Questions FP2 (1157 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE FP2 2011 June Q8
8 In a crossword competition the times, \(x\) minutes, taken by a random sample of 6 entrants to complete a crossword are summarised as follows. $$\Sigma x = 210.9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 151.2$$ The time to complete a crossword has a normal distribution with mean \(\mu\) minutes. Calculate a \(95 \%\) confidence interval for \(\mu\). Assume now that the standard deviation of the population is known to be 5.6 minutes. Find the smallest sample size that would lead to a \(95 \%\) confidence interval for \(\mu\) of width at most 5 minutes.
CAIE FP2 2011 June Q9
9 Mr Lee asserts that boys are slower than girls at completing a particular mathematical puzzle. In order to test his assertion, a random sample of 40 boys and a random sample of 60 girls are selected from a large group of students who attempted the puzzle. The times taken by the boys, \(b\) minutes, and the times taken by the girls, \(g\) minutes, are summarised as follows. $$\Sigma b = 92.0 \quad \Sigma b ^ { 2 } = 216.5 \quad \Sigma g = 129.8 \quad \Sigma g ^ { 2 } = 288.8$$ Test at the \(2.5 \%\) significance level whether this evidence supports Mr Lee's assertion.
CAIE FP2 2011 June Q10
10 The mid-day temperature, \(x ^ { \circ } \mathrm { C }\), and the amount of sunshine, \(y\) hours, were recorded at a winter holiday resort on each of 12 days, chosen at random during the winter season. The results are summarised as follows. $$\Sigma x = 18.7 \quad \Sigma x ^ { 2 } = 106.43 \quad \Sigma y = 34.7 \quad \Sigma y ^ { 2 } = 133.43 \quad \Sigma x y = 92.01$$
  1. Find the product moment correlation coefficient for the data.
  2. Stating your hypotheses, test at the \(1 \%\) significance level whether there is a non-zero correlation between mid-day temperature and amount of sunshine.
  3. Use the equation of a suitable regression line to estimate the number of hours of sunshine on a day when the mid-day temperature is \(2 ^ { \circ } \mathrm { C }\).
CAIE FP2 2011 June Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{e8a16ec8-b6b7-4b0c-b0c1-8f5f7a9e4fa6-5_511_508_392_817}
A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.
CAIE FP2 2011 June Q11 OR
\includegraphics[max width=\textwidth, alt={}]{e8a16ec8-b6b7-4b0c-b0c1-8f5f7a9e4fa6-5_383_839_1635_651}
The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 ,
\frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 ,
x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find
  1. the probability density function of \(Y\),
  2. the median value of \(Y\).
CAIE FP2 2011 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{020ebd88-b920-40ce-84cf-5c26d45e2935-2_355_695_1073_726} A uniform solid hemisphere, of radius \(a\) and mass \(M\), is placed with its curved surface in contact with a rough plane that is inclined at an angle \(\alpha\) to the horizontal. A particle \(P\) of mass \(m\) is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and \(P\) at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is \(\frac { 1 } { 2 }\), show that
  1. \(\tan \alpha \leqslant \frac { 1 } { 2 }\),
  2. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).
CAIE FP2 2011 June Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{020ebd88-b920-40ce-84cf-5c26d45e2935-5_511_508_392_817}
A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.
CAIE FP2 2011 June Q11 OR
\includegraphics[max width=\textwidth, alt={}]{020ebd88-b920-40ce-84cf-5c26d45e2935-5_383_839_1635_651}
The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 ,
\frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 ,
x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find
  1. the probability density function of \(Y\),
  2. the median value of \(Y\).
CAIE FP2 2011 June Q1
1 Three small spheres, \(A , B\) and \(C\), of masses \(m , k m\) and \(6 m\) respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\) and the coefficient of restitution between \(B\) and \(C\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u\) and is brought to rest by the subsequent collision. Show that \(k = 2\). Given that there are no further collisions after \(B\) has collided with \(C\), show that \(e \leqslant \frac { 1 } { 3 }\).
