Questions FP2 (1157 questions)

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OCR FP2 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{72a1330a-c6dc-4f3a-9b0e-333b099f4509-4_782_1065_251_500} The diagram shows the curve \(y = \frac { 1 } { x }\) for \(x > 0\) and a set of \(( n - 1 )\) rectangles of unit width below the curve. These rectangles can be used to obtain an inequality of the form $$\frac { 1 } { a } + \frac { 1 } { a + 1 } + \frac { 1 } { a + 2 } + \ldots + \frac { 1 } { b } < \int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x$$ Another set of rectangles can be used similarly to obtain $$\int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x < \frac { 1 } { c } + \frac { 1 } { c + 1 } + \frac { 1 } { c + 2 } + \ldots + \frac { 1 } { d }$$
  1. Write down the values of the constants \(a\) and \(c\), and express \(b\) and \(d\) in terms of \(n\). The function f is defined by \(\mathrm { f } ( n ) = 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } - \ln n\), for positive integers \(n\).
  2. Use your answers to part (i) to obtain upper and lower bounds for \(\mathrm { f } ( n )\).
  3. By using the first 2 terms of the Maclaurin series for \(\ln ( 1 + x )\) show that, for large \(n\), $$f ( n + 1 ) - f ( n ) \approx - \frac { n - 1 } { 2 n ^ { 2 } ( n + 1 ) } .$$
OCR FP2 2012 June Q8
8 The curve \(C _ { 1 }\) has equation \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are polynomials of degree 2 and 1 respectively. The asymptotes of the curve are \(x = - 2\) and \(y = \frac { 1 } { 2 } x + 1\), and the curve passes through the point \(\left( - 1 , \frac { 17 } { 2 } \right)\).
  1. Express the equation of \(C _ { 1 }\) in the form \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\).
  2. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
  3. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
    Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
  4. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of intersection. Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
  5. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of
    intersection. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR FP2 2013 June Q1
1 By using the substitution \(t = \tan \frac { 1 } { 2 } \theta\), find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 1 + \cos \theta } \mathrm { d } \theta\).
OCR FP2 2013 June Q2
2
  1. Using the definitions for \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
  2. Hence solve the equation \(\sinh ^ { 2 } x = 5 \cosh x - 7\), giving your answers in logarithmic form.
OCR FP2 2013 June Q3
3 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 3 + x } \right)\) for \(x > - 1\).
  1. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 1 } { 2 ( x + 1 ) ^ { 2 } }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).
OCR FP2 2013 June Q4
4 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
  2. Hence find \(I _ { 11 }\) as an exact fraction.
OCR FP2 2013 June Q5
5 You are given that the equation \(x ^ { 3 } + 4 x ^ { 2 } + x - 1 = 0\) has a root, \(\alpha\), where \(- 1 < \alpha < 0\).
  1. Show that the Newton-Raphson iterative formula for this equation can be written in the form $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 4 x _ { n } ^ { 2 } + 1 } { 3 x _ { n } ^ { 2 } + 8 x _ { n } + 1 } .$$
  2. Using the initial value \(x _ { 1 } = - 0.7\), find \(x _ { 2 }\) and \(x _ { 3 }\) and find \(\alpha\) correct to 5 decimal places.
  3. The diagram shows a sketch of the curve \(y = x ^ { 3 } + 4 x ^ { 2 } + x - 1\) for \(- 1.5 \leqslant x \leqslant 1\).
    \includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-3_602_926_749_566} Using the copy of the diagram in your answer book, explain why the initial value \(x _ { 1 } = 0\) will fail to find \(\alpha\).
OCR FP2 2013 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-4_656_1017_251_525} The diagram shows part of the curve \(y = \ln ( \ln ( x ) )\). The region between the curve and the \(x\)-axis for \(3 \leqslant x \leqslant 6\) is shaded.
  1. By considering \(n\) rectangles of equal width, show that a lower bound, \(L\), for the area of the shaded region is \(\frac { 3 } { n } \sum _ { r = 0 } ^ { n - 1 } \ln \left( \ln \left( 3 + \frac { 3 r } { n } \right) \right)\).
  2. By considering another set of \(n\) rectangles of equal width, find a similar expression for an upper bound, \(U\), for the area of the shaded region.
  3. Find the least value of \(n\) for which \(U - L < 0.001\).
OCR FP2 2013 June Q7
7 The equation of a curve is \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ( x - 7 ) }\).
  1. Write down the equations of the asymptotes.
  2. Find the coordinates of the stationary points on the curve.
  3. Find the coordinates of the point where the curve meets one of its asymptotes.
  4. Sketch the curve.
OCR FP2 2013 June Q8
8 The equation of a curve is \(x ^ { 2 } + y ^ { 2 } - x = \sqrt { x ^ { 2 } + y ^ { 2 } }\).
  1. Find the polar equation of this curve in the form \(r = \mathrm { f } ( \theta )\).
  2. Sketch the curve.
  3. The line \(x + 2 y = 2\) divides the region enclosed by the curve into two parts. Find the ratio of the two areas.
OCR FP2 2014 June Q1
1 Find \(\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x\), giving your answer exactly in logarithmic form.
OCR FP2 2014 June Q2
2 It is given that \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\).
  1. Using the standard Maclaurin expansion for \(\ln ( 1 + x )\), write down the first four terms in the expansion of \(\mathrm { f } ( x )\), stating the set of values of \(x\) for which the expansion is valid.
