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Edexcel AS Paper 2 2021 November Q2
10 marks Moderate -0.8
  1. A particle \(P\) moves along a straight line.
At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.
  1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
  2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)
Edexcel AS Paper 2 2021 November Q3
13 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a022ec0-7640-4664-87a6-1963309cad6a-08_761_595_210_735} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit the pulley. The balls are modelled as particles.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Air resistance is modelled as being negligible.
Using this model,
    1. write down an equation of motion for \(P\),
    2. write down an equation of motion for \(Q\),
  2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
  3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
  4. State how this would affect the accelerations of the particles.
    VIAV SIHI NI III IM ION OCVIIN SIHI NI III M M O N OOVIIV SIHI NI IIIYM ION OC
Edexcel AS Paper 2 2021 November Q1
2 marks Easy -1.2
1. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-02_399_743_248_662} The Venn diagram, where \(p\) is a probability, shows the 3 events \(A , B\) and \(C\) with their associated probabilities.
  1. Find the value of \(p\).
  2. Write down a pair of mutually exclusive events from \(A , B\) and \(C\).
Edexcel AS Paper 2 2021 November Q2
9 marks Moderate -0.3
  1. The partially completed table and partially completed histogram give information about the ages of passengers on an airline.
There were no passengers aged 90 or over.
Age ( \(x\) years)\(0 \leqslant x < 5\)\(5 \leqslant x < 20\)\(20 \leqslant x < 40\)\(40 \leqslant x < 65\)\(65 \leqslant x < 80\)\(80 \leqslant x < 90\)
Frequency545901
\includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}
  1. Complete the histogram.
  2. Use linear interpolation to estimate the median age. An outlier is defined as a value greater than \(Q _ { 3 } + 1.5 \times\) interquartile range.
    Given that \(Q _ { 1 } = 27.3\) and \(Q _ { 3 } = 58.9\)
  3. determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
    (2)
Edexcel AS Paper 2 2021 November Q3
4 marks Easy -1.8
  1. Helen is studying one of the qualitative variables from the large data set for Heathrow from 2015.
She started with the data from 3rd May and then took every 10th reading.
There were only 3 different outcomes with the following frequencies
Outcome\(A\)\(B\)\(C\)
Frequency1621
  1. State the sampling technique Helen used.
  2. From your knowledge of the large data set
    1. suggest which variable was being studied,
    2. state the name of outcome \(A\). George is also studying the same variable from the large data set for Heathrow from 2015. He started with the data from 5th May and then took every 10th reading and obtained the following
      Outcome\(A\)\(B\)\(C\)
      Frequency1611
      Helen and George decided they should examine all of the data for this variable for Heathrow from 2015 and obtained the following
      Outcome\(A\)\(B\)\(C\)
      Frequency155263
  3. State what inference Helen and George could reliably make from their original samples about the outcomes of this variable at Heathrow, for the period covered by the large data set in 2015.
Edexcel AS Paper 2 2021 November Q4
10 marks Standard +0.3
  1. A nursery has a sack containing a large number of coloured beads of which \(14 \%\) are coloured red.
Aliya takes a random sample of 18 beads from the sack to make a bracelet.
  1. State a suitable binomial distribution to model the number of red beads in Aliya's bracelet.
  2. Use this binomial distribution to find the probability that
    1. Aliya has just 1 red bead in her bracelet,
    2. there are at least 4 red beads in Aliya's bracelet.
  3. Comment on the suitability of a binomial distribution to model this situation. After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.
  4. Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.
  5. Find the \(p\)-value in this case.
Edexcel AS Paper 2 2021 November Q5
5 marks Standard +0.8
  1. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.
Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).
Edexcel PMT Mocks Q1
6 marks Standard +0.3
  1. a. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use \(x = \frac { 1 } { 36 }\) in the expansion to find an approximation for \(\sqrt { 2 }\). There is no need to carry out the calculation.
Edexcel PMT Mocks Q2
3 marks Moderate -0.8
2. The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 } \\ & C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the \(x\)-coordinate of the point where \(C _ { 1 }\) and \(C _ { 2 }\) intersect is \(\frac { - 2 } { 5 }\).
Edexcel PMT Mocks Q3
5 marks Moderate -0.8
3. Relative to a fixed origin,
  • point \(A\) has position vector \(- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }\)
  • point \(B\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }\)
  • point \(C\) has position vector \(\mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
  • point \(D\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) a. Show that \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\) are parallel and the ratio \(\overrightarrow { A B } : \overrightarrow { C D }\) in its simplest form.
    b. Hence describe the quadrilateral \(A B C D\).
Edexcel PMT Mocks Q4
6 marks Moderate -0.8
  1. Ben starts a new company.
  • In year 1 his profits will be \(\pounds 24000\).
  • In year 11 his profit is predicted to be \(\pounds 64000\).
Model \(\boldsymbol { P }\) assumes that his profit will increase by the same amount each year.
a. According to model \(\boldsymbol { P }\), determine Ben's profit in year 5. Model \(\boldsymbol { Q }\) assumes that his profit will increase by the same percentage each year.
b. According to model \(\boldsymbol { Q }\), determine Ben's profit in year 5 . Give your answer to the nearest £10.