CAIE FP2 2011 June Q2
5 marks
2
\includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696} A uniform circular disc with centre \(A\) has mass \(M\) and radius \(3 a\). A second uniform circular disc with centre \(B\) has mass \(\frac { 1 } { 9 } M\) and radius \(a\). The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at \(P\) and the circumference of the smaller disc at \(O\). A particle of mass \(\frac { 1 } { 3 } M\) is attached at \(P\) (see diagram). Show that the moment of inertia of the system about an axis through \(O\), perpendicular to the plane of the discs, is \(51 M a ^ { 2 }\). The system is free to rotate about a fixed horizontal axis through \(O\), perpendicular to the plane of the discs. The system is held with \(O P\) horizontal and is then released from rest. Given that \(a = 0.5 \mathrm {~m}\), find the greatest speed of \(P\) in the subsequent motion, giving your answer correct to 2 significant figures.
[0pt] [5]
CAIE FP2 2011 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_419_1102_1859_520} The diagram shows two uniform rods \(B A\) and \(A C\), smoothly hinged at \(A\). The rod \(B A\) has length \(8 a\) and weight \(W\); the rod \(A C\) has length \(6 a\) and weight \(2 W\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) resting on a rough horizontal floor and angle \(C A B\) equal to \(90 ^ { \circ }\). Show that the normal contact force at \(B\) is \(\frac { 26 } { 25 } W\). The coefficient of friction between each rod and the floor is \(\mu\). Find the least possible value of \(\mu\).
CAIE FP2 2011 June Q4
4 A particle \(P\) of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). When hanging at rest under gravity, \(P\) is given a horizontal velocity of magnitude \(\sqrt { } ( 3 a g )\) and subsequently moves freely in a vertical circle. Show that the tension \(T\) in the string when \(O P\) makes an angle \(\theta\) with the downward vertical is given by $$T = m g ( 1 + 3 \cos \theta )$$ When the string is horizontal, it comes into contact with a small smooth peg \(Q\) which is at the same horizontal level as \(O\) and at a distance \(x\) from \(O\), where \(x < a\). Given that \(P\) completes a vertical circle about \(Q\), find the least possible value of \(x\).
CAIE FP2 2011 June Q5
5 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.01 \mathrm { e } ^ { - 0.01 x } & x \geqslant 0
0 & x < 0 \end{cases}$$
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find the median value of \(X\).
  3. Find the probability that \(X\) lies between the median and the mean.
CAIE FP2 2011 June Q6
6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).
CAIE FP2 2011 June Q7
7 A fair die is thrown until a 6 appears for the first time. Assuming that the throws are independent, find
  1. the probability that exactly 5 throws are needed,
  2. the probability that fewer than 8 throws are needed,
  3. the least integer \(n\) such that the probability of obtaining a 6 before the \(n\)th throw is at least 0.99 .
CAIE FP2 2011 June Q8
8 A company decides that its employees should follow an exercise programme for 30 minutes each day, with the aim that they lose weight and increase productivity. The weights, in kg , of a random sample of 8 employees at the start of the programme and after following the programme for 6 weeks are shown in the table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Weight before \(( \mathrm { kg } )\)98.687.390.485.2100.592.489.991.3
Weight after \(( \mathrm { kg } )\)93.585.288.284.695.489.386.087.6
Assuming that loss in weight is normally distributed, find a 95\% confidence interval for the mean loss in weight of the company's employees. Test at the \(5 \%\) significance level whether, after the exercise programme, there is a reduction of more than 2.5 kg in the population mean weight.
