Questions — OCR (4907 questions)

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OCR FP1 2010 June Q6
6 marks Moderate -0.3
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z - 3 + 4\text{i}| = 5\), [2]
    2. \(|z| = |z - 6|\). [2]
  2. Indicate, by shading, the region of the Argand diagram for which $$|z - 3 + 4\text{i}| \leq 5 \quad \text{and} \quad |z| \geq |z - 6|.$$ [2]
OCR FP1 2010 June Q7
7 marks Standard +0.8
The quadratic equation \(x^2 + 2kx + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac{\alpha + \beta}{\alpha}\) and \(\frac{\alpha + \beta}{\beta}\). [7]
OCR FP1 2010 June Q8
9 marks Standard +0.3
  1. Show that \(\frac{1}{\sqrt{r + 2} + \sqrt{r}} = \frac{\sqrt{r + 2} - \sqrt{r}}{2}\). [2]
  2. Hence find an expression, in terms of \(n\), for $$\sum_{r=1}^{n} \frac{1}{\sqrt{r + 2} + \sqrt{r}}.$$ [6]
  3. State, giving a brief reason, whether the series \(\sum_{r=1}^{\infty} \frac{1}{\sqrt{r + 2} + \sqrt{r}}\) converges. [1]
OCR FP1 2010 June Q9
9 marks Standard +0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & a & -1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{pmatrix}\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf{A}\). [3]
  2. Three simultaneous equations are shown below. \begin{align} ax + ay - z &= -1
    ay + 2z &= 2a
    x + 2y + z &= 1 \end{align} For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
    1. \(a = 0\)
    2. \(a = 1\)
    3. \(a = 2\) [6]
OCR FP1 2010 June Q10
11 marks Standard +0.8
The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).
  1. Use an algebraic method to find \(z\). [5]
  2. Show that \(z^3 = 2 + 11\text{i}\). [1]
The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).
  1. Find \(w\). [5]
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]
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]