Questions — SPS (686 questions)

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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 FM Pure 2025 January Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]
SPS SPS FM Pure 2025 January Q2
8 marks Standard +0.3
  1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]
SPS SPS FM Pure 2025 January Q3
8 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).
  1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
  2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
  3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]
SPS SPS FM Pure 2025 January Q4
12 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]
It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).
  1. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]
The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).
  1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
  2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]
SPS SPS FM Pure 2025 January Q5
11 marks Standard +0.3
With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
  2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]
The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)
  1. Find the position vector of \(B\). [3]
SPS SPS FM Pure 2025 January Q6
12 marks Standard +0.3
You are given the complex number \(w = 2 + 2\sqrt{3}i\).
  1. Express \(w\) in modulus-argument form. [3]
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]
SPS SPS FM Pure 2025 January Q7
8 marks Standard +0.8
A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}
  1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
  2. Hence find the volume of metal needed to create the candlestick. [4]
SPS SPS FM 2025 February Q1
7 marks Challenging +1.2
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 \(x = 2\). \includegraphics{figure_1} 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 FM 2025 February Q2
5 marks Moderate -0.3
  1. Find the first three terms in the expansion of \((1-2x)^{-1}\) in ascending powers of \(x\), where \(|x| < \frac{1}{2}\). [3]
  2. Hence find the coefficient of \(x^2\) in the expansion of \(\frac{x+3}{\sqrt{1-2x}}\). [2]
SPS SPS FM 2025 February Q3
4 marks Moderate -0.3
Express \(\frac{9x^2+43x+8}{(3+x)(1-x)(2x+1)}\) in partial fractions. [4]
SPS SPS FM 2025 February Q4
5 marks Moderate -0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a constant.
  1. Find \(\mathbf{A}^{-1}\). [2]
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\).
  1. Given that \(\mathbf{PA} = \mathbf{B}\), find the matrix \(\mathbf{P}\). [3]
SPS SPS FM 2025 February Q5
10 marks Moderate -0.8
  1. \(P\), \(Q\) and \(T\) are three transformations in 2-D. \(P\) is a reflection in the \(x\)-axis. \(\mathbf{A}\) is the matrix that represents \(P\). Write down the matrix \(\mathbf{A}\). [1]
  2. \(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). \(\mathbf{B}\) is the matrix that represents \(Q\). Find the matrix \(\mathbf{B}\). [2]
  3. \(T\) is \(P\) followed by \(Q\). \(\mathbf{C}\) is the matrix that represents \(T\). Determine the matrix \(\mathbf{C}\). [2]
  4. \(L\) is the line whose equation is \(y = x\). Explain whether or not \(L\) is a line of invariant points under \(T\). [2]
  5. An object parallelogram, \(M\), is transformed under \(T\) to an image parallelogram, \(N\). Explain what the value of the determinant of \(\mathbf{C}\) means about • the area of \(N\) compared to the area of \(M\). • the orientation of \(N\) compared to the orientation of \(M\). [3]
SPS SPS FM 2025 February Q6
8 marks Standard +0.3
The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).
  1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
  2. Find the acute angle between these lines. [3]