Questions — SPS SPS SM (125 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
SPS SPS SM 2020 June Q1
6 marks Moderate -0.8
A curve has equation $$y = 2x^3 - 2x^2 - 2x + 8$$
  1. Find \(\frac{dy}{dx}\) [2]
  2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation. [4]
SPS SPS SM 2020 June Q2
9 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that - \(AFE\) is a straight line - \(AF = FE = 10.7\)m - \(BF = FD = 9.2\)m - angle \(BFD = 1.82\) radians find
  1. the perimeter of the stage, in metres, to one decimal place, [5]
  2. the area of the stage, in m², to one decimal place. [4]
SPS SPS SM 2020 June Q3
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 is a sketch showing the line \(l_1\) with equation \(y = 2x - 1\) and the point \(A\) with coordinates \((-2, 3)\). The line \(l_2\) passes through \(A\) and is perpendicular to \(l_1\)
  1. Find the equation of \(l_2\) writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [3]
The point \(B\) and the point \(C\) lie on \(l_1\) such that \(ABC\) is an isosceles triangle with \(AB = AC = 2\sqrt{13}\)
  1. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5x^2 - 12x - 32 = 0$$ [4]
Given that \(B\) lies in the 3rd quadrant
  1. find, using algebra and showing your working, the coordinates of \(B\). [4]
SPS SPS SM 2020 June Q4
4 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{g}(x)\). The curve has a single turning point, a minimum, at the point \(M(4, -1.5)\). The curve crosses the \(x\)-axis at two points, \(P(2, 0)\) and \(Q(7, 0)\). The curve crosses the \(y\)-axis at a single point \(R(0, 5)\).
  1. State the coordinates of the turning point on the curve with equation \(y = 2\text{g}(x)\). [1]
  2. State the largest root of the equation $$\text{g}(x + 1) = 0$$ [1]
  3. State the range of values of \(x\) for which \(\text{g}'(x) \leqslant 0\) [1]
Given that the equation \(\text{g}(x) + k = 0\), where \(k\) is a constant, has no real roots,
  1. state the range of possible values for \(k\). [1]
SPS SPS SM 2020 June Q5
4 marks Easy -1.2
Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of $$\left(1 + \frac{3}{4}x\right)^6$$ simplifying each term. [4]
SPS SPS SM 2020 June Q6
3 marks Moderate -0.8
A company which makes batteries for electric cars has a 10-year plan for growth. • In year 1 the company will make 2600 batteries • In year 10 the company aims to make 12000 batteries In order to calculate the number of batteries it will need to make each year, from year 2 to year 9, the company considers the following model: *the number of batteries made will increase by the same percentage each year* Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10. [3]
SPS SPS SM 2020 June Q7
9 marks Moderate -0.3
  1. Solve, for \(-90° \leqslant \theta < 270°\), the equation, $$\sin(2\theta + 10°) = -0.6$$ giving your answers to one decimal place. [5]
    1. A student's attempt at the question "Solve, for \(-90° < x < 90°\), the equation \(7\tan x = 8\sin x\)" is set out below. \begin{align} 7\tan x &= 8\sin x
      7 \times \frac{\sin x}{\cos x} &= 8\sin x
      7\sin x &= 8\sin x \cos x
      7 &= 8\cos x
      \cos x &= \frac{7}{8}
      x &= 29.0° \text{ (to 3 sf)} \end{align} Identify two mistakes made by this student, giving a brief explanation of each mistake. [2]
    2. Find the smallest positive solution to the equation $$7\tan(4\alpha + 199°) = 8\sin(4\alpha + 199°)$$ [2]
SPS SPS SM 2020 June Q8
4 marks Challenging +1.2
Prove by contradiction that there are no positive integers \(a\) and \(b\) with \(a\) odd such that $$a + 2b = \sqrt{8ab}$$ [4]
SPS SPS SM 2020 June Q9
4 marks Moderate -0.5
\includegraphics{figure_1} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = ab^t$$ where \(a\) and \(b\) are constants. The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log_{10} P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model, find a complete equation for the model giving the value of \(b\) to 4 significant figures. [4]
SPS SPS SM 2020 June Q10
8 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3x - 2\sqrt{x}\), \(x \geqslant 0\) and the line \(l\) with equation \(y = 8x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\). [5]
  2. \includegraphics{figure_4} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\). [3]
SPS SPS SM 2020 June Q11
9 marks Standard +0.3
  1. Sketch the curve with equation $$y = k - \frac{1}{2x}$$ where \(k\) is a positive constant State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. [3]
The straight line \(l\) has equation \(y = 2x + 3\) Given that \(l\) cuts the curve in two distinct places,
  1. find the range of values of \(k\), writing your answer in set notation. [6]
SPS SPS SM 2020 June Q12
8 marks Standard +0.3
\includegraphics{figure_6} **In this question you must show all stages of your working.** **Solutions relying on calculator technology are not acceptable.** Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2^{2x}$$ The point \(P\left(a, 96\sqrt{2}\right)\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3 \times 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is $$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]
SPS SPS SM 2020 June Q13
6 marks Standard +0.3
\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\). *(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]
SPS SPS SM 2020 October Q1
2 marks Easy -1.8
Simplify fully the following expressions:
  1. \(\frac{7y^{13}}{35y^7}\) [1]
  2. \(6x^{-2} \div x^{-5}\) [1]
SPS SPS SM 2020 October Q2
3 marks Easy -1.8
A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).
  1. State what type of sequence this is. [1]
  2. Find \(u_{17}\). [2]
SPS SPS SM 2020 October Q3
6 marks Moderate -0.3
  1. Write \(3x^2 - 6x + 1\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [2]
  2. Solve \(3x^2 - 6x + 1 \leq 0\), giving your answer in set notation. [4]
SPS SPS SM 2020 October Q4
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
  2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]
SPS SPS SM 2020 October Q5
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS SM 2020 October Q6
5 marks Moderate -0.8
  1. A student was asked to solve the equation \(2^{2x+4} - 9(2^x) = 0\). The student's attempt is written out below. $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ $$\text{Let } y = 2^x$$ $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$ Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2^{2x+4} - 9(2^x) = 0\), giving your answer in exact form. [3]
SPS SPS SM 2020 October Q7
11 marks Moderate -0.3
  1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
  3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]
SPS SPS SM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q9
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]
SPS SPS SM 2020 October Q10
8 marks Standard +0.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]
SPS SPS SM 2021 February Q1
1 marks Easy -2.5
Which of the options below best describes the correlation shown in the diagram below? \includegraphics{figure_1} Tick \((\checkmark)\) one box. [1 mark] moderate positive \(\square\) strong positive \(\square\) moderate negative \(\square\) strong negative \(\square\)
SPS SPS SM 2021 February Q2
1 marks Easy -2.5
Lenny is one of a team of people interviewing shoppers in a town centre. He is asked to survey 50 women between the ages of 18 and 29 Identify the name of this type of sampling. Circle your answer. [1 mark] simple random stratified quota systematic