Questions — SPS SPS SM (125 questions)

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SPS SPS SM 2025 February Q5
7 marks Standard +0.3
A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]
SPS SPS SM 2025 February Q6
9 marks Standard +0.3
For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.
  1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
  2. If \(fg(2) = 144\), find the value of \(a\). [3]
  3. Determine whether the function \(fg\) has an inverse. [2]
SPS SPS SM 2025 February Q7
8 marks Standard +0.3
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ [3]
  2. Hence solve the equation $$2 \sin 2\theta \tan 2\theta = \cos 2\theta + 5,$$ giving all values of \(\theta\) between \(0°\) and \(180°\), correct to \(1\) decimal place. [5]
SPS SPS SM 2025 February Q8
7 marks Challenging +1.2
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS SM 2025 October Q1
8 marks Easy -1.8
Express each of the following in the form \(px^q\), where \(p\) and \(q\) are constants.
  1. \(\frac{2}{\sqrt[3]{x}}\) [1]
  2. \((5x\sqrt{x})^3\) [1]
  3. \(\sqrt{2x^3} \times \sqrt{8x^5}\) [1]
  4. \(x^5(27x^6)^{\frac{1}{3}}\) [2]
SPS SPS SM 2025 October Q2
3 marks Moderate -0.3
In this question you must show detailed reasoning. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
SPS SPS SM 2025 October Q3
5 marks Moderate -0.8
The line \(l\) passes through the points \(A(-3, 0)\) and \(B\left(\frac{5}{3}, 22\right)\)
  1. Find the equation of \(l\) giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are constants. [3]
\includegraphics{figure_2} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\). Given that
  • \(C\) has equation \(y = 2x^2 + 5x - 3\)
  • the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)
  1. use inequalities to define \(R\). [2]
SPS SPS SM 2025 October Q4
6 marks Moderate -0.8
  1. A sequence has terms \(u_1, u_2, u_3, \ldots\) defined by \(u_1 = 3\) and \(u_{n+1} = u_n^2 - 5\) for \(n \geq 1\).
    1. Find the values of \(u_2\), \(u_3\) and \(u_4\). [2]
    2. Describe the behaviour of the sequence. [1]
  2. The second, third and fourth terms of a geometric progression are 12, 8 and \(\frac{16}{3}\). Determine the sum to infinity of this geometric progression. [3]
SPS SPS SM 2025 October Q5
5 marks Standard +0.3
In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the cuboid \(ABCDEFGH\) where \(AD = 3\) cm, \(AF = (2x + 1)\) cm and \(DC = (x - 2)\) cm. The volume of the cuboid is at most 9 cm³. Find the range of possible values of \(x\). Give your answer in interval notation. [5]
SPS SPS SM 2025 October Q6
3 marks Moderate -0.8
Sketch the graph of $$y = (x - k)^2(x + 2k)$$ where \(k\) is a positive constant. Label the coordinates of the points where the graph meets the axes. \includegraphics{figure_6} [3]
SPS SPS SM 2025 October Q7
7 marks Standard +0.3
In this question you must show detailed reasoning. Solve the following equations.
  1. \(y^6 + 7y^3 - 8 = 0\) [3]
  2. \(9^{x+1} + 3^x = 8\) [4]
SPS SPS SM 2025 October Q8
7 marks Moderate -0.3
In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.
  1. \(2\log_3(x + 1) = 1 + \log_3(x + 7)\) [4]
  2. \(\log_y\left(\frac{1}{x}\right) = -\frac{3}{2}\) [3]
SPS SPS SM 2025 October Q9
4 marks Moderate -0.8
  1. Show that the equation \(x^2 + kx - k^2 = 0\) has real roots for all real values of \(k\). [2]
  2. Show that the roots of the equation \(x^2 + kx - k^2 = 0\) are \(\left(\frac{-1 \pm \sqrt{5}}{2}\right)k\). [2]
SPS SPS SM 2025 October Q10
7 marks Moderate -0.3
\(f(x) = x^4 + bx + c\) \((x-2)\) is a factor of \(f(x)\). \(f(-3) = 35\).
  1. Find \(b\) and \(c\). [4]
  2. Hence express \(f(x)\) as the product of linear and cubic factors. [3]
SPS SPS SM 2025 October Q11
9 marks Moderate -0.8
A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.
  1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]
The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.
\(t\) minutes813213550
\(M\) kg0.40.30.20.10.05
The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.
    1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
    2. Interpret the value of \(k\) in context. [1]
  2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
  3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
  4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]
SPS SPS SM 2025 October Q12
5 marks Standard +0.3
An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).
    1. Show that \(r = \frac{a + 2d}{a}\). [1]
    2. Find \(d\) in terms of \(a\). [2]
  2. Find the common ratio of the geometric progression. [2]
SPS SPS SM 2025 October Q13
9 marks Standard +0.3
The circle \(C\) has equation $$x^2 + y^2 + 10x - 4y + 1 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the exact radius of \(C\) [2]
    The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\), giving your answer in set notation. [2]
  3. The line with equation \(y = mx + 4\) is a tangent to \(C\). Find possible exact values of \(m\). [5]
SPS SPS SM 2025 November Q1
7 marks Easy -1.3
Do not use a calculator for this question
  1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
    1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
    2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]
SPS SPS SM 2025 November Q2
12 marks Easy -1.3
  1. Write \(4x^2 - 24x + 27\) in the form \(a(x - b)^2 + c\). [4]
  2. State the coordinates of the minimum point on the curve \(y = 4x^2 - 24x + 27\). [2]
  3. Solve the equation \(4x^2 - 24x + 27 = 0\). [3]
  4. Sketch the graph of the curve \(y = 4x^2 - 24x + 27\). [3]
SPS SPS SM 2025 November Q3
5 marks Standard +0.3
The equation \(kx^2 + 4x + (5 - k) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\). Write your answer using set notation. [5]
SPS SPS SM 2025 November Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left(\frac{x^4}{2}\right)\) [3]
  3. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]
(Total 9 marks)
SPS SPS SM 2025 November Q5
Moderate -0.8
An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 29 terms is 1102.
  1. Show that \(a + 14d = 38\). (3 marks)
  2. The sum of the second term and the seventh term is 13. Find the value of \(a\) and the value of \(d\). (4 marks)
SPS SPS SM 2025 November Q6
8 marks Standard +0.8
A sequence \(t_1, t_2, t_3, t_4, t_5, \ldots\) is given by $$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$ where \(a\) is a non zero constant.
  1. Given that \(\sum_{r=1}^{3} (r^3 + t_r) = 12\), determine the possible values of \(a\). [4]
  2. Evaluate \(\sum_{r=8}^{31} (t_{r+1} - at_r)\). [4]
SPS SPS SM 2025 November Q7
11 marks Moderate -0.8
There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
  2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]
\(t\)12345678910
\(\log_{10} y\)1.511.581.982.68
SPS SPS SM 2025 November Q8
11 marks Standard +0.3
The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$
  1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
  2. Determine the coordinates of \(P\). [3]
  3. Find the equation of the common tangent to the circles at P. [4]