Questions FP2 (1279 questions)

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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]
OCR MEI FP2 2011 January Q1
19 marks Standard +0.3
  1. A curve has polar equation \(r = 2(\cos \theta + \sin \theta)\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{3}{4}\pi\).
    1. Show that a cartesian equation of the curve is \(x^2 + y^2 = 2x + 2y\). Hence or otherwise sketch the curve. [5]
    2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{2}\pi\). Give your answer in terms of \(\pi\). [7]
    1. Given that \(f(x) = \arctan(\frac{1}{2}x)\), find \(f'(x)\). [2]
    2. Expand \(f'(x)\) in ascending powers of \(x\) as far as the term in \(x^4\). Hence obtain an expression for \(f(x)\) in ascending powers of \(x\) as far as the term in \(x^5\). [5]
OCR MEI FP2 2011 January Q2
19 marks Standard +0.3
    1. Given that \(z = \cos \theta + j \sin \theta\), express \(z^n + z^{-n}\) and \(z^n - z^{-n}\) in simplified trigonometrical form. [2]
    2. By considering \((z + z^{-1})^6\), show that $$\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6 \cos 4\theta + 15 \cos 2\theta + 10).$$ [3]
    3. Obtain an expression for \(\cos^6 \theta - \sin^6 \theta\) in terms of \(\cos 2\theta\) and \(\cos 6\theta\). [5]
  2. The complex number \(w\) is \(8e^{i\pi/3}\). You are given that \(z_1\) is a square root of \(w\) and that \(z_2\) is a cube root of \(w\). The points representing \(z_1\) and \(z_2\) in the Argand diagram both lie in the third quadrant.
    1. Find \(z_1\) and \(z_2\) in the form \(re^{i\theta}\). Draw an Argand diagram showing \(w\), \(z_1\) and \(z_2\). [6]
    2. Find the product \(z_1z_2\), and determine the quadrant of the Argand diagram in which it lies. [3]
OCR MEI FP2 2011 January Q3
16 marks Standard +0.3
  1. Show that the characteristic equation of the matrix $$\mathbf{M} = \begin{pmatrix} 1 & -4 & 5 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \end{pmatrix}$$ is \(\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0\). [4]
  2. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf{M}\), and determine whether or not \(\mathbf{M}\) has any other real eigenvalues. [4]
  3. Find an eigenvector, \(\mathbf{v}\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf{M}^n\mathbf{v}\), where \(n\) is an integer. [5]
  4. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [3]
OCR MEI FP2 2011 January Q4
18 marks Standard +0.8
  1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]
A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).
  1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
  2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]
OCR MEI FP2 2011 January Q5
18 marks Challenging +1.2
A curve has parametric equations $$x = t + a \sin t, \quad y = 1 - a \cos t,$$ where \(a\) is a positive constant.
  1. Draw, on separate diagrams, sketches of the curve for \(-2\pi < t < 2\pi\) in the cases \(a = 1\), \(a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
    1. loops,
    2. cusps. [7]
  2. Suppose that the point P\((x, y)\) lies on the curve. Show that the point P\('(-x, y)\) also lies on the curve. What does this indicate about the symmetry of the curve? [3]
  3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(-2\pi < t < 2\pi\) and \(a \neq 1\). [5]
  4. In the case \(a = \frac{1}{2}\pi\), show that \(t = \frac{1}{3}\pi\) and \(t = \frac{5}{3}\pi\) give the same point. Find the angle at which the curve crosses itself at this point. [3]
OCR MEI FP2 2009 June Q1
16 marks Standard +0.3
    1. Use the Maclaurin series for \(\ln(1 + x)\) and \(\ln(1 - x)\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln\left(\frac{1 + x}{1 - x}\right)\). State the range of validity of this series. [4]
    2. Find the value of \(x\) for which \(\frac{1 + x}{1 - x} = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places. [4]
  2. A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
    1. By plotting suitable points, or otherwise, sketch the curve. [3]
    2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]
OCR MEI FP2 2009 June Q2
19 marks Standard +0.3
  1. Obtain the characteristic equation for the matrix \(\mathbf{M}\) where $$\mathbf{M} = \begin{pmatrix} 3 & 1 & -2 \\ 6 & -1 & 0 \\ 2 & 0 & 1 \end{pmatrix}.$$ Hence or otherwise obtain the value of \(\det(\mathbf{M})\). [3]
  2. Show that \(-1\) is an eigenvalue of \(\mathbf{M}\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue \(-1\). Hence or otherwise write down the solution to the following system of equations. [9] $$3x + y - 2z = -0.1$$ $$-y = 0.6$$ $$2x + z = 0.1$$
  3. State the Cayley-Hamilton theorem and use it to show that $$\mathbf{M}^3 = 3\mathbf{M}^2 - 3\mathbf{M} - 7\mathbf{I}.$$ Obtain an expression for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [4]
  4. Find the numerical values of the elements of \(\mathbf{M}^{-1}\), showing your working. [3]
OCR MEI FP2 2009 June Q3
19 marks Standard +0.8
    1. Sketch the graph of \(y = \arcsin x\) for \(-1 \leq x \leq 1\). [1] Find \(\frac{dy}{dx}\), justifying the sign of your answer by reference to your sketch. [4]
    2. Find the exact value of the integral \(\int_0^1 \frac{1}{\sqrt{2 - x^2}} dx\). [3]
  2. The infinite series \(C\) and \(S\) are defined as follows. $$C = \cos \theta + \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta + \ldots$$ $$S = \sin \theta + \frac{1}{3}\sin 3\theta + \frac{1}{5}\sin 5\theta + \ldots$$ By considering \(C + jS\), show that $$C = \frac{3\cos \theta}{5 - 3\cos 2\theta},$$ and find a similar expression for \(S\). [11]
OCR MEI FP2 2009 June Q4
18 marks Standard +0.8
  1. Prove, from definitions involving exponentials, that $$\cosh 2u = 2\cosh^2 u - 1.$$ [3]
  2. Prove that \(\arsinh y = \ln\left(y + \sqrt{y^2 + 1}\right)\). [4]
  3. Use the substitution \(x = 2\sinh u\) to show that $$\int \sqrt{x^2 + 4} dx = 2\arsinh \frac{x}{2} + \frac{x}{2}\sqrt{x^2 + 4} + c,$$ where \(c\) is an arbitrary constant. [6]
  4. By first expressing \(t^2 + 2t + 5\) in completed square form, show that $$\int_{-1}^1 \sqrt{t^2 + 2t + 5} dt = 2\left(\ln(1 + \sqrt{2}) + \sqrt{2}\right).$$ [5]
OCR MEI FP2 2009 June Q5
18 marks Challenging +1.3
Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}
  1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
  2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
  3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
  4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]
CAIE FP2 2013 November Q3
Standard +0.8
3 hours
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value is necessary, take the acceleration due to gravity to be \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of a calculator is expected, where appropriate.
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
CAIE FP2 2013 November Q1
Challenging +1.2
1 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-2_547_423_260_861} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).
CAIE FP2 2013 November Q4
Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-3_563_572_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),
  1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
  2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).
CAIE FP2 2013 November Q9
Standard +0.3
9 For a random sample of 10 observations of pairs of values \(( x , y )\), the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are $$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36 ,$$ respectively.
  1. Find the value of the product moment correlation coefficient for the sample.
  2. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
  3. Find the mean values of \(x\) and \(y\) for this sample.
  4. Estimate the value of \(x\) when \(y = 2.3\) and comment on the reliability of your answer.