Questions — SPS (686 questions)

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SPS SPS SM 2023 October Q6
8 marks Standard +0.3
In part (ii) of this question you must show detailed reasoning.
  1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
  2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]
SPS SPS SM 2023 October Q7
5 marks Easy -1.3
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]
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 FM 2023 October Q1
4 marks Moderate -0.8
This question requires detailed reasoning. Express \(\frac{3 + \sqrt{20}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [4]
SPS SPS FM 2023 October Q2
6 marks Moderate -0.8
Solve each of the following equations, for \(0° < x < 360°\).
  1. \(\sin \frac{1}{2}x = 0.8\) [3]
  2. \(\sin x = 3 \cos x\) [3]
SPS SPS FM 2023 October Q3
6 marks Easy -1.3
  1. Sketch the curve \(y = -\frac{1}{x}\). [2]
  2. The curve \(y = -\frac{1}{x}\) is translated by 2 units parallel to the x-axis in the positive direction. State the equation of the transformed curve. [2]
  3. Describe a transformation that transforms the curve \(y = -\frac{1}{x}\) to the curve \(y = -\frac{1}{3x}\). [2]
SPS SPS FM 2023 October Q4
7 marks Moderate -0.3
In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4\sqrt{x - 3x + 1}\) at the point on the curve where x = 4. Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]
SPS SPS FM 2023 October Q5
6 marks Moderate -0.8
  1. Find the binomial expansion of \((3 + kx)^3\), simplifying the terms. [4]
  2. It is given that, in the expansion of \((3 + kx)^3\), the coefficient of \(x^2\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form. [2]
SPS SPS FM 2023 October Q6
8 marks Standard +0.3
In this question you must show detailed reasoning. The functions f and g are defined for all real values of \(x\) by $$f(x) = x^3 \text{ and } g(x) = x^2 + 2.$$
  1. Write down expressions for
    1. \(fg(x)\), [1]
    2. \(gf(x)\). [1]
  2. Hence find the values of \(x\) for which \(fg(x) - gf(x) = 24\). [6]
SPS SPS FM 2023 October Q7
6 marks Standard +0.8
The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q\sqrt{r}\) where \(p\), \(q\) and \(r\) are integers. [6]
SPS SPS FM 2023 October Q8
5 marks Standard +0.8
Prove that \(2^{3n} - 3^n\) is divisible by 5 for all integers \(n \geq 1\). [5]
SPS SPS FM 2023 October Q9
12 marks Standard +0.3
  1. \includegraphics{figure_9} The shape ABC shown in the diagram is a student's design for the sail of a small boat. The curve AC has equation \(y = 2 \log_2 x\) and the curve BC has equation \(y = \log_2\left(x - \frac{3}{2}\right) + 3\). State the x-coordinate of point A. [1]
  2. Determine the x-coordinate of point B. [3]
  3. By solving an equation involving logarithms, show that the x-coordinate of point C is 2. [4] It is given that, correct to 3 significant figures, the area of the sail is 0.656 units\(^2\).
  4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines. [4]
SPS SPS FM Pure 2024 January Q1
7 marks Standard +0.8
Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(P(1, 2)\). \includegraphics{figure_1} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7]
SPS SPS FM Pure 2024 January Q2
6 marks Standard +0.3
  1. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
  2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ where $$f(x) = k - |2x - 3k|$$ [2]
SPS SPS FM Pure 2024 January Q3
4 marks Moderate -0.5
Find the value of the integral: $$\int_0^1 \frac{x^{\frac{1}{2}} + x^{-\frac{1}{3}}}{x} \, dx$$ [4]
SPS SPS FM Pure 2024 January Q4
13 marks Standard +0.3
The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.
  1. Find the value of \(c\) and the value of \(d\). [3]
The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.
  1. Find the position vector of \(P\). [6]
  2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]
SPS SPS FM Pure 2024 January Q5
13 marks Standard +0.8
Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$
  1. Express \(f(x)\) in terms of partial fractions [5]
  2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
  3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]
SPS SPS FM Pure 2024 January Q6
10 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$ where \(k\) is a constant. Given that $$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$ where \(a\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,
    1. determine the value of \(a\)
    2. show that \(k = -9\) [3]
  2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf{M}\). [6]
  3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]
SPS SPS FM Pure 2024 January Q7
14 marks Standard +0.3
The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]
The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg(z - 6) = -\frac{3\pi}{4}$$
  1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. [4]
  2. Find the complex number for which both \(|z - 6i| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]
SPS SPS FM Pure 2023 September Q1
5 marks Moderate -0.8
$$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}$$
  1. Find \(\mathbf{A}^{-1}\) [2 marks]
  2. The determinant of \(\mathbf{A}^2\) is equal to 4. Find the possible values of \(k\). [3 marks]
SPS SPS FM Pure 2023 September Q2
5 marks Moderate -0.3
A sequence \(u_n\) is defined by \(u_{n+1} = 2u_n + 3\) and \(u_1 = 1\). Prove by induction that \(u_n = 4 \times 2^{n-1} - 3\) for all positive integers \(n\). [5]
SPS SPS FM Pure 2023 September Q3
5 marks Standard +0.3
A finite region is bounded by the curve with equation \(y = x + x^{-\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]
SPS SPS FM Pure 2023 September Q4
13 marks Standard +0.8
The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$
  1. Find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [2]
The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).
  1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]
The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]