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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
SPS SPS SM Pure 2022 June Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-34_606_737_146_760} \captionsetup{labelformat=empty} \caption{Figure 6
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.}
\end{figure} Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2 ^ { 2 x } .$$ The point \(P ( a , 96 \sqrt { } 2 )\) lies on the curve.
  1. Find the exact value of \(a\). The curve with equation \(y = 3 \times 2 ^ { 2 x }\) meets the curve with equation \(y = 6 ^ { 3 - x }\) at the point \(Q\).
  2. Show that the \(x\) coordinate of \(Q\) is \(\frac { 3 + 2 \log _ { 2 } 3 } { 3 + \log _ { 2 } 3 }\).
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SPS SPS SM Pure 2022 June Q17
17. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-36_613_860_189_653} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = 2 ^ { x ^ { 2 } } - x\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = - 0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).
  1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). The estimate for R is found to be 2.58 .
    Using this value, and showing your working,
  2. estimate the value of \(\int _ { - 0.5 } ^ { 1.5 } \left( 2 ^ { x ^ { 2 } + 1 } + 2 x \right) d x\).
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SPS SPS FM Mechanics 2021 September Q1
  1. A car is initially travelling with a constant velocity of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T \mathrm {~s}\). It then decelerates at a constant rate for \(\frac { T } { 2 } \mathrm {~s}\), reaching a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then immediately accelerates at a constant rate for \(\frac { 3 T } { 2 } \mathrm {~s}\) reaching a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    a Sketch a velocity-time graph to illustrate the motion.
    b Given that the car travels a total distance of 1312.5 m over the journey described, find the value of \(T\).
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  2. A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\) the displacement \(s \mathrm {~cm}\) from a fixed point \(O\) is given by: \(s = \frac { 1 } { 6 } \left( 8 t ^ { 3 } - 105 t ^ { 2 } + 144 t + 540 \right)\).
    Find the distance between the points at which the particle is instantaneously at rest.
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  3. A cylindrical object with mass 8 kg rests on two cylindrical bars of equal radius. The lines connecting the centre of each of the bars to the centre of the object make an angle of \(40 ^ { \circ }\) to the vertical.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3c44f549-39f9-4b51-9aa3-b918c39c5e5b-06_647_506_333_694}
\end{figure} a Draw a diagram showing all the forces acting on the object. Describe each of the forces using words.
b Calculate the magnitude of the force on each of the bars due to the cylindrical object.
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SPS SPS FM Mechanics 2021 September Q4
4. A box \(A\) of mass 0.8 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a sphere \(B\) of mass 1.2 kg , which hangs freely below the pulley. The magnitude of the frictional force between \(A\) and the table is \(F N\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.9 m in 0.8 s . Modelling \(A\) and \(B\) as particles, calculate
a the acceleration of \(B\),
b the tension in the string,
c the value of \(F\).
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SPS SPS FM Mechanics 2021 September Q5
5. In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal.
a Find the exact value of the coefficient of friction between the particle and the plane. A horizontal force of 36 N is now applied to the particle.
b Find how far down the plane the particle travels after the force has been applied for 4 s .
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SPS SPS FM Pure 2021 September Q1
  1. (a) The equation \(\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0\) has a single root, \(\alpha\).
Show that \(\alpha\) lies between 3 and 4 .
(b) Use the recurrence relation \(x _ { n + 1 } = \left( 2 - \mathrm { e } ^ { - x _ { n } } \right) ^ { 2 }\), with \(x _ { 1 } = 3.5\), to find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
(c) The diagram below shows parts of the graphs of \(y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }\) and \(y = x\), and a position of \(x _ { 1 }\). On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{f5cae2a4-a0f4-4227-a773-fcdecd87cb46-04_1180_1502_808_374}
SPS SPS FM Pure 2021 September Q2
2. (a) Find the binomial expansion of \(( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
(2 marks)
(b) (i) Find the binomial expansion of \(( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
(ii) Given that \(\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }\), use your binomial expansion from part (b)(i) to obtain an approximation to \(\sqrt [ 3 ] { \frac { 2 } { 7 } }\), giving your answer to six decimal places.
(2 marks)
SPS SPS FM Pure 2021 September Q3
3. The diagram below shows the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
\includegraphics[max width=\textwidth, alt={}, center]{f5cae2a4-a0f4-4227-a773-fcdecd87cb46-08_645_1256_246_497}
  1. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
  2. Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$
SPS SPS FM Pure 2021 September Q4
4. By forming and solving a quadratic equation, solve the equation $$8 \sec x - 2 \sec ^ { 2 } x = \tan ^ { 2 } x - 2$$ in the interval \(0 < x < 2 \pi\), giving the values of \(x\) in radians to three significant figures.
SPS SPS FM Pure 2021 September Q5
5.
    1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
    2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
  2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$
SPS SPS FM Pure 2021 September Q6
  1. The functions \(f\) and \(g\) are defined with their respective domains by
$$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } - 6 x + 5 , & \text { for } x \geqslant 3
\mathrm {~g} ( x ) = | x - 6 | , & \text { for all real values of } x \end{array}$$
  1. Find the range of f .
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\). Give your answer in its simplest form.
    1. Find \(\mathrm { gf } ( x )\).
    2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
SPS SPS FM Pure 2021 September Q7
7. The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 1 , - 5,6 )\) and \(( - 4,5 , - 1 )\) respectively.
The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3
- 2
4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7
- 7
5 \end{array} \right]\).
  1. Show that the point \(C\) lies on the line \(l\).
  2. Find a vector equation of the line that passes through points \(A\) and \(B\).
  3. The point \(D\) lies on the line through \(A\) and \(B\) such that the angle \(C D A\) is a right angle. Find the coordinates of \(D\).
