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OCR FP1 2015 June Q7
10 marks Standard +0.3
7
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\). You must show sufficient working to justify your answers.
  2. Hence solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0\).
OCR FP1 2015 June Q8
10 marks Challenging +1.3
8
  1. Show that \(\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
  3. Hence find the value of \(\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
OCR FP1 2015 June Q9
10 marks Standard +0.3
9 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4 \\ 2 & a & 3 \\ 0 & 1 & a \end{array} \right)\).
  1. Find the values of \(a\) for which \(\mathbf { D }\) is singular.
  2. Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3 \\ 2 x + a y + 3 z = 2 \\ y + a z = 0 \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
    (a) \(a = 3\) (b) \(a = 1\)
OCR FP1 2015 June Q10
10 marks Standard +0.8
10 The cubic equation \(x ^ { 3 } + 4 x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \sqrt { u }\) to obtain a cubic equation in \(u\).
  2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma\).
OCR FP1 2016 June Q1
5 marks Moderate -0.5
1 Find \(\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )\), giving your answer in a fully factorised form.
OCR FP1 2016 June Q2
7 marks Standard +0.3
2 The complex number \(z\) has modulus \(2 \sqrt { 3 }\) and argument \(- \frac { 1 } { 3 } \pi\). Giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers, and showing clearly how you obtain them, find
  1. \(z\),
  2. \(\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }\).
OCR FP1 2016 June Q3
6 marks Standard +0.3
3 The quadratic equation \(k x ^ { 2 } + x + k = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)\) in terms of \(k\).
OCR FP1 2016 June Q4
6 marks Easy -1.2
4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r } 6 \\ 4 \\ - 1 \end{array} \right)\). Find
  1. \(5 \mathbf { A } - 3 \mathbf { B }\),
  2. BC,
  3. CA .
OCR FP1 2016 June Q5
4 marks Standard +0.3
5 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 5 \text { and } u _ { n + 1 } = 3 u _ { n } + 2 \text { for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 2 \times 3 ^ { n } - 1\).
OCR FP1 2016 June Q6
9 marks Standard +0.3
6 In an Argand diagram the points \(A\) and \(B\) represent the complex numbers \(5 + 4 \mathrm { i }\) and \(1 + 2 \mathrm { i }\) respectively.
  1. Given that \(A\) and \(B\) are the ends of a diameter of a circle \(C\), find the equation of \(C\) in complex number form. The perpendicular bisector of \(A B\) is denoted by \(l\).
  2. Sketch \(C\) and \(l\) on a single Argand diagram.
  3. Find the complex numbers represented by the points of intersection of \(C\) and \(l\).
OCR FP1 2016 June Q7
8 marks Standard +0.3
7 The matrix \(\left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\) represents a transformation P .
  1. Describe fully the transformation P . The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1 \\ - 1 & 0 \end{array} \right)\).
  2. Given that \(\mathbf { M }\) represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .
OCR FP1 2016 June Q9
6 marks Standard +0.3
9
  1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 \\ 0 & a & 5 \\ 1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
  2. Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1 \\ 3 y + 5 z = 5 \\ x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
  3. Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
  4. Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
  5. By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR FP2 2009 January Q1
6 marks Standard +0.3
1
  1. Write down and simplify the first three terms of the Maclaurin series for \(\mathrm { e } ^ { 2 x }\).
  2. Hence show that the Maclaurin series for $$\ln \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$$ begins \(\ln a + b x ^ { 2 }\), where \(a\) and \(b\) are constants to be found.
OCR FP2 2009 January Q2
6 marks Standard +0.3
2 It is given that \(\alpha\) is the only real root of the equation \(x ^ { 5 } + 2 x - 28 = 0\) and that \(1.8 < \alpha < 2\).
  1. The iteration \(x _ { n + 1 } = \sqrt [ 5 ] { 28 - 2 x _ { n } }\), 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.
  2. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 1.8915749\), 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 \(y = \sqrt [ 5 ] { 28 - 2 x }\) at \(x = \alpha\).
  3. Prove that the derivative of \(\sin ^ { - 1 } x\) is \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
  4. Given that $$\sin ^ { - 1 } 2 x + \sin ^ { - 1 } y = \frac { 1 } { 2 } \pi$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\).
  5. By means of a suitable substitution, show that $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x$$ can be transformed to \(\int \cosh ^ { 2 } \theta \mathrm {~d} \theta\).
  6. Hence show that \(\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } x \sqrt { x ^ { 2 } - 1 } + \frac { 1 } { 2 } \cosh ^ { - 1 } x + c\).
OCR FP2 2009 January Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-3_661_734_267_703} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 4.36$$ 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 \(\mathrm { f } ( x ) = 0\), with \(x _ { 1 } = k\).
    (a) To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\) ?
    (b) What happens if \(1 < k < 2\) ?
