Questions FP2 (1279 questions)

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OCR FP2 2014 June Q3
7 marks Challenging +1.2
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
6 marks Standard +0.8
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
9 marks Standard +0.8
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
9 marks Standard +0.3
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
11 marks Challenging +1.8
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
10 marks Standard +0.8
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
12 marks Standard +0.3
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
3 marks Standard +0.8
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
4 marks Standard +0.3
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
5 marks Standard +0.3
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
9 marks Challenging +1.2
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
9 marks Standard +0.8
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
12 marks Challenging +1.8
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.
OCR FP2 2015 June Q7
10 marks Challenging +1.2
7 It is given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 25 } { ( x - 1 ) ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Write down the equations of the asymptotes of the curve \(y = \mathrm { f } ( x )\).
  3. Find the value of \(x\) where the graph of \(y = \mathrm { f } ( x )\) cuts the horizontal asymptote.
  4. Sketch the graph of \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP2 2015 June Q8
9 marks Standard +0.3
8 It is given that \(\mathrm { f } ( x ) = 2 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm { f } ( x )\) has a stationary point at \(x = - \frac { 1 } { 2 } \ln 5\) and find the value of \(y\) at this point.
  2. Solve the equation \(\mathrm { f } ( x ) = 5\), giving your answers exactly. \section*{Question 9 begins on page 4.}
OCR FP2 2015 June Q9
11 marks Standard +0.8
9 The equation of a curve in polar coordinates is \(r = 2 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
  1. Sketch the curve.
  2. Find the area of the region enclosed by this curve.
  3. By expressing \(\sin 3 \theta\) in terms of \(\sin \theta\), show that a cartesian equation for the curve is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } y - 2 y ^ { 3 } .$$ \section*{END OF QUESTION PAPER}
CAIE FP2 2009 June Q1
5 marks Challenging +1.2
1 A line \(O P\) of fixed length \(l\) rotates in a plane about the fixed point \(O\). At time \(t = 0\), the line is at the position \(O A\). At time \(t\), angle \(A O P = \theta\) radians and \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta\). Show that, for all \(t\), the magnitude of the acceleration of \(P\) is equal to the magnitude of its velocity.
CAIE FP2 2009 June Q2
7 marks Moderate -0.5
2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is \(2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
CAIE FP2 2009 June Q3
8 marks Challenging +1.8
3 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-2_513_711_890_717} A uniform lamina of mass \(m\) is bounded by concentric circles with centre \(O\) and radii \(a\) and \(2 a\). The lamina is free to rotate about a fixed smooth horizontal axis \(T\) which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about \(T\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). When hanging at rest, with \(O\) vertically below \(T\), the lamina is given an angular speed \(\omega\) about \(T\). The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of \(\omega\).
CAIE FP2 2009 June Q4
11 marks Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre \(O\), radius 0.6 m and weight 36 N . A uniform rod \(A B\), of weight 14 N and length 1 m , rests with \(A\) in contact with the plane and \(B\) in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is \(C\), and \(A , B , C\) and \(O\) lie in the same vertical plane (see diagram). The contacts at \(A , B\) and \(C\) are rough and the system is in equilibrium. By taking moments about \(C\) for the system, show that the magnitude of the normal contact force at \(A\) is 10 N . Show that the magnitudes of the frictional forces at \(A , B\) and \(C\) are equal. The coefficients of friction at \(A , B\) and \(C\) are all equal to \(\mu\). Find the smallest possible value of \(\mu\).
CAIE FP2 2009 June Q5
12 marks Challenging +1.2
5 Two spheres \(A\) and \(B\), of equal radius, have masses \(m _ { 1 }\) and \(m _ { 2 }\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards sphere \(B\) with speed \(u\) and, as a result of the collision, \(A\) is brought to rest. Show that
  1. the speed of \(B\) immediately after the collision cannot exceed \(u\),
  2. \(m _ { 1 } \leqslant m _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, \(B\) hits a smooth vertical wall which is at an angle of \(60 ^ { \circ }\) to the direction of motion of \(B\) (see diagram). In the impact with the wall \(B\) loses \(\frac { 2 } { 3 }\) of its kinetic energy. Find the coefficient of restitution between \(B\) and the wall and show that the direction of motion of \(B\) turns through \(90 ^ { \circ }\).
CAIE FP2 2009 June Q6
6 marks Standard +0.8
6 The times taken by employees in a factory to complete a certain task have a normal distribution with mean \(\mu\) seconds and standard deviation \(\sigma\) seconds, both of which are unknown. Based on a random sample of 20 employees, the symmetric \(95 \%\) confidence interval for \(\mu\) is \(( 481,509 )\). Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
[0pt] [6]
CAIE FP2 2009 June Q7
8 marks Standard +0.3
7 An experiment was carried out to determine how much weedkiller to apply per \(100 \mathrm {~m} ^ { 2 }\) in a large field. Ten \(100 \mathrm {~m} ^ { 2 }\) areas of the field were randomly chosen and sprayed with predetermined volumes of the weedkiller. The volume of the weedkiller is denoted by \(x\) litres and the number of weeds that survived is denoted by \(y\). The results are given in the table.
\(x\)0.100.150.200.250.300.350.400.450.500.55
\(y\)484044353924101396
$$\left[ \Sigma x = 3.25 , \Sigma x ^ { 2 } = 1.2625 , \Sigma y = 268 , \Sigma y ^ { 2 } = 9548 , \Sigma x y = 66.10 . \right]$$ It is given that the product moment correlation coefficient for the data is - 0.951 , correct to 3 decimal places.
  1. Calculate the equation of a suitable regression line, giving a reason for your choice of line.
  2. Estimate the best volume of weedkiller to apply, and comment on the reliability of your estimate.
CAIE FP2 2009 June Q8
8 marks Standard +0.3
8 Part of a research study of identical twins who had been separated at birth involved a random sample of 9 pairs, in which one twin had been raised by the natural parents and the other by adoptive parents. The IQ scores of these twins were measured, with the following results.
Twin pair123456789
IQ of twin raised by natural parents8292115132889511283123
IQ of twin raised by adoptive parents9288115134979410788130
It may be assumed that the difference in IQ scores has a normal distribution. The mean IQ scores of separated twins raised by natural parents and by adoptive parents are denoted by \(\mu _ { N }\) and \(\mu _ { A }\) respectively. Obtain a \(90 \%\) confidence interval for \(\mu _ { N } - \mu _ { A }\). One of the researchers claimed that there was no evidence of a difference between the two population means. State, giving a reason, whether the confidence interval supports this claim.
CAIE FP2 2009 June Q9
9 marks Standard +0.3
9 The proportions of blood types \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) and O in the Australian population are \(38 \% , 10 \% , 3 \%\) and \(49 \%\) respectively. In order to test whether the population in Sydney conforms to these figures, a random sample of 200 residents is selected. The table shows the observed frequencies of these types in the sample.
Blood TypeABABO
Frequency57249110
Carry out a suitable test at the 5\% significance level. Find the smallest sample size that could be used for the test.