  2. Hence find the exact value of $$1 - \frac { 1 } { 2 } \left( \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 3 } \left( \frac { 1 } { 2 } \right) ^ { 4 } - \frac { 1 } { 4 } \left( \frac { 1 } { 2 } \right) ^ { 6 } + \ldots .$$
OCR FP2 2014 June Q3
3 The diagram shows the curve \(y = \frac { 1 } { x ^ { 3 } }\) for \(1 \leqslant x \leqslant n\) where \(n\) is an integer. A set of ( \(n - 1\) ) rectangles of unit width is drawn under the curve.
\includegraphics[max width=\textwidth, alt={}, center]{736932f1-4007-4a04-a08b-2551db0b136c-2_611_947_1103_557}
  1. Write down the sum of the areas of the rectangles.
  2. Hence show that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 3 } } < \frac { 3 } { 2 }\).
OCR FP2 2014 June Q4
4 The curves \(y = \cos ^ { - 1 } x\) and \(y = \tan ^ { - 1 } ( \sqrt { 2 } x )\) intersect at a point \(A\).
  1. Verify that the coordinates of \(A\) are \(\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 4 } \pi \right)\).
  2. Determine whether the tangents to the curves at \(A\) are perpendicular.
OCR FP2 2014 June Q5
5 A curve has equation \(y = \frac { x ^ { 2 } - 8 } { x - 3 }\).
  1. Find the equations of the asymptotes of the curve.
  2. Prove that there are no points on the curve for which \(4 < y < 8\).
  3. Sketch the curve. Indicate the asymptotes in your sketch.
OCR FP2 2014 June Q6
6
  1. Given that \(y = \cosh ^ { - 1 } x\), show that \(y = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cosh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  3. Solve the equation \(\cosh x = 3\), giving your answers in logarithmic form.
OCR FP2 2014 June Q7
7 It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } x \mathrm {~d} x\).
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
  2. Explain why \(I _ { 2 n + 1 } < I _ { 2 n - 1 }\).
  3. It is given that \(I _ { 2 n + 1 } < I _ { 2 n } < I _ { 2 n - 1 }\). Take \(n = 5\) to find an interval within which the value of \(\pi\) lies.
OCR FP2 2014 June Q8
8 A curve has polar equation \(r = a ( 1 + \cos \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
  1. Find the equation of the tangent at the pole.
  2. Sketch the curve.
  3. Find the area enclosed by the curve.
OCR FP2 2014 June Q9
9 The equation \(10 x - 8 \ln x = 28\) has a root \(\alpha\) in the interval [3,4]. The iteration \(x _ { n + 1 } = \mathrm { g } \left( x _ { n } \right)\), where \(\mathrm { g } ( x ) = 2.8 + 0.8 \ln x\) and \(x _ { 1 } = 3.8\), is to be used to find \(\alpha\).
  1. Find the value of \(\alpha\) correct to 5 decimal places. You should show the result of each step of the iteration to 6 decimal places.
  2. Illustrate this iteration by means of a sketch.
  3. The difference, \(\delta _ { r }\), between successive approximations is given by \(\delta _ { r } = x _ { r + 1 } - x _ { r }\). Find \(\delta _ { 3 }\).
  4. Given that \(\delta _ { n + 1 } \approx \mathrm {~g} ^ { \prime } ( \alpha ) \delta _ { n }\), for all positive integers \(n\), estimate the smallest value of \(n\) such that \(\delta _ { n } < 10 ^ { - 6 } \delta _ { 1 }\). \section*{OCR}
OCR FP2 2015 June Q1
1 By first expressing \(\tanh y\) in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
OCR FP2 2015 June Q2
2 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \sin x )\). Using standard series, find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q3
3 By first completing the square, find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } } \mathrm {~d} x\).
OCR FP2 2015 June Q4
4 It is given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = n I _ { n - 1 } + k\) for \(n \geqslant 1\), where \(k\) is a constant to be determined.
  2. Find the exact value of \(I _ { 3 }\).
  3. Find the exact value of \(990 I _ { 8 } - I _ { 11 }\).
OCR FP2 2015 June Q5
5 It is given that \(y = \sin ^ { - 1 } 2 x\).
  1. Using the derivative of \(\sin ^ { - 1 } x\) given in the List of Formulae (MF1), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x }\).
  3. Hence show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } - 12 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\).
  4. Using your results from parts (i), (ii) and (iii), find the Maclaurin series for \(\sin ^ { - 1 } 2 x\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q6
6 It is given that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - x - 1 = 0\) has three roots, one of which is positive.
  1. Show that the Newton-Raphson iterative formula for finding this root can be written $$x _ { n + 1 } = \frac { 6 x _ { n } ^ { 3 } + 5 x _ { n } ^ { 2 } + 1 } { 9 x _ { n } ^ { 2 } + 10 x _ { n } - 1 } .$$
  2. A sequence of iterates \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) which will find the positive root is such that the magnitude of the error in \(x _ { 2 }\) is greater than the magnitude of the error in \(x _ { 1 }\). On the graph given in the Printed Answer Book, mark a possible position for \(x _ { 1 }\).
  3. Apply the iterative formula in part (i) when the initial value is \(x _ { 1 } = - 1\). Describe the behaviour of the iterative sequence, illustrating your answer on the graph given in the Printed Answer Book.
  4. A sequence of approximations to the positive root is given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\). Successive differences \(x _ { r } - x _ { r - 1 } = d _ { r }\), where \(r \geqslant 2\), are such that \(d _ { r } \approx k \left( d _ { r - 1 } \right) ^ { 2 }\) where \(k\) is a constant. Show that \(d _ { 4 } \approx \frac { d _ { 3 } ^ { 3 } } { d _ { 2 } ^ { 2 } }\) and demonstrate this numerically when \(x _ { 1 } = 1\).
  5. Find the value of the positive root correct to 5 decimal places.