  2. Sketch the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points.
  3. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$1 + 2 \sinh ^ { 2 } x \equiv \cosh 2 x .$$
  4. Solve the equation $$\cosh 2 x - 5 \sinh x = 4$$ giving your answers in logarithmic form.
OCR FP2 2009 January Q7
8 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-4_511_609_264_769} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2 \cos \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi .$$ The points \(P , Q , R\) and \(S\) on the curve are such that the straight lines \(P O R\) and \(Q O S\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates ( \(r , \alpha\) ).
  1. Show that \(O P + O Q + O R + O S = k\), where \(k\) is a constant to be found.
  2. Given that \(\alpha = \frac { 1 } { 4 } \pi\), find the exact area bounded by the curve and the lines \(O P\) and \(O Q\) (shaded in the diagram).
OCR FP2 2009 January Q8
11 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-5_579_1363_267_390} 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 $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n + 1 } < \ln ( n + 1 ) .$$
  2. By considering the areas of another set of rectangles, show that $$1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } > \ln ( n + 1 ) .$$
  3. Hence show that $$\ln ( n + 1 ) + \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n + 1 } \frac { 1 } { r } < \ln ( n + 1 ) + 1$$
  4. State, with a reason, whether \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r }\) is convergent.
OCR FP2 2009 January Q9
12 marks Challenging +1.2
9 A curve has equation $$y = \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) }$$ where \(a\) is a positive constant.
  1. Explain why the curve has no asymptotes parallel to the \(y\)-axis.
  2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve.
  3. Find the exact value of \(\int _ { a } ^ { 2 a } \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) } \mathrm { d } x\), showing that it is independent of \(a\).
OCR FP2 2010 January Q1
5 marks Standard +0.3
1 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \sin x\).
  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 \(\mathrm { f } ( x ) = 0\). Find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the answers correct to 6 decimal places.
  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 \(\mathrm { g } ( x ) = \sqrt { \sin x }\), use \(e _ { 3 }\) and \(e _ { 4 }\) to estimate \(\mathrm { g } ^ { \prime } ( \alpha )\).
OCR FP2 2010 January Q2
6 marks Standard +0.3
2 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )\).
  1. Find \(\mathrm { f } ( 0 )\) and \(\mathrm { f } ^ { \prime } ( 0 )\), and show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).
OCR FP2 2010 January Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-2_597_1006_973_571} A curve with no stationary points has equation \(y = \mathrm { f } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) 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 \(\left( x _ { 1 } , \mathrm { f } \left( x _ { 1 } \right) \right)\) meets the \(x\)-axis where \(x = x _ { 2 }\) (see diagram).
  1. Show that \(x _ { 2 } = x _ { 1 } - \frac { \mathrm { f } \left( x _ { 1 } \right) } { \mathrm { f } ^ { \prime } \left( x _ { 1 } \right) }\).
  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 \(\alpha\).
  3. Use the Newton-Raphson method, with a first approximation of 1 , to find a second approximation to the root of \(x ^ { 2 } - 2 \sinh x + 2 = 0\).
OCR FP2 2010 January Q4
7 marks Standard +0.3
4 The equation of a curve, in polar coordinates, is $$r = \mathrm { e } ^ { - 2 \theta } , \quad \text { for } 0 \leqslant \theta \leqslant \pi .$$
  1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value.
  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 \(O P\) and \(O Q\) can be expressed as \(k \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right)\), where \(k\) is a constant to be found.
OCR FP2 2010 January Q7
8 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-3_591_1131_986_507} The diagram shows the curve with equation \(y = \sqrt [ 3 ] { x }\), together with a set of \(n\) rectangles of unit width.
  1. By considering the areas of these rectangles, explain why $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } > \int _ { 0 } ^ { n } \sqrt [ 3 ] { x } \mathrm {~d} x$$
  2. By drawing another set of rectangles and considering their areas, show that $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } < \int _ { 1 } ^ { n + 1 } \sqrt [ 3 ] { x } \mathrm {~d} x$$
  3. Hence find an approximation to \(\sum _ { n = 1 } ^ { 100 } \sqrt [ 3 ] { n }\), giving your answer correct to 2 significant figures.
OCR FP2 2010 January Q8
10 marks Standard +0.8
8 The equation of a curve is $$y = \frac { k x } { ( x - 1 ) ^ { 2 } } ,$$ where \(k\) is a positive constant.
  1. Write down the equations of the asymptotes of the curve.
  2. Show that \(y \geqslant - \frac { 1 } { 4 } k\).
  3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve.
OCR FP2 2010 January Q9
12 marks Standard +0.8
9
  1. Given that \(y = \tanh ^ { - 1 } x\), for \(- 1 < x < 1\), prove that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  2. It is given that \(\mathrm { f } ( x ) = a \cosh x - b \sinh x\), where \(a\) and \(b\) are positive constants.
    (a) Given that \(b \geqslant a\), show that the curve with equation \(y = \mathrm { f } ( x )\) has no stationary points.
    (b) In the case where \(a > 1\) and \(b = 1\), show that \(\mathrm { f } ( x )\) has a minimum value of \(\sqrt { a ^ { 2 } - 1 }\).