Questions FP2 (1157 questions)

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CAIE FP2 2012 November Q8
8 The yield of a particular crop on a farm is thought to depend principally on the amount of sunshine during the growing season. For a random sample of 8 years, the average yield, \(y\) kilograms per square metre, and the average amount of sunshine per day, \(x\) hours, are recorded. The results are given in the following table.
\(x\)12.210.45.26.311.810.014.22.3
\(y\)159107811126
$$\left[ \Sigma x = 72.4 , \Sigma x ^ { 2 } = 769.9 , \Sigma y = 78 , \Sigma y ^ { 2 } = 820 , \Sigma x y = 761.3 . \right]$$
  1. Find the equation of the regression line of \(y\) on \(x\).
  2. Find the product moment correlation coefficient.
  3. Test, at the \(5 \%\) significance level, whether there is positive correlation between the average yield and the average amount of sunshine per day.
CAIE FP2 2012 November Q9
9 marks
9 The leaves from oak trees growing in two different areas \(A\) and \(B\) are being measured. The lengths, in cm , of a random sample of 7 oak leaves from area \(A\) are $$6.2 , \quad 8.3 , \quad 7.8 , \quad 9.3 , \quad 10.2 , \quad 8.4 , \quad 7.2$$ Assuming that the distribution is normal, find a 95\% confidence interval for the mean length of oak leaves from area \(A\). The lengths, in cm, of a random sample of 5 oak leaves from area \(B\) are $$5.9 , \quad 7.4 , \quad 6.8 , \quad 8.2 , \quad 8.7$$ Making suitable assumptions, which should be stated, test, at the \(5 \%\) significance level, whether the mean length of oak leaves from area \(A\) is greater than the mean length of oak leaves from area \(B\). [9]
CAIE FP2 2012 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-5_389_702_484_719}
Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$
CAIE FP2 2012 November Q10 OR
A continuous random variable \(X\) is believed to have the probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 10 } \left( 5 x - x ^ { 2 } - 4 \right) & 2 \leqslant x < 4
0 & \text { otherwise } \end{cases}$$ A random sample of 60 observations was taken and these values are summarised in the following grouped frequency table.
Interval\(2 \leqslant x < 2.4\)\(2.4 \leqslant x < 2.8\)\(2.8 \leqslant x < 3.2\)\(3.2 \leqslant x < 3.6\)\(3.6 \leqslant x < 4\)
Observed frequency19171680
The estimated mean, based on the grouped data in the table above, is 2.69 , correct to 2 decimal places. It is decided that a goodness of fit test will only be conducted if the mean predicted from the probability density function is within \(10 \%\) of the estimated mean. Show that this condition is satisfied. The relevant expected frequencies are as follows.
Interval\(2 \leqslant x < 2.4\)\(2.4 \leqslant x < 2.8\)\(2.8 \leqslant x < 3.2\)\(3.2 \leqslant x < 3.6\)\(3.6 \leqslant x < 4\)
Expected frequency15.45616.03214.30410.2723.936
Show how the expected frequency for the interval \(3.2 \leqslant x < 3.6\) is obtained. Carry out the goodness of fit test at the 10\% significance level.
CAIE FP2 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_216_1205_253_470} A rigid body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), the centres of which are rigidly attached to the ends \(A\) and \(B\) of a uniform rod of mass \(3 m\) and length \(10 a\). The discs and the rod are in the same plane and \(O\) is the point on the rod such that \(A O = 4 a\) (see diagram). Show that the moment of inertia of the body about an axis through \(O\) perpendicular to the plane of the discs is \(81 m a ^ { 2 }\).
CAIE FP2 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find
  1. the angular acceleration of the disc,
  2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).
CAIE FP2 2012 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-5_389_702_484_719}
Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$
CAIE FP2 2012 November Q1
1 A particle \(P\) is moving in a circle of radius 1.5 m . At time \(t \mathrm {~s}\) its velocity is \(\left( k - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a positive constant. When \(t = 3\), the magnitudes of the radial and transverse components of the acceleration of \(P\) are equal. Find the possible values of \(k\).
