Questions — SPS SPS SM (125 questions)

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SPS SPS SM 2023 October Q8
7 marks Standard +0.3
In this question you must show detailed reasoning. The curve \(C_1\) has equation \(y = 8 - 10x + 6x^2 - x^3\) The curve \(C_2\) has equation \(y = x^2 - 12x + 14\)
  1. Verify that when \(x = 1\) the curves \(C_1\) and \(C_2\) intersect. [2]
The curves also intersect when \(x = k\). Given that \(k < 0\)
  1. use algebra to find the exact value of \(k\). [5]
SPS SPS SM 2023 October Q9
10 marks Moderate -0.8
The first term of a geometric progression is \(10\) and the common ratio is \(0.8\).
  1. Find the fourth term. [2]
  2. Find the sum of the first \(20\) terms, giving your answer correct to \(3\) significant figures. [2]
  3. The sum of the first \(N\) terms is denoted by \(S_N\), and the sum to infinity is denoted by \(S_\infty\). Show that the inequality \(S_\infty - S_N < 0.01\) can be written as $$0.8^N < 0.0002,$$ and use logarithms to find the smallest possible value of \(N\). [6]
SPS SPS SM 2023 October Q10
7 marks Standard +0.8
In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]
SPS SPS SM 2024 October Q1
3 marks Easy -1.2
A is inversely proportional to B. B is inversely proportional to the square of C. When A is 2, C is 8. Find C when A is 12. [3]
SPS SPS SM 2024 October Q2
5 marks Moderate -0.8
  1. Write \(3x^2 + 24x + 5\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
The finite region R is enclosed by the curve \(y = 3x^2 + 24x + 5\) and the \(x\)-axis.
  1. State the inequalities that define R, including its boundaries. [2]
SPS SPS SM 2024 October Q3
5 marks Moderate -0.8
The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]
SPS SPS SM 2024 October Q4
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q5
8 marks Moderate -0.8
\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.
  1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]
Given that
  • lines \(l_1\) and \(l_2\) intersect at the point C
  • line \(l_1\) crosses the \(x\)-axis at the point A
  1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]
SPS SPS SM 2024 October Q6
8 marks Moderate -0.8
In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$
  1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
  2. Find the time when the mass is 30 grams. [3]
  3. Sketch the graph of \(m\) against \(t\). [2]
SPS SPS SM 2024 October Q7
4 marks Moderate -0.8
Express \(\frac{a^{\frac{1}{2}} - a^{\frac{2}{3}}}{a^{\frac{1}{3}} - a}\) in the form \(a^m + \sqrt{a^n}\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1. [4]
SPS SPS SM 2024 October Q8
5 marks Standard +0.3
A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:
  • The line through the centres of circle C and circle D has gradient 1.
  • Circle D touches circle C at exactly one point.
  • The centre of circle D lies in the first quadrant.
  • Circle D has the same radius as circle C.
Find the coordinates of the centre of circle D. [5]
SPS SPS SM 2024 October Q9
9 marks Moderate -0.3
In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of f(x). [1]
    2. Hence find the exact roots of the equation f(x) = 0. [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS SM 2024 October Q10
7 marks Standard +0.3
The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of k, giving a reason for your answer. [4]
  2. Find the value of \(\sum_{r=2}^{\infty} u_r\) [3]
SPS SPS SM 2024 October Q1
3 marks Moderate -0.8
The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v\)ms\(^{-1}\). When \(v = 3.6\), \(P = 50\). Determine the wind speed that will give a power output of 225 watts. [3]
SPS SPS SM 2024 October Q2
7 marks Easy -1.2
Solve the inequalities
  1. \(3 - 8x > 4\), [2]
  2. \((2x - 4)(x - 3) < 12\). [5]
SPS SPS SM 2024 October Q3
6 marks Standard +0.3
The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]
SPS SPS SM 2024 October Q4
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
SPS SPS SM 2024 October Q5
11 marks Moderate -0.3
A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).
  1. Show that the line does not meet the circle. [3]
    1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
    2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]
SPS SPS SM 2024 October Q6
6 marks Moderate -0.8
The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).
\(t\)051025
\(m\)200160
  1. Find the values missing from the table. [2]
  2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams. [4]
SPS SPS SM 2024 October Q7
6 marks Moderate -0.3
A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below. \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\) \(4\log_3 x - 3 \log_3 x - 2 = 0\) \(\log_3 x - 2 = 0\) \(\log_3 x = 2\) \(x = 8\)
  1. Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]
SPS SPS SM 2024 October Q8
8 marks Standard +0.3
In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$ is convergent.
  1. Find the set of possible values of \(x\), giving your answer in set notation. [5]
  2. Given that the sum to infinity of the series is \(\frac{2}{3}\), find the value of \(x\). [3]
SPS SPS SM 2025 February Q1
2 marks Easy -1.2
Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]
SPS SPS SM 2025 February Q2
13 marks Standard +0.3
  1. \includegraphics{figure_2} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(ABCDEF\). The base \(ACFD\) of the roof is a horizontal rectangle, and the cross-section \(ABC\) of the roof is an isosceles triangle with \(AB = BC\). The lengths of \(AC\) and \(CF\) are \(2x\) cm and \(y\) cm respectively, and the height of \(BE\) above the base of the roof is \(x\) cm. The total surface area of the five faces of the roof is \(600\) cm\(^2\) and the volume of the roof is \(V\) cm\(^3\). Show that \(V = kx (300 - x^2)\), where \(k = \sqrt{a + b}\) and \(a\) and \(b\) are integers to be determined. [6]
  2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum. [4]
  3. Find the maximum volume of the roof. Give your answer in cm\(^3\), correct to the nearest integer. [1]
  4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state. [2]
SPS SPS SM 2025 February Q3
6 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]
SPS SPS SM 2025 February Q4
6 marks Moderate -0.3
  1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
  2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]