Questions — SPS (1106 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
SPS SPS FM Pure 2021 May Q1
  1. Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
    1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
    2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).
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    3. The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
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    4. The greatest value of \(r\) on the curve occurs at the point \(P\).
      1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
      2. Find the value of \(r\) at the point \(P\).
      3. Mark the point \(P\) on a copy of the graph.
    5. In this question you must show detailed reasoning.
    Find the exact area of the region enclosed by the curve.
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SPS SPS FM Pure 2021 May Q3
3. You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & 0 & 1
0 & - 1 & 0 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { 4 }\).
  2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
  3. Write down the matrix B. The point \(P\) has coordinates \(( 2,3,4 )\). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
  4. Find the coordinates of \(P ^ { \prime }\).
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SPS SPS FM Pure 2021 May Q4
4. Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
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SPS SPS FM Pure 2021 May Q5
5. Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
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SPS SPS FM Pure 2021 May Q6
6. $$\mathbf { A } = \left( \begin{array} { r r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is a constant. }$$
  1. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(k\). A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
    The point \(P\) has position vector \(\binom { a } { 2 a }\) relative to an origin \(O\).
    The point \(Q\) has position vector \(\binom { 7 } { - 3 }\) relative to \(O\).
    Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),
  2. determine the value of \(a\) and the value of \(k\). Given that, for a different value of \(k , T\) maps the line \(y = 2 x\) onto itself,
  3. determine this value of \(k\).
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SPS SPS FM Pure 2021 May Q7
7. Given that \(y = \arcsin x , - 1 \leq x < 1\),
  1. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\). Given that \(f ( x ) = \frac { 3 x + 2 } { \sqrt { 4 - x ^ { 2 } } }\),
  2. show that the mean value of \(f ( x )\) over the interval \([ 0 , \sqrt { } 2 ]\), is $$\frac { \pi \sqrt { } 2 } { 4 } + A \sqrt { } 2 - A$$ where \(A\) is a constant to be determined.
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SPS SPS FM Pure 2021 May Q8
8.
  1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$
  2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.
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SPS SPS FM Pure 2021 May Q9
9.
  1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
  2. Find the range of \(x\) for which the expansion is valid.
  3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots$$ [BLANK PAGE]
SPS SPS FM Pure 2021 May Q10
10. A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t$$
  1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). Initially there was 10 g of the substance.
  2. Find the particular solution of the differential equation.
  3. Find to 6 significant figures the amount of substance that would be predicted by the model at
    (a) 6 hours,
    (b) 6.25 hours.
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SPS SPS FM Pure 2021 May Q1
  1. In this question you must show detailed reasoning.
    1. By using partial fractions show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } + 3 r + 2 } = \frac { 1 } { 2 } - \frac { 1 } { n + 2 }\).
    2. Hence determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } + 3 r + 2 }\).
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  2. A plane \(\Pi\) has the equation \(\mathbf { r } . \left( \begin{array} { r } 3
    6
    - 2 \end{array} \right) = 15 . C\) is the point \(( 4 , - 5,1 )\). Find the shortest distance between \(\Pi\) and \(C\).
  3. Lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following equations.
    \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4
    3
    1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
    4
    - 2 \end{array} \right)\)
    \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
    2
    4 \end{array} \right) + \mu \left( \begin{array} { r } 1
    - 2
    1 \end{array} \right)\)
    Find, in exact form, the distance between \(l _ { 1 }\) and \(l _ { 2 }\).
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SPS SPS FM Pure 2021 May Q3
3. In this question you must show detailed reasoning. Show that $$\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } d x = 1$$ [BLANK PAGE]
SPS SPS FM Pure 2021 May Q4
4. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0
0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.
  1. State the plane of reflection of R .
  2. Determine the value of \(a\).
  3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
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SPS SPS FM Pure 2021 May Q5
5 marks
5. Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.
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SPS SPS FM Pure 2021 May Q6
6. A circle \(C\) in the complex plane has equation \(| z - 2 - 5 i | = a\). The point \(z _ { 1 }\) on \(C\) has the least argument of any point on \(C\), and \(\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }\).