CAIE FP2 2012 November Q2
2 A small bead of mass \(m\) is threaded on a thin smooth wire which forms a circle of radius \(a\). The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass \(4 m\) which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed \(\sqrt { } ( k g a )\). Show that, when the angle between the two parts of the string is \(\theta\), the normal force exerted on the bead by the wire is \(m g ( 3 \cos \theta + k - 6 )\), towards the centre. Given that the bead reaches the highest point of the wire, find an inequality which must be satisfied by \(k\).
CAIE FP2 2012 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{bcd7ee99-e382-4cb6-aa39-d8b385b01319-2_506_623_977_760} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\) and mass \(m\), are smoothly hinged at \(B\). They rest in equilibrium with \(C\) in contact with a smooth vertical wall and \(A\) in contact with a rough horizontal floor. The rods are in a vertical plane perpendicular to the wall. The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) respectively with the horizontal (see diagram). Show that
  1. the reaction at \(C\) has magnitude \(\frac { 1 } { 2 } m g \cot \beta\),
  2. \(\tan \alpha = 3 \tan \beta\). The coefficient of friction at \(A\) is \(\mu\). Given that \(\alpha = 60 ^ { \circ }\), find the least possible value of \(\mu\).
CAIE FP2 2012 November Q4
4 Three particles \(A , B\) and \(C\) have masses \(m , 2 m\) and \(m\) respectively. The particles are able to move on a smooth horizontal surface in a straight line, and \(B\) is between \(A\) and \(C\). Initially \(A\) is moving towards \(B\) with speed \(2 u\) and \(C\) is moving towards \(B\) with speed \(u\). The particle \(B\) is at rest. The coefficient of restitution between any pair of particles is \(e\). The first collision is between \(A\) and \(B\).
  1. Show that the speed of \(B\) immediately before its collision with \(C\) is \(\frac { 2 } { 3 } u ( 1 + e )\).
  2. Find the velocity of \(B\) immediately after its collision with \(C\).
  3. Given that \(e > \frac { 1 } { 2 }\), show that there are no further collisions between the particles.
CAIE FP2 2012 November Q5
5 Four identical uniform rods, each of mass \(m\) and length \(2 a\), are rigidly joined to form a square frame \(A B C D\). Show that the moment of inertia of the frame about an axis through \(A\) perpendicular to the plane of the frame is \(\frac { 40 } { 3 } m a ^ { 2 }\). The frame is suspended from \(A\) and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through \(A\). When the frame is at rest with \(C\) vertically below \(A\), it is given an angular velocity \(\sqrt { } \left( \frac { 6 g } { 5 a } \right)\). Find the angular velocity of the frame when \(A C\) makes an angle \(\theta\) with the downward vertical through \(A\). When \(A C\) is horizontal, the speed of \(C\) is \(k \sqrt { } ( g a )\). Find the value of \(k\) correct to 3 significant figures.
CAIE FP2 2012 November Q6
6 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 6 } \mathrm { e } ^ { - \frac { 1 } { 6 } x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$ Find
  1. the distribution function of \(X\),
  2. the probability that \(X\) lies between the median and the mean.
CAIE FP2 2012 November Q7
7 The speed \(v\) at which a javelin is thrown by an athlete is measured in \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results for 10 randomly chosen throws are summarised by $$\Sigma v = 1110.8 , \quad \Sigma ( v - \bar { v } ) ^ { 2 } = 333.9$$ where \(\bar { v }\) is the sample mean.
  1. Stating any necessary assumption, calculate a \(99 \%\) confidence interval for the mean speed of a throw. The results for a further 5 randomly chosen throws are now combined with the above results. It is found that the sample variance is smaller than that used in part (i).
  2. State, with reasons, whether a \(95 \%\) confidence interval calculated from the combined 15 results will be wider or less wide than that found in part (i).
CAIE FP2 2012 November Q8
8 Drinking glasses are sold in packs of 4. The manufacturer conducts a survey to assess the quality of the glasses. The results from a sample of 50 randomly chosen packs are summarised in the following table.
