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CAIE FP2 2018 November Q11 OR
In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. $$\Sigma x = 10.56 \quad \Sigma x ^ { 2 } = 14.1775 \quad \Sigma y = 12.39 \quad \Sigma y ^ { 2 } = 15.894$$ A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\).
  1. Test, at the \(10 \%\) significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal.
    A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as $$\Sigma x = 10.24 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.294$$ where \(\bar { x }\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p \mathrm {~kg}\). A test of this claim is carried out at the \(10 \%\) significance level, using only this second sample from lake \(A\). This test supports the scientist's claim.
  2. Find the greatest possible value of \(p\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2018 November Q2
2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
    In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
  2. Find the value of \(e\).
  3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{f2073c6e-0f76-4246-89a7-2f3a9f7aaff8-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
  4. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
    The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
  5. Find the possible values of \(x\).
CAIE FP2 2019 November Q1
5 marks
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5]
\includegraphics[max width=\textwidth, alt={}, center]{76ab909c-b34d-4a48-84e8-8df6f0255a86-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q3
3 Three uniform small spheres \(A , B\) and \(C\) have equal radii and masses \(5 m , 5 m\) and \(3 m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).
  1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac { 1 } { 2 } u ( 1 - e )\) and find the speed of \(B\).
    Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.
  2. Find the set of possible values of \(e\).
CAIE FP2 2019 November Q4
4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{76ab909c-b34d-4a48-84e8-8df6f0255a86-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2019 November Q6
6 A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95\% confidence interval for the population mean height, \(\mu\) metres, is calculated from the data as \(1.65 \leqslant \mu \leqslant 1.85\).
  1. Find an unbiased estimate for the population variance.
  2. Denoting the height of a member of the club by \(x\) metres, find \(\Sigma x ^ { 2 }\) for this sample of 9 members.
CAIE FP2 2019 November Q7
7 The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - a t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200 .
  1. Write down the value of \(a\).
  2. Find the probability that an electrical component of this type develops a fault in less than 150 days.
    A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99 .
  3. Find the smallest possible value of \(n\).
CAIE FP2 2019 November Q8
8 A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. ( 1 tonne \(= 1000 \mathrm {~kg}\).) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x ^ { 2 } = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes \({ } ^ { 2 }\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the \(10 \%\) significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\).
CAIE FP2 2019 November Q9
9 A random sample of five pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The values are shown in the following table, where \(p\) and \(q\) are constants.
\(x\)12345
\(y\)4\(p\)\(q\)21
The equation of the regression line of \(y\) on \(x\) is \(y = - 0.5 x + 3.5\).
  1. Find the values of \(p\) and \(q\).
  2. Find the value of the product moment correlation coefficient.
CAIE FP2 2019 November Q10
10 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 30 } \left( \frac { 8 } { x ^ { 2 } } + 3 x ^ { 2 } - 14 \right) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  2. Find the probability density function of \(Y\).
  3. Find the value of \(y\) such that \(\mathrm { P } ( Y < y ) = 0.8\).
CAIE FP2 2019 November Q11 EITHER
The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac { 2 } { 3 } \mathrm {~kg}\) is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N . The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N , has one end attached to \(P\) and the other end attached to \(B\).
  1. Show that when \(P\) is in equilibrium \(A P = 0.75 \mathrm {~m}\).
    The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
  2. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. Find the speed of \(P\) when it passes through the equilibrium position.
  4. Find the speed of \(P\) when its acceleration is equal to half of its maximum value.
CAIE FP2 2019 November Q11 OR
The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
Number of repairs in a week012345\(\geqslant 6\)
Number of weeks61596310
  1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data.
    Records over a longer period of time indicate that the mean number of repairs in a week is 1.6 . The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
    Number of repairs in a week012345\(\geqslant 6\)
    Expected frequency8.07612.92110.3375.5132.205\(a\)\(b\)
  2. Show that \(a = 0.706\) and find the value of the constant \(b\).
  3. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a \(10 \%\) significance level.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2019 November Q1
5 marks
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5]
\includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2019 November Q1
5 marks
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5]
\includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2017 Specimen Q2
2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\).
  1. Find expressions for the speeds of \(A\) and \(B\) immediately after the collision.
    Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal.
  2. Find \(e\).
  3. Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\).
    \(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a \mathrm {~m}\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). It is given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\).
CAIE FP2 2017 Specimen Q4
4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\).
  1. Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\).
    It is now given that \(u = \sqrt { } ( a g )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\).
  2. Find the speed of the combined particle immediately after the collision.
    In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\).
  3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
  4. Find the value of \(\cos \theta\) when the string becomes slack.
CAIE FP2 2017 Specimen Q5
5 A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar { x }\) is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a \(95 \%\) confidence interval for the population mean.
CAIE FP2 2017 Specimen Q6
6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\).
  1. Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328}
  2. Find \(\mathrm { P } ( X = 4 )\).
  3. Find \(\mathrm { P } ( X > 4 )\).
  4. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).
CAIE FP2 2017 Specimen Q7
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  1. Show that \(Y\) has probability density function given by $$g ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16
    0 & \text { otherwise } \end{cases}$$
  2. Find the median value of \(Y\).
  3. Find the expected value of \(Y\).
CAIE FP2 2017 Specimen Q8
8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.
CAIE FP2 2017 Specimen Q9
9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \quad \Sigma x ^ { 2 } = 29950 \quad \Sigma y = 400 \quad \Sigma y ^ { 2 } = 21226 \quad \Sigma x y = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination.
  1. Estimate the mark that this student would have obtained in the French examination.
  2. Test, at the \(5 \%\) significance level, whether there is non-zero correlation between marks in Mathematics and marks in French.
CAIE FP2 2017 Specimen Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-18_598_601_440_772}
An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram).
  1. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\).
  2. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\).
  3. Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).
CAIE FP2 2017 Specimen Q10 OR
A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$
  1. Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make.
    A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\) 's Crown plants.
  2. Test, at the \(5 \%\) significance level, whether Farmer \(A\) 's claim is justified.