Questions — SPS (1106 questions)

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SPS SPS SM Pure 2021 May Q2
2. (a) Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
(b) Explain how the trapezium rule could be used to obtain a more accurate estimate.
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SPS SPS SM Pure 2021 May Q3
3. Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
  2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
  3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).
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SPS SPS SM Pure 2021 May Q4
4. Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
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SPS SPS SM Pure 2021 May Q5
5.
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
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SPS SPS SM Pure 2021 May Q6
6. Shona makes the following claim.
" \(n\) is an even positive integer greater than \(2 \Rightarrow 2 ^ { n } - 1\) is not prime"
Prove that Shona's claim is true.
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SPS SPS SM Pure 2021 May Q7
7. A curve has parametric equations $$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$ for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
  2. Find the cartesian equation of the curve.
  3. State the set of values that \(x\) can take and hence sketch the curve.
    [0pt] [BLANK PAGE] \section*{8. In this question you must show detailed reasoning.} The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
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SPS SPS SM Pure 2021 May Q9
9. In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{5795db3d-2fcb-444e-a878-79e83c846334-20_747_481_233_826} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
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SPS SPS SM Pure 2021 May Q1
1.
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta\).
  2. Hence determine an approximate solution to \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta\).
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SPS SPS SM Pure 2021 May Q2
3 marks
2. Solve the equation \(| 2 x - 1 | = | x + 3 |\).
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SPS SPS SM Pure 2021 May Q3
3. Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).
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SPS SPS SM Pure 2021 May Q4
4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
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SPS SPS SM Pure 2021 May Q5
5. A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
  2. Find the equation of the normal to the curve at the point \(( 1,2 )\).
    [0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} A circle touches the lines \(y = \frac { 1 } { 2 } x\) and \(y = 2 x\) at \(( 6,3 )\) and \(( 3,6 )\) respectively.
    \includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-14_515_524_338_790} Find the equation of the circle.
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SPS SPS SM Pure 2021 May Q7
7. It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3 \tan 2 x - 8 \tan x = 4$$
  1. Show that \(t = \sqrt [ 3 ] { \frac { 1 } { 2 } + \frac { 1 } { 4 } t - \frac { 1 } { 2 } t ^ { 2 } }\), where \(t = \tan x\).
  2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8 .
  3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration.
  4. Solve the equation \(3 \tan 4 y - 8 \tan 2 y = 4\) for \(0 < y < \frac { 1 } { 2 } \pi\).
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SPS SPS SM Pure 2021 May Q8
8. Find the general solution of the differential equation $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 20 x - 35 )$$ Give your answer in the form \(y = \mathrm { f } ( x )\).
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SPS SPS SM Pure 2021 May Q9
9. (i) Show that the two non-stationary points of inflection on the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\) are at \(x = \pm \frac { 1 } { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-20_492_1064_237_513} The diagram shows the curve \(y = \ln \left( 1 + 4 x ^ { 2 } \right)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac { 1 } { 2 }\) and \(x = - \frac { 1 } { 2 }\).
(ii) Show that the area of the shaded region is given by $$\int _ { 0 } ^ { \ln 2 } \sqrt { \mathrm { e } ^ { y } - 1 } \mathrm {~d} y$$ (iii) Show that the substitution \(\mathrm { e } ^ { y } = \sec ^ { 2 } \theta\) transforms the integral in part (ii) to \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } 2 \tan ^ { 2 } \theta \mathrm {~d} \theta\).
(iv) Hence find the exact area of the shaded region.
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SPS SPS SM Mechanics 2021 May Q1
1.
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-04_607_894_150_644} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate
  1. \(\alpha\),
  2. \(F\).
SPS SPS SM Mechanics 2021 May Q2
2. A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
  2. Find the speed of \(P\) at time \(t = 0\) seconds.
SPS SPS SM Mechanics 2021 May Q3
3. A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-08_184_1266_283_402} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find
  1. the magnitude of the reaction of the support on the plank at \(D\),
  2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
  3. Find the weight of the stone block.
  4. Explain the limitation of modelling
    (a) the stone block as a particle,
    (b) the plank as a rigid rod.
SPS SPS SM Mechanics 2021 May Q4
4. A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after 2.4 seconds. The horizontal component of the initial velocity of \(P\) is \(\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }\).
  1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground.
  2. Find the vertical component of the initial velocity of \(P\).
    \(P\) just clears a vertical wall which is situated at a horizontal distance \(d \mathrm {~m}\) from \(O\).
  3. Find the height of the wall. The speed of \(P\) as it passes over the wall is \(16 \mathrm {~ms} ^ { - 1 }\).
  4. Find the value of \(d\) correct to 3 significant figures.
SPS SPS SM Mechanics 2021 May Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-12_501_880_132_614} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) has mass 0.2 kg and is held at rest on the plane. \(Q\) has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is 0.4 . The particle \(P\) is released.
  1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released.
    \(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of 0.8 m before it comes to rest. \(P\) does not reach the pulley.
  2. Find the speed of the particles immediately before \(Q\) strikes the floor.
  3. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. End of Examination Extra Answer Space
SPS SPS SM Statistics 2021 May Q1
  1. Three Bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.
\begin{displayquote} Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only \end{displayquote} Sasha selects at random one marble from \(\operatorname { Bag } A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(C\).
  1. Draw a tree diagram to represent this information.
  2. Find the probability that Sasha selects 3 green marbles.
  3. Find the probability that Sasha selects at least 1 marble of each colour.
  4. Given that Sasha selects a red marble, find the probability that he selects it from Bag \(B\).
SPS SPS SM Statistics 2021 May Q2
2.
  1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
    (a) Find \(\mathrm { P } ( X > 25 )\).
    (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
    (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
  2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).
SPS SPS SM Statistics 2021 May Q3
3. Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.
  1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
  2. Use the table below to carry out the test at the \(5 \%\) significance level. Critical values of Pearson's product-moment correlation coefficient.
    \cline{2-5}
    1-tail
    test
    		\(5 \%\)\(2.5 \%\)\(1 \%\)
    2-tail
    test
    		\(10 \%\)\(5 \%\)\(2.5 \%\)\(1 \%\)
    380.27090.32020.37600.4128
    390.26730.31600.37120.4076
    400.26380.31200.36650.4026
    410.26050.30810.36210.3978
SPS SPS SM Statistics 2021 May Q4
  1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml
Given that \(15 \%\) of bottles contain less than 24.63 ml
  1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
  2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\)
SPS SPS SM Statistics 2021 May Q5
5. Two events C and D are such that \(P ( C \mid D ) = 3 \times P ( C )\) where \(P ( C ) \neq 0\).
  1. Explain whether or not events C and D could be independent events. Given also that $$P ( C \cap D ) = \frac { 1 } { 2 } \times P ( C ) \text { and } P \left( C ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$$
  2. find \(P ( C )\), showing your working clearly.