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SPS SPS SM Pure 2021 September Q3
3. A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).
  1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
  3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
SPS SPS SM Pure 2021 September Q4
4.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-10_656_776_210_721} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. If \(f ( x ) = x ^ { 3 }\), find \(f ^ { \prime } ( x )\) from first principles.
SPS SPS SM Pure 2021 September Q5
5. A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$
  1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
    1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
    3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.
SPS SPS SM Pure 2021 September Q6
8 marks
6. The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_499_718_219_703} The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
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  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
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  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_488_700_1409_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
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SPS SPS SM Pure 2021 September Q7
7. The \(n\)th term of a sequence is \(u _ { n }\). The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first two terms of the sequence are given by \(u _ { 1 } = 96\) and \(u _ { 2 } = 72\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 24 .
  1. Show that \(p = \frac { 2 } { 3 }\).
  2. Find the value of \(u _ { 3 }\).
SPS SPS SM Pure 2021 September Q8
8. (i) Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
(3 marks)
(ii) Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
(2 marks)
SPS SPS SM Pure 2021 September Q9
4 marks
9. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      [0pt] [2 marks]
    2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
      [0pt] [2 marks]
    1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
    2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).
SPS SPS SM Pure 2021 September Q10
10. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
SPS SPS SM Pure 2021 September Q11
11.
  1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
  2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.
SPS SPS SM Statistics 2021 September Q1
  1. A random sample of distances travelled to work for 120 commuters from a train station in Devon is recorded. The distances travelled, to the nearest mile, are summarised below.
Distance (to the nearest mile)Number of commuters
0-910
10-1919
20-2943
30-3925
40-498
50-596
60-695
70-793
80-891
For this distribution:
a estimate the median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\). The mid-point of the lowest class was taken to be 4.75 giving: $$\Sigma f x = 3552.5 \text { and } \Sigma f x ^ { 2 } = 138043.125$$ b Estimate the mean and the standard deviation of this distribution.
c Explain why the median is less than the mean for these data.
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SPS SPS SM Statistics 2021 September Q2
2. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a grouped frequency table. The mid-point of each class in the table was represented by \(x\) and the corresponding frequency for that class by \(f\). The data were then coded using: $$y = \frac { ( x - 755.0 ) } { 2.5 }$$ and summarised as follows: $$\sum f y = - 467 , \sum f y ^ { 2 } = 9179$$ Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
(4 marks)
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SPS SPS SM Statistics 2021 September Q3
3. There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. \begin{displayquote} 112 take systems support,
70 take developing software,
81 take networking,
35 take developing software and systems support,
28 take networking and developing software,
40 take systems support and networking,
4 take all three extra options. \end{displayquote} a Draw a Venn diagram to represent this information. A student from the course is chosen at random.
b Find the probability that this student takes
i none of the three extra options
ii networking only. Students who take systems support and networking are eligible to become technicians.
c Given that the randomly chosen student is eligible to become a technician, find the probability that this student takes all three extra options.
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SPS SPS SM Statistics 2021 September Q4
4. A company assembles drills using components from two sources. Goodbuy supplies the components for \(85 \%\) of the drills whilst Amart supplies the components for the rest.
It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
a Represent this information on a tree diagram. An assembled drill is selected at random.
b Find the probability that the drill is not faulty.
SPS SPS SM Statistics 2021 September Q5
5. Figure 2 is a histogram showing the distribution of the time taken in minutes, \(t\), by a group of people to swim 500 m . \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a58a2c41-3a53-41cc-b80f-84adb04a5f5c-11_547_1120_333_374}
\end{figure} a Find the probability that a person chosen at random from the group takes longer than 18 minutes.
SPS SPS SM Statistics 2021 September Q6
6. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2
k ( x - 2 ) & x = 3
0 & \text { otherwise } \end{cases}\)
where \(k\) is a positive constant.
a Show that \(k = 0.25\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)
SPS SPS SM Statistics 2021 September Q7
7. Emma throws a fair coin 15 times and records the number of times it shows a head.
a State the appropriate distribution to model the number of times the coin shows a head giving any relevant parameter values.
b Find the probability that Emma records:
i exactly 8 heads
ii at least 4 heads.