CAIE FP2 2011 June Q9
9 The marks achieved by a random sample of 15 college students in a Physics examination ( \(x\) ) and in a General Studies examination (y) are summarised as follows. $$\Sigma x = 752 \quad \Sigma x ^ { 2 } = 38814 \quad \Sigma y = 773 \quad \Sigma y ^ { 2 } = 45351 \quad \Sigma x y = 40236$$
  1. Find the mean values, \(\bar { x }\) and \(\bar { y }\).
  2. Another college student achieved a mark of 56 in the General Studies examination, but was unable to take the Physics examination. Use the equation of a suitable regression line to estimate the mark that the student would have obtained in the Physics examination.
  3. Find the product moment correlation coefficient for the given data.
  4. Stating your hypotheses, test at the \(5 \%\) level of significance whether there is a non-zero product moment correlation coefficient between examination marks in Physics and in General Studies achieved by college students.
CAIE FP2 2011 June Q10 EITHER
One end of a light elastic string is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and hangs freely under gravity. In the equilibrium position, the extension of the string is \(d\). Show that the period of small vertical oscillations about the equilibrium position is \(2 \pi \sqrt { } \left( \frac { d } { g } \right)\). The particle is now pulled down and released from rest at a distance \(2 d\) below the equilibrium position. Given that the particle does not reach \(O\) in the subsequent motion, show that the time taken until the particle first comes to instantaneous rest is \(\left( \sqrt { } 3 + \frac { 2 } { 3 } \pi \right) \sqrt { } \left( \frac { d } { g } \right)\).
CAIE FP2 2011 June Q10 OR
A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table.
Number of letters012345\(\geqslant 6\)
Number of days57605325410
It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.
Number of letters012345\(\geqslant 6\)
Expected number of days53.96470.693\(p\)\(q\)6.6221.7350.463
  1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
  2. Carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2012 June Q1
1 A circular flywheel of radius 0.3 m , with moment of inertia about its axis \(18 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is rotating freely with angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A tangential force of constant magnitude 48 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE FP2 2012 June Q2
2 Two particles, of masses \(3 m\) and \(m\), are moving in the same straight line towards each other with speeds \(2 u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4 m u\). Show that the total loss in kinetic energy is \(\frac { 4 } { 3 } m u ^ { 2 }\).
CAIE FP2 2012 June Q3
3 A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { } \left( \frac { 7 } { 2 } g a \right)\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(O P\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac { 3 } { 2 } m g ( 1 + 2 \cos \theta )\). Find the speed of \(P\)
  1. when it loses contact with the sphere,
  2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.)
    \(4 \quad A B\) is a diameter of a uniform circular disc \(D\) of mass \(9 m\), radius \(3 a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112 m a ^ { 2 }\). A particle of mass \(3 m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k \sqrt { } ( g a )\). Find the value of \(k\), correct to 3 significant figures.
CAIE FP2 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{39282b82-5229-484a-beb9-7a845dbb5727-2_478_867_1816_641} Two uniform rods \(A B\) and \(B C\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(A B\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(A B\) has mass \(3 m\) and length \(3 a\), the rod \(B C\) has mass \(5 m\) and length \(5 a\), and \(C\) is at a distance \(6 a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(B C\) at \(C\) has magnitude \(\frac { 13 } { 2 } m g\). The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\).
CAIE FP2 2012 June Q6
6 The probability that a particular type of light bulb is defective is 0.01 . A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. The first defective bulb is the \(N\) th to be tested. Write down the value of \(\mathrm { E } ( N )\). Find the least value of \(n\) such that \(\mathrm { P } ( N \leqslant n )\) is greater than 0.9 .
CAIE FP2 2012 June Q7
7 A random sample of 8 swimmers from a swimming club were timed over a distance of 100 metres, once in an outdoor pool and once in an indoor pool. Their times, in seconds, are given in the following table.
Swimmer\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Outdoor time66.262.460.865.468.864.365.267.2
Indoor time66.160.360.965.266.463.862.469.8
Assuming a normal distribution, test, at the \(5 \%\) significance level, whether there is a non-zero difference between mean time in the outdoor pool and mean time in the indoor pool.