Edexcel PMT Mocks Q5
9 marks Moderate -0.3
5. The function f is defined by $$\mathrm { f } : x \rightarrow \frac { 2 x - 3 } { x - 1 } \quad x \in R , x \neq 1$$ a. Find \(f ^ { - 1 } ( 3 )\).
b. Show that $$\mathrm { ff } ( x ) = \frac { x + p } { x - 2 } \quad x \in R , \quad x \neq 2$$ where \(p\) is an integer to be found. The function g is defined by $$g : x \rightarrow x ^ { 2 } - 5 x \quad x \in R , 0 \leq x \leq 6$$ c. Find the range of g .
d. Explain why the function g does not have an inverse.
Edexcel PMT Mocks Q6
7 marks Standard +0.3
6. a. Express \(4 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\), and give the value of \(\alpha\), in degrees, to 2 decimal places. $$T = \frac { 8400 } { 19 + ( 4 \sin x - 5 \cos x ) ^ { 2 } } , x > 0$$ b. Use your answer to part \(a\) to calculate
i. the minimum value of \(T\).
ii. the smallest value of \(x , x > 0\), at which this minimum value occurs.
Edexcel PMT Mocks Q7
5 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-09_928_1093_258_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\). The curve \(C\) meets \(l\) at the points \(( 2,4 )\) and \(( 6,0 )\) as shown. The shaded region \(R\), shown shaded in Figure 1, is bounded by \(\mathrm { C } , l\) and the \(y\)-axis. Given that \(\mathrm { f } ( x )\) is a quadratic function in \(x\), use inequalities to define region \(R\).
Edexcel PMT Mocks Q8
9 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-11_1112_1211_280_386} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\) c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\) Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).
Edexcel PMT Mocks Q9
8 marks Standard +0.8
9. a. Use the substitution \(t ^ { 2 } = 2 x - 5\) to show that $$\int \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x = \int \frac { 2 t } { t ^ { 2 } + 6 t + 5 } \mathrm {~d} t$$ b. Hence find the exact value of $$\int _ { 3 } ^ { 27 } \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x$$
Edexcel PMT Mocks Q10
6 marks Challenging +1.2
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-16_1116_1433_360_420} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 64\) with centre \(O _ { 1 }\).
Circle \(C _ { 2 }\) has equation \(( x - 6 ) ^ { 2 } + y ^ { 2 } = 100\) with centre \(O _ { 2 }\).
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
a. Show that angle \(A O _ { 2 } B = 1.85\) radians to 3 significant figures.
(3)
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.
Edexcel PMT Mocks Q11
2 marks Easy -1.2
11. In a science experiment, a radio active particle, \(N\), decays over time, \(t\), measured in minutes. The rate of decay of a particle is proportional to the number of particles remaining. Write down a suitable equation for the rate of change of the number of particles, \(N\) in terms of \(t\).
Edexcel PMT Mocks Q12
8 marks Standard +0.8
12. a. Show that $$\sec \theta - \cos \theta = \sin \theta \tan \theta \quad \theta \neq ( \pi n ) ^ { 0 } \quad n \in Z$$ b. Hence, or otherwise, solve for \(0 < x \leq \pi\) $$\sec x - \cos x = \sin x \tan \left( 3 x - \frac { \pi } { 9 } \right)$$
Edexcel PMT Mocks Q13
6 marks Standard +0.8
  1. A sequence \(a _ { 1 } , a _ { 2 } a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = 5 - p a _ { n } \quad n \geq 1$$ where \(p\) is a constant.
Given that
  • \(a _ { 1 } = 4\)
  • the sequence is a periodic sequence of order 2.
    a. Write down an expression for \(a _ { 2 }\) and \(a _ { 3 }\).
    b. Find the value of \(p\).
    c. Find \(\sum _ { r = 1 } ^ { 21 } a _ { r }\)
Edexcel PMT Mocks Q14
10 marks Standard +0.3
  1. A circular stain is growing.
The rate of increase of its radius is inversely proportional to the square of the radius. At time \(t\) seconds the circular stain has radius \(r \mathrm {~cm}\) and area \(A \mathrm {~cm} ^ { 2 }\).
a. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { k } { \sqrt { A } }\). Given that
  • the initial area of the circular stain is \(0.09 \mathrm {~cm} ^ { 2 }\).
  • after 10 seconds the area of the circular stain is \(0.36 \mathrm {~cm} ^ { 2 }\).
    b. Solve the differential equation to find a complete equation linking \(A\) and \(t\).
Edexcel PMT Mocks Q15
6 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x\) b. Find the coordinates of the points of inflection of the curve.
Edexcel PMT Mocks Q16
4 marks Easy -1.2
16. Use algebra to prove that the product of any two consecutive odd numbers is an odd number.
Edexcel PMT Mocks Q1
3 marks Standard +0.8
  1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of
$$\frac { \theta \tan 3 \theta } { \cos 2 \theta - 1 }$$
Edexcel PMT Mocks Q2
4 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-03_442_552_351_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sector \(P O Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(P O Q\) is 0.5 radians.
The area of the sector is \(9 \mathrm {~cm} ^ { 2 }\).
Show that the perimeter of the sector is \(k\) times the length of the arc, where \(k\) is an integer.