Number of perfect glasses01234
Number of packs13101719
Fit a binomial distribution to the data and carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2012 November Q9
9 Experiments are conducted to test the breaking strength of each of two types of rope, \(P\) and \(Q\). A random sample of 50 ropes of type \(P\) and a random sample of 70 ropes of type \(Q\) are selected. The breaking strengths, \(p\) and \(q\), measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$ Test, at the \(10 \%\) significance level, whether the mean breaking strengths of type \(P\) and type \(Q\) ropes are the same.
CAIE FP2 2012 November Q10
10 Delegates who travelled to a conference were asked to report the distance, \(y \mathrm {~km}\), that they had travelled and the time taken, \(x\) minutes. The values reported by a random sample of 8 delegates are given in the following table.
Delegate\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(x\)90467298526510582
\(y\)90556985455011074
$$\left[ \Sigma x = 610 , \Sigma x ^ { 2 } = 49682 , \Sigma y = 578 , \Sigma y ^ { 2 } = 45212 , \Sigma x y = 47136 . \right]$$ Find the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\). Estimate the time taken by a delegate who travelled 100 km to the conference. Calculate the product moment correlation coefficient for this sample.
CAIE FP2 2012 November Q11 EITHER
A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(8 m g\) and natural length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is pulled vertically downwards a distance \(\frac { 1 } { 4 } a\) from its equilibrium position and released from rest. Show that the string first becomes slack after a time \(\frac { 2 \pi } { 3 } \sqrt { } \left( \frac { a } { 8 g } \right)\). Find, in terms of \(a\), the total distance travelled by \(P\) from its release until it subsequently comes to instantaneous rest for the first time.
CAIE FP2 2012 November Q11 OR
\includegraphics[max width=\textwidth, alt={}]{bcd7ee99-e382-4cb6-aa39-d8b385b01319-5_453_807_1041_667}
The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 5\) only. For \(0 \leqslant x \leqslant 5\) the graph of its probability density function f consists of two straight line segments, as shown in the diagram. Find \(k\) and show that f is given by $$f ( x ) = \begin{cases} \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 2
\frac { 1 } { 4 } & 2 < x \leqslant 5
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
  1. Find the probability density function of \(Y\).
  2. Show that \(\mathrm { E } ( Y ) = 10.25\).
  3. Show that the median of \(Y\) is the square of the median of \(X\).
CAIE FP2 2013 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_553_435_258_854} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).
CAIE FP2 2013 November Q2
2 The point \(O\) is on the fixed line \(l\). The point \(A\) on \(l\) is such that \(O A = 3 \mathrm {~m}\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\) and period \(\pi\) seconds. When \(P\) is at \(A\) its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(P\) when it is at the point \(B\) on \(l\), where \(O B = 6 \mathrm {~m}\) and \(B\) is on the same side of \(O\) as \(A\). Find, correct to 2 decimal places, the time, in seconds, taken for \(P\) to travel directly from \(A\) to \(B\).
CAIE FP2 2013 November Q3
9 marks
3
\includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_570_419_1539_863} A uniform disc, of mass 2 kg and radius 0.2 m , is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a small block of mass 4 kg , which hangs freely (see diagram). The system is released from rest. During the subsequent motion, the block experiences a constant resistance to its motion, of magnitude \(R \mathrm {~N}\). Given that the angular speed of the disc after it has turned through 2 radians is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find \(R\) and the tension in the string.
[0pt] [9]
CAIE FP2 2013 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-3_567_575_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),
  1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
  2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).
CAIE FP2 2013 November Q5
5 Two uniform small smooth spheres \(A\) and \(B\), of equal radii, have masses \(2 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). After the collision \(B\) goes on to collide directly with a fixed smooth vertical barrier, before colliding with \(A\) again. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\) and the coefficient of restitution between \(B\) and the barrier is \(e\). After the second collision between \(A\) and \(B\), the speed of \(B\) is five times the speed of \(A\). Find the two possible values of \(e\).
CAIE FP2 2013 November Q6
6 A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by the random variable \(X\). State the mean value of \(X\). Find the probability that obtaining a 5 or a 6 takes more than 8 throws. Find the least integer \(n\) such that the probability of obtaining a 5 or a 6 in fewer than \(n\) throws is more than 0.99.