Questions FP2 (1157 questions)

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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.
AQA FP2 2010 January Q1
1
  1. Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
    1. Express $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$ in terms of \(\cosh x\).
    2. Sketch the curve \(y = \cosh x\).
    3. Hence solve the equation $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$ giving your answers in logarithmic form.
AQA FP2 2010 January Q2
2
  1. On the same Argand diagram, draw:
    1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 4\);
    2. the locus of points satisfying \(| z | = | z - 2 \mathrm { i } |\).
  2. Indicate on your sketch the set of points satisfying both $$| z - 4 + 2 i | \leqslant 4$$ and $$| z | \geqslant | z - 2 \mathrm { i } |$$
AQA FP2 2010 January Q3
3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).
    1. Write down another root, \(\beta\), of the equation.
    2. Find the third root, \(\gamma\).
    3. Find the values of \(p\) and \(q\).
    1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
    3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.
AQA FP2 2010 January Q4
4 A curve \(C\) is given parametrically by the equations $$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$
  1. Express $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$ in terms of \(\cosh t\).
  2. The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
    1. Show that \(S\), the area of the curved surface generated, is given by $$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
    2. Find the exact value of \(S\).
AQA FP2 2010 January Q5
5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:
    1. \(u _ { 1 } = 6\);
    2. \(u _ { 2 } = 42\);
    3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$
AQA FP2 2010 January Q6
6
  1. Show that the substitution \(t = \tan \theta\) transforms the integral $$\int \frac { \mathrm { d } \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$$ into $$\int \frac { \mathrm { d } t } { 9 + t ^ { 2 } }$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { d \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } = \frac { \pi } { 18 }$$
AQA FP2 2010 January Q7
7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$
  1. Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
  2. Show that $$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$
AQA FP2 2010 January Q8
8
    1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
    2. Write down the five other non-real roots in terms of \(\omega\).
  2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
  3. Show that:
    1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
    2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\).
AQA FP2 2011 January Q1
1
  1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 4 + 3 \mathrm { i } | = 5$$
    1. Indicate on your diagram the point \(P\) representing \(z _ { 1 }\), where both $$\left| z _ { 1 } - 4 + 3 \mathrm { i } \right| = 5 \quad \text { and } \quad \arg z _ { 1 } = 0$$
    2. Find the value of \(\left| z _ { 1 } \right|\).
AQA FP2 2011 January Q2
2
  1. Given that $$u _ { r } = \frac { 1 } { 6 } r ( r + 1 ) ( 4 r + 11 )$$ show that $$u _ { r } - u _ { r - 1 } = r ( 2 r + 3 )$$
  2. Hence find the sum of the first hundred terms of the series $$1 \times 5 + 2 \times 7 + 3 \times 9 + \ldots + r ( 2 r + 3 ) + \ldots$$
AQA FP2 2011 January Q3
3
  1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
  2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
    It is given that \(\alpha = 1 + \mathrm { i }\).
    1. Find the value of \(k\).
    2. Show that \(\beta + \gamma = 4\).
    3. Find the values of \(\beta\) and \(\gamma\).
AQA FP2 2011 January Q4
4
  1. Prove that the curve $$y = 12 \cosh x - 8 \sinh x - x$$ has exactly one stationary point.
  2. Given that the coordinates of this stationary point are \(( a , b )\), show that \(a + b = 9\).
AQA FP2 2011 January Q5
5
  1. Given that \(u = \sqrt { 1 - x ^ { 2 } }\), find \(\frac { \mathrm { d } u } { \mathrm {~d} x }\).
  2. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$ where \(a\) and \(b\) are rational numbers.
AQA FP2 2011 January Q6
6
  1. Given that $$x = \ln ( \sec t + \tan t ) - \sin t$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
  2. A curve is given parametrically by the equations $$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$ The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\). Show that \(s = \ln p\), where \(p\) is an integer.
AQA FP2 2011 January Q7
7
  1. Given that $$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$ show that $$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$ where \(a\) is an integer.
  2. Prove by induction that \(12 ^ { n } + 2 \times 5 ^ { n - 1 }\) is divisible by 7 for all integers \(n \geqslant 1\).
AQA FP2 2011 January Q8
8
  1. Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
    1. \(\quad 4 ( 1 + i \sqrt { 3 } )\);
    2. \(4 ( 1 - i \sqrt { 3 } )\).
  2. The complex number \(z\) satisfies the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
    1. Solve the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. Illustrate the roots on an Argand diagram.
    1. Explain why the sum of the roots of the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ is zero.
    2. Deduce that \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).
AQA FP2 2012 January Q1
1
  1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
  2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).