \(0 < \phi < \pi\). [2 marks]
  3. Use the results from part (b) to find the values of:
    1. \(\tan^2 \frac{\pi}{8} \tan^2 \frac{3\pi}{8}\); [4 marks]
    2. \(\tan^2 \frac{\pi}{8} + \tan^2 \frac{3\pi}{8}\). [4 marks]
OCR FP2 2009 January Q1
6 marks Standard +0.3
  1. Write down and simplify the first three terms of the Maclaurin series for \(e^{2x}\). [2]
  2. Hence show that the Maclaurin series for $$\ln(e^{2x} + e^{-2x})$$ begins \(\ln a + bx^2\), where \(a\) and \(b\) are constants to be found. [4]
OCR FP2 2009 January Q2
12 marks Standard +0.8
It is given that \(\alpha\) is the only real root of the equation \(x^3 + 2x - 28 = 0\) and that \(1.8 < \alpha < 2\).
  1. The iteration \(x_{n+1} = \sqrt[3]{28 - 2x_n}\), with \(x_1 = 1.9\), is to be used to find \(\alpha\). Find the values of \(x_2\), \(x_3\) and \(x_4\), giving the answers correct to 7 decimal places. [3]
  2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 1.891 574 9\), correct to 7 decimal places, evaluate \(\frac{e_3}{e_2}\) and \(\frac{e_4}{e_3}\). Comment on these values in relation to the gradient of the curve with equation \(y = \sqrt[3]{28 - 2x}\) at \(x = \alpha\). [3]
OCR FP2 2009 January Q3
7 marks Standard +0.3
  1. Prove that the derivative of \(\sin^{-1} x\) is \(\frac{1}{\sqrt{1-x^2}}\). [3]
  2. Given that $$\sin^{-1} 2x + \sin^{-1} y = \frac{1}{2}\pi,$$ find the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{4}\). [4]
OCR FP2 2009 January Q4
6 marks Standard +0.8
  1. By means of a suitable substitution, show that $$\int \frac{x^2}{\sqrt{x^2-1}} dx$$ can be transformed to \(\int \cosh^2 \theta \, d\theta\). [2]
  2. Hence show that \(\int \frac{x^2}{\sqrt{x^2-1}} dx = \frac{1}{2}\sqrt{x^2-1} + \frac{1}{2}\cosh^{-1} x + c\). [4]
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]