SPS SPS FM Mechanics 2022 February Q1
  1. One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(0.4 \mathrm {~kg} . O\) is a vertical distance of 1 m below a horizontal ceiling. \(P\) is held at a point 1.5 m vertically below \(O\) and released from rest (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-02_480_375_370_274}
Assuming that there is no obstruction to the motion of \(P\) as it passes \(O\), find the speed of \(P\) when it first hits the ceiling.
SPS SPS FM Mechanics 2022 February Q2
2. A particle \(P\) of mass 2 kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is \(( 5 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision its velocity is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. The impulse imparted on \(P\) by the wall is denoted by INs. Find the following.
    • The magnitude of \(\mathbf { I }\)
    • The angle between I and i
    • Find the loss of kinetic energy of \(P\) as a result of the collision.
SPS SPS FM Mechanics 2022 February Q3
  1. A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW .
In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R \mathrm {~N}\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).
  1. Show that \(R = 430\).
  2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be \(60 v\) where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
  3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.
SPS SPS FM Mechanics 2022 February Q4
4. Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-08_518_830_319_283} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-08_597_780_1281_287} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
  2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple.
SPS SPS FM Mechanics 2022 February Q5
5. Two smooth circular discs \(A\) and \(B\) of masses \(m _ { A } \mathrm {~kg}\) and \(m _ { B } \mathrm {~kg}\) respectively are moving on a horizontal plane. At the instant before they collide the velocities of \(A\) and \(B\) are as follows, as shown in the diagram below.
  • The velocity of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the line of centres, where \(\tan \alpha = \frac { 4 } { 3 }\).
  • The velocity of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\frac { 1 } { 3 } \pi\) radians to the line of centres.
    \includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-10_442_808_443_312}
The direction of motion of \(B\) after the collision is perpendicular to the line of centres.
  1. Show that \(\frac { 3 } { 2 } \leqslant \frac { m _ { B } } { m _ { A } } \leqslant 4\).
  2. Given that \(m _ { A } = 2\) and \(m _ { B } = 6\), find the total loss of kinetic energy as a result of the collision.
SPS SPS FM Mechanics 2022 February Q6
6. A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string of length 0.8 m . The other end of the string is attached to a fixed point \(O . P\) is at rest vertically below \(O\) when it experiences a horizontal impulse of magnitude 20 Ns . In the subsequent motion the angle the string makes with the downwards vertical through \(O\) is denoted by \(\theta\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-12_373_476_365_285}
  1. Find the magnitude of the acceleration of \(P\) at the first instant when \(\theta = \frac { 1 } { 3 } \pi\) radians.
  2. Determine the value of \(\theta\) at which the string first becomes slack. End of Examination
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SPS SPS FM Pure 2022 February Q1
  1. (a) Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
    (b) Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.
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$$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.
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SPS SPS FM Pure 2022 February Q3
3. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1
- 3
3 \end{array} \right) + \lambda \left( \begin{array} { r } 3
2
- 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } . \left( \begin{array} { r } 2
- 5
- 3 \end{array} \right) = 4\).
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
  2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\).
    \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\).
    \(l _ { 2 }\) is the line with the following properties.
    • \(l _ { 2 }\) passes through \(A\)
    • \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
    • \(l _ { 2 }\) is parallel to \(\Pi\)
    • Find, in vector form, the equation of \(l _ { 2 }\).
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SPS SPS FM Pure 2022 February Q4
    1. \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
  1. \(\mathbf { A B }\)
  2. \(\mathbf { A } + \mathbf { B }\)
    (ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
    a & 0 & 0
    b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
    2 & 12 & - 1
    - 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where \(a , b\) and \(\lambda\) are constants,
  3. determine
    • the value of \(\lambda\)
    • the value of \(a\)
    • the value of \(b\)
    • Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1
      a & 0 & 0
      b & a & b \end{array} \right)\)
      (iii) Given that
    $$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
    0 & \sin \theta & \cos \theta
    0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.
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