Prove that \(a = \frac { 3 \sqrt { 2 } } { 2 }\).
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SPS SPS FM Pure 2021 May Q7
7. The region \(R\) between the \(x\)-axis, the curve \(y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }\) and the lines \(x = \sqrt { p }\) and \(x = \sqrt { 3 p }\), where \(p\) is a positive parameter, is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\).
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt { 48 }\) find in exact form
    • the greatest possible value of the volume of \(S\)
    • the least possible value of the volume of \(S\).
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SPS SPS FM Pure 2021 May Q8
8 marks
  1. Let \(C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos ( r \theta )\). Show that \(C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos ( 10 \theta )\).
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  2. During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by
$$\frac { d y } { d t } = 0.3 x - 0.2 y \quad \text { and } \quad \frac { d z } { d t } = 0.2 y + 0.1 x$$ where \(x , y\) and \(z\) are the amounts in kg of \(X , Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = A e ^ { - 0.4 t }\), stating the value of \(A\).
  2. Show that \(\frac { d x } { d t } + \frac { d y } { d t } + \frac { d z } { d t } = 0\). Comment on this result in the context of the industrial process.
  3. Express \(y\) in terms of \(t\).
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SPS SPS FM Mechanics 2021 May Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-06_662_540_376_742} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac { 1 } { 3 }\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta ^ { 0 }\) with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of \(\theta\).
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SPS SPS FM Mechanics 2021 May Q3
3. A car of mass 800 kg is driven with its engine generating a power of 15 kW .
  1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
  2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac { 1 } { 20 }\), find the speed of the car.
  3. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
    1. the resistance to motion of the car,
    2. the magnitude of the tension in the towbar.
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SPS SPS FM Mechanics 2021 May Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-14_357_840_445_552} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and B has mass 0.4 kg . Immediately before the collision \(A\) is moving with speed \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision \(A\) is stationary. Find:
  1. the coefficient of restitution between \(A\) and \(B\),
  2. the angle turned through by the direction of motion of B as a result of the collision.
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    [0pt] [Question 4 Continued] \section*{5.} A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg .
    \(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .
  3. Find the tension in the string.
  4. Find the speed of \(P\).
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    [0pt] [Question 5 Continued] \section*{6.} The figure below shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\), and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leq y \leq 8\).
    \includegraphics[max width=\textwidth, alt={}, center]{ba21d750-a058-43c7-b602-2bafe545b94a-22_595_643_523_680} Find the coordinates of the centre of mass of a uniform lamina occupying this region. No marks will be deducted for using the numerical integration function of your calculator for this question.
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SPS SPS FM Statistics 2021 May Q1
1. The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the \(1 \%\) level whether there has been a change in the mean crop yield.
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SPS SPS FM Statistics 2021 May Q2
2. The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
a) Find the probability that the weight of a randomly selected peeled banana is at most 150 g . Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
b) Find the probability that the total weight of a smoothie is less than 700 g .
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SPS SPS FM Statistics 2021 May Q3
3. A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\). The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method (continuity correction is not required), which should be justified, to find \(P ( \bar { X } \leq 6.10 )\).
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SPS SPS FM Statistics 2021 May Q4
10 marks
4. A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726 . Test at the \(5 \%\) significance level whether there is a difference in the ranks of the students taught by the two professors.
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SPS SPS FM Statistics 2021 May Q5
5. The continuous random variable \(T\) has cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 ,
1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}$$
  1. Find the cumulative distribution function of \(2 T\).
  2. Show that, for constant \(k , \quad \mathrm { E } \left( \mathrm { e } ^ { k T } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
  3. \(T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\).
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SPS SPS SM Pure 2021 May Q1
  1. The function f is defined for all non-negative values of \(x\) by
$$\mathrm { f } ( x ) = 3 + \sqrt { x }$$
  1. Evaluate \(\mathrm { ff } ( 169 )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\).
  3. On a single diagram sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating how the two graphs are related.
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