Questions — SPS (1106 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2023 September Q9
9. The equations of two lines are $$\mathbf { r } = \left( \begin{array} { l } 3
0
2 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
3 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { r } - 1
8
2 \end{array} \right) + \mu \left( \begin{array} { c } - 3
1
- 5 \end{array} \right)$$ Find the coordinates of the point where these lines intersect.
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q10
10. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geq 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2
0 & 1 \end{array} \right) .$$ [BLANK PAGE]
SPS SPS FM Pure 2023 September Q11
11. A curve has parametric equations \(x = \frac { 1 } { t } - 1\) and \(y = 2 t + \frac { 1 } { t ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the coordinates of the stationary point and, by considering the gradient of the curve on either side of this point, determine its nature.
  3. Find a cartesian equation of the curve.
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q12
12.
  1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
  2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A e ^ { n }\).
    [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q13
13. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 }
\frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)$$
  1. The diagram below shows the unit square \(O A B C\). The image of the unit square under the transformation represented by M is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
    \includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-28_778_1095_513_557}
  2. Find the equation of the line of invariant points of this transformation.
    1. Find the determinant of M .
    2. Describe briefly how this value relates to the transformation represented by M .
      [0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 September Q14
14. Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q1
  1. technology.
    1. Differentiate with respect to \(x\)
      1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
      2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
        [0pt] [BLANK PAGE]
    (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
    (ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q3
3.
  1. Given that \(\cos A = \frac { 3 } { 4 }\), where \(270 ^ { \circ } < A < 360 ^ { \circ }\), find the exact value of \(\sin 2 A\).
    1. Show that \(\cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right) \equiv \cos 2 x\). Given that $$y = 3 \sin ^ { 2 } x + \cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right)$$
    2. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q4
4. $$\mathrm { f } ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),
  1. find the value of \(R\) and the value of \(\alpha\).
    (4)
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs.
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q5
5. The curve \(C\) has equation $$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
  2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q1
1. a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left( 1 + \frac { x } { 2 } \right) ^ { 7 }\), giving each coefficient in exact simplified form.
b) Hence determine the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 2 } \right) ^ { 7 }$$ [BLANK PAGE]
SPS SPS SM Pure 2023 September Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-06_562_1118_185_614} The figure above shows a triangle with vertices at \(A ( 2,6 ) , B ( 11,6 )\) and \(C ( p , q )\).
a) Given that the point \(D ( 6,2 )\) is the midpoint of \(A C\), determine the value of \(p\) and the value of \(q\). The straight line \(l\), passes through \(D\) and is perpendicular to \(A C\).
The point \(E\) is the intersection of \(l\) and \(A B\).
b) Find the coordinates of \(E\).
(4)
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q3
3. $$x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).
a) Determine the value of \(r\) and the coordinates of \(C\). The point \(P ( 4,2 )\) lies on the circle.
b) Show that an equation of the tangent to the circle at \(P\) is $$y = 14 - 3 x$$ [BLANK PAGE]
SPS SPS SM Pure 2023 September Q4
4. $$\begin{aligned} & f ( x ) = \mathrm { e } ^ { x } , x \in \mathbb { R } , x > 0
& g ( x ) = 2 x ^ { 3 } + 11 , x \in \mathbb { R } \end{aligned}$$ a) Find and simplify an expression for the composite function \(g f ( x )\).
b) State the domain and range of \(g f ( x )\).
c) Solve the equation $$g f ( x ) = 27$$ The equation \(g f ( x ) = k\), where \(k\) is a constant, has solutions.
d) State the range of the possible values of \(k\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q5
5. Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors \(4 \mathbf { i } + 2 \mathbf { j } , 3 \mathbf { i } + 4 \mathbf { j }\) and \(- \mathbf { i } + 12 \mathbf { j }\), respectively.
