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SPS SPS SM 2022 October Q1
2 marks Easy -1.8
Simplify \(\left(\frac{x^{12}}{16}\right)^{-\frac{3}{4}}\) [2]
SPS SPS SM 2022 October Q2
5 marks Easy -1.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = -3x^2 + 12x + 8$$
  1. Write \(f(x)\) in the form $$a(x + b)^2 + c$$ where \(a\), \(b\) and \(c\) are constants to be found. [3]
The curve \(C\) has a maximum turning point at \(M\).
  1. Find the coordinates of \(M\). [2]
SPS SPS SM 2022 October Q3
5 marks Moderate -0.8
In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Simplify $$\frac{\sqrt{32} + \sqrt{18}}{3 + \sqrt{2}}$$ giving your answer in the form \(b\sqrt{2} + c\), where \(b\) and \(c\) are integers. [5]
SPS SPS SM 2022 October Q4
6 marks Moderate -0.3
The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
  2. Hence find the set of possible values of \(k\). [2]
SPS SPS SM 2022 October Q5
7 marks Moderate -0.8
  1. Given that $$y = \log_3 x$$ find expressions in terms of \(y\) for
    1. \(\log_3\left(\frac{x}{9}\right)\)
    2. \(\log_3 \sqrt{x}\)
    Write each answer in its simplest form. [3]
  2. Hence or otherwise solve $$2\log_3\left(\frac{x}{9}\right) - \log_3 \sqrt{x} = 2$$ [4]
SPS SPS SM 2022 October Q6
6 marks Moderate -0.8
An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54
  1. show that $$a + 4d = 6$$ [2]
Given also that the 8th term is half the 7th term,
  1. find the values of \(a\) and \(d\). [4]
SPS SPS SM 2022 October Q7
6 marks Moderate -0.3
In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Using algebra, find all solutions of the equation $$3x^3 - 17x^2 - 6x = 0$$ [3]
  2. Hence find all real solutions of $$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]
SPS SPS SM 2022 October Q8
7 marks Standard +0.3
\includegraphics{figure_2} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = pm^q$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg. Figure 2 illustrates the linear relationship between \(\log_{10} h\) and \(\log_{10} m\) The line meets the vertical \(\log_{10} h\) axis at 2.25 and has a gradient of \(-0.235\)
  1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). [3]
A particular mammal has a mass of 5kg and a resting heart rate of 119 beats per minute.
  1. Comment on the suitability of the model for this mammal. [3]
  2. With reference to the model, interpret the value of the constant \(p\). [1]
SPS SPS SM 2022 October Q9
7 marks Challenging +1.2
A sequence of numbers \(a_1, a_2, a_3, \ldots\) is defined by $$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, \(n \in \mathbb{N}\) where \(k\) is a constant. Given that
  • the sequence is a periodic sequence of order 3
  • \(a_1 = 2\)
  1. show that $$k^2 + k - 2 = 0$$ [3]
  2. For this sequence explain why \(k \neq 1\) [1]
  3. Find the value of $$\sum_{r=1}^{80} a_r$$ [3]
SPS SPS SM 2022 October Q10
7 marks Standard +0.3
A circle \(C\) with radius \(r\)
  • lies only in the 1st quadrant
  • touches the \(x\)-axis and touches the \(y\)-axis
The line \(l\) has equation \(2x + y = 12\)
  1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]
Given also that \(l\) is a tangent to \(C\),
  1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]
SPS SPS FM 2023 January Q1
5 marks Easy -1.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\). \(\mathbf{I}\) denotes the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf{A} + 3\mathbf{B} - 4\mathbf{I}\). [3]
  2. \(\mathbf{AB}\). [2]
SPS SPS FM 2023 January Q2
4 marks Moderate -0.3
The transformations \(\mathbf{R}\), \(\mathbf{S}\) and \(\mathbf{T}\) are defined as follows. \begin{align} \mathbf{R} &: \quad \text{reflection in the } x\text{-axis}
\mathbf{S} &: \quad \text{stretch in the } x\text{-direction with scale factor } 3
\mathbf{T} &: \quad \text{translation in the positive } x\text{-direction by } 4 \text{ units} \end{align}
  1. The curve \(y = \ln x\) is transformed by \(\mathbf{R}\) followed by \(\mathbf{T}\). Find the equation of the resulting curve. [2]
  2. Find, in terms of \(\mathbf{S}\) and \(\mathbf{T}\), a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]
SPS SPS FM 2023 January Q3
5 marks Standard +0.8
Express \(\frac{x^2}{(x-1)^2(x-2)}\) in partial fractions. [5]
SPS SPS FM 2023 January Q4
5 marks Challenging +1.2
$$\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}$$ The matrix \(\mathbf{A}\) represents the linear transformation \(M\). Prove that, for the linear transformation \(M\), there are no invariant lines. [5]
SPS SPS FM 2023 January Q5
7 marks Moderate -0.3
  1. Expand \((2+x)^{-2}\) in ascending powers of \(x\) up to and including the term in \(x^3\), and state the set of values of \(x\) for which the expansion is valid. [5]
  2. