Questions CP1 (58 questions)

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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
Edexcel CP1 Specimen Q2
  1. Prove by induction that for all positive integers \(n\),
$$f ( n ) = 2 ^ { 3 n + 1 } + 3 \left( 5 ^ { 2 n + 1 } \right)$$ is divisible by 17
Edexcel CP1 Specimen Q3
3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.
Edexcel CP1 Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-08_492_1063_214_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 4 + \cos 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 9 } { 2 }\)
The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.
Edexcel CP1 Specimen Q5
  1. A pond initially contains 1000 litres of unpolluted water.
The pond is leaking at a constant rate of 20 litres per day.
It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water. It is assumed that the pollutant instantly dissolves throughout the pond upon entry.
Given that there are \(x\) grams of the pollutant in the pond after \(t\) days,
  1. show that the situation can be modelled by the differential equation, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 50 - \frac { 4 x } { 200 + t }$$
  2. Hence find the number of grams of pollutant in the pond after 8 days.
  3. Explain how the model could be refined.
Edexcel CP1 Specimen Q6
6. $$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$
  1. Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
  2. Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is $$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
  3. Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of $$\mathrm { f } ( x ) + \ln k$$ where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).
Edexcel CP1 Specimen Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-14_259_327_214_868} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the image of a gold pendant which has height 2 cm . The pendant is modelled by a solid of revolution of a curve \(C\) about the \(y\)-axis. The curve \(C\) has parametric equations $$x = \cos \theta + \frac { 1 } { 2 } \sin 2 \theta , \quad y = - ( 1 + \sin \theta ) \quad 0 \leqslant \theta \leqslant 2 \pi$$
  1. Show that a Cartesian equation of the curve \(C\) is $$x ^ { 2 } = - \left( y ^ { 4 } + 2 y ^ { 3 } \right)$$
  2. Hence, using the model, find, in \(\mathrm { cm } ^ { 3 }\), the volume of the pendant.
Edexcel CP1 Specimen Q8
  1. The line \(l _ { 1 }\) has equation \(\frac { x - 2 } { 4 } = \frac { y - 4 } { - 2 } = \frac { z + 6 } { 1 }\)
The plane \(\Pi\) has equation \(x - 2 y + z = 6\)
The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi\).
Find a vector equation of the line \(l _ { 2 }\)
Edexcel CP1 Specimen Q9
  1. A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line.
The vertical displacement, \(x\) metres, of the top of the capsule below its initial position at time \(t\) seconds is modelled by the differential equation, $$m \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 200 \cos t , \quad t \geqslant 0$$ where \(m\) is the mass of the capsule including its passengers, in thousands of kilograms.
The maximum permissible weight for the capsule, including its passengers, is 30000 N .
Taking the value of \(g\) to be \(10 \mathrm {~ms} ^ { - 2 }\) and assuming the capsule is at its maximum permissible weight,
    1. explain why the value of \(m\) is 3
    2. show that a particular solution to the differential equation is $$x = 40 \sin t - 20 \cos t$$
    3. hence find the general solution of the differential equation.
  2. Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released.