  1. Find the magnitude of the vector \(\overrightarrow { O C }\)
  2. Find the angle that the vector \(\overrightarrow { O B }\) makes with the vector \(\mathbf { j }\) to the nearest degree
  3. Show that the points \(A\), \(B\) and \(C\) are collinear
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q6
6. Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by \(10 \%\) compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.
a) Show that the volume of the liquid in the container at the end of the second month is 202.5 litres.
b) Find the volume of the liquid in the container at the end of the twelfth month. (2) At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.
c) Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-16_581_978_210_699} The figure above shows a circular sector \(O A B\) whose centre is at \(O\). The radius of the sector is 60 cm . The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively, so that \(| O C | = | O D | = 24 \mathrm {~cm}\). Given that the length of the arc \(A B\) is 48 cm , find the area of the shaded region \(A B D C\), correct to the nearest \(\mathrm { cm } ^ { 2 }\).
[0pt] [BLANK PAGE] $$y = ( 3 - x ) ( 4 + x ) ^ { 2 }$$ a) Sketch the graph of \(C\). The sketch must include any points where the graph meets the coordinate axes.
b) Sketch in separate diagrams the graph of ...
i. \(\quad \ldots \quad y = ( 3 - 2 x ) ( 4 + 2 x ) ^ { 2 }\).
ii. \(\ldots y = ( 3 + x ) ( 4 - x ) ^ { 2 }\).
iii. ... \(y = ( 2 - x ) ( 5 + x ) ^ { 2 }\). Each of the sketches must include any points where the graph meets the coordinate axes.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q9
6 marks
9. Solve the following trigonometric equation in the range given. $$4 \tan ^ { 2 } \theta \cos \theta = 15,0 \leq \theta < 360 ^ { \circ } .$$ [6 marks]
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q10
10.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-22_451_506_178_767} The figure above shows solid right prism of height \(h \mathrm {~cm}\). The cross section of the prism is a circular sector of radius \(r \mathrm {~cm}\), subtending an angle of 2 radians at the centre.
a) Given that the volume of the prism is \(1000 \mathrm {~cm} ^ { 3 }\), show clearly that $$S = 2 r ^ { 2 } + \frac { 4000 } { r } ,$$ where \(S \mathrm {~cm} ^ { 2 }\) is the total surface area of the prism.
b) Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q11
11. It is given that $$f ( x ) = x ^ { 2 } - k x + ( k + 3 ) ,$$ where \(k\) is a constant. If the equation \(f ( x ) = 0\) has real roots find the range of the possible values of \(k\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q12
12.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-26_504_856_239_657} The figure above shows the curve \(C\) with equation $$f ( x ) = \frac { x + 4 } { \sqrt { x } } , x > 0 .$$ a) Determine the coordinates of the minimum point of \(C\), labelled as \(M\). The point \(N\) lies on the \(x\) axis so that \(M N\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(M N\) and the straight line with equation \(x = 1\).
b) Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q13
13. Prove or disprove each of the following statements:
  1. If \(n\) is an integer, then \(3 n ^ { 2 } - 11 n + 13\) is a prime number.
  2. If \(x\) is a real number, then \(x ^ { 2 } - 8 x + 17\) is positive.
  3. If \(p\) and \(q\) are irrational numbers, then \(p q\) is irrational.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q14
14.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-30_490_992_226_573} The diagram above shows the curve with equation $$y = ( x - 4 ) ^ { 2 } , x \in \mathbb { R }$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(A B\) and \(B C\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
SPS SPS FM Pure 2023 February Q1
4 marks
  1. Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
    [0pt] [4]
    [0pt] [BLANK PAGE]
  2. Solve the equation \(2 z - 5 i z ^ { * } = 12\).
    [0pt] [BLANK PAGE]
\section*{3. In this question you must show detailed reasoning.} Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-06_538_723_296_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
[0pt] [BLANK PAGE]
4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3
SPS SPS FM Pure 2023 February Q4
4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \mu \left( \begin{array} { l } 2
0
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) is perpendicular to \(\Pi\).
  2. Hence find a Cartesian equation of \(\Pi\). The line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { r } 4
    - 5
    2 \end{array} \right) + t \left( \begin{array} { r } 1
    6
    - 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
    The point \(A\) lies on \(l\).
    Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
  3. determine the possible coordinates of \(A\).
    [0pt] [BLANK PAGE]