  4. The point \(E\) lies on the line through \(A\) and \(B\) such that the area of triangle \(A C E\) is three times the area of triangle \(A C D\). Find the coordinates of the two possible positions of \(E\).
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SPS SPS FM Pure 2021 September Q8
8. (a) It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
  1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
    (1 mark)
  2. Solve the equation $$( z - 2 i ) ^ { * } = 4 i z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
    (b) It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q \mathrm { i }\) is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
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SPS SPS FM Pure 2021 September Q9
9. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { n + 1 } = \frac { 5 u _ { n } - 3 } { 3 u _ { n } - 1 }$$ Prove by induction that, for all integers \(n \geqslant 1\), $$u _ { n } = \frac { 3 n + 1 } { 3 n - 1 }$$ (6 marks)
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SPS SPS FM Pure 2021 September Q10
10. The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$ where \(x\) metres is the height of the platform above the ground after time \(t\) seconds.
At \(t = 0\), the height of the platform above the ground is 4 metres.
Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.
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SPS SPS FM Pure 2021 September Q11
11. The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f5cae2a4-a0f4-4227-a773-fcdecd87cb46-24_986_993_228_623}
  1. Find the matrix which represents the stretch that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 2 }\).
  2. The triangle \(T _ { 2 }\) is reflected in the line \(y = \sqrt { 3 } x\) to give a third triangle, \(T _ { 3 }\). Find, using surd forms where appropriate:
    1. the matrix which represents the reflection that maps triangle \(T _ { 2 }\) onto triangle \(T _ { 3 }\);
    2. the matrix which represents the combined transformation that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 3 }\).
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SPS SPS FM Statistics 2021 September Q1
  1. a) 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy.
    (3 marks)
    b) A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers.
    How many different teams can be selected?
    (3 marks)
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  2. \(\quad \mathrm { P } ( E ) = 0.25 , \mathrm { P } ( F ) = 0.4\) and \(\mathrm { P } ( E \cap F ) = 0.12\)
    a Find \(P \left( E ^ { \prime } \mid F ^ { \prime } \right)\)
    b Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer.
The event \(G\) has \(\mathrm { P } ( G ) = 0.15\)
The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
c Draw a Venn diagram to illustrate the events \(E , F\) and \(G\), giving the probabilities for each region.
d Find \(\mathrm { P } \left( [ F \cup G ] ^ { \prime } \right)\)
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SPS SPS FM Statistics 2021 September Q3
3. A group of students were surveyed by a principal and \(\frac { 2 } { 3 }\) were found to always hand in assignments on time. When questioned about their assignments \(\frac { 3 } { 5 }\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac { 11 } { 20 }\) hand them in on time.
a Draw a tree diagram to represent this information.
b Find the probability that a randomly selected student:
i always start their assignments on the day they are issued and hand them in on time.
ii does not always hand in assignments on time and does not start their assignments on the day they are issued.
c Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer.
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SPS SPS FM Statistics 2021 September Q4
4. In a town, \(54 \%\) of the residents are female and \(46 \%\) are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female.
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SPS SPS FM Statistics 2021 September Q5
5. The heights of a population of men are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). It is known that \(20 \%\) of the men are taller than 180 cm and \(5 \%\) are shorter than 170 cm .
a Sketch a diagram to show the distribution of heights represented by this information.
b Find the value of \(\mu\) and \(\sigma\).
c Three men are selected at random, find the probability that they are all taller than 175 cm .
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SPS SPS SM Mechanics 2021 September Q1
  1. A racing car starts from rest at the point \(A\) and moves with constant acceleration.of \(11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 8 s . The velocity it has reached after 8 s is then maintained for \(T \mathrm {~s}\). The racing car then decelerates from this velocity to \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a further 2 s , reaching point \(B\).
    a Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch.
    b Given that the distance between \(A\) and \(B\) is 1404 m , find the value of \(T\).
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  2. A particle \(P\) is acted upon by three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) given by \(\mathbf { F } _ { 1 } = ( 6 \mathbf { i } - 4 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,
    a find the value of \(a\) and the value of \(b\).
The force \(\mathbf { F } _ { 2 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 3 }\) is \(\mathbf { R }\).
b Find the magnitude of \(\mathbf { R }\).
c Find the angle, to \(0.1 ^ { \circ }\), that \(\mathbf { R }\) makes with \(\mathbf { i }\).
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SPS SPS SM Mechanics 2021 September Q3
3. A car of mass 1200 kg pulls a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of 3200 N . The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
a find the resistance to motion on the trailer,
b find the tension in the tow-rope. When the car and trailer are travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,
c find the distance the trailer travels before coming to a stop,
d state how you have used the modelling assumption that the tow-rope is inextensible.
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SPS SPS SM Mechanics 2021 September Q4
4. A car starts from the point \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the distance of the car from \(A\) is \(s \mathrm {~m}\), where \(s = 30 t - 0.4 t ^ { 2 } , 0 \leqslant t \leqslant 25\). The car reaches the point \(B\) when \(t = 25\).
a Find the distance \(A B\).
b Show that the car travels with a constant acceleration and state the value of this acceleration. A runner passes through \(B\) when \(t = 0\) with an initial velocity of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) running directly towards \(A\). The runner has a constant acceleration of \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
c Find the distance from \(A\) at which the runner and the car pass one another.
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SPS SPS SM Pure 2021 September Q1
5 marks
1.
  1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
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  2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
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SPS SPS SM Pure 2021 September Q2
2. (a) (i) Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
(ii) Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
(b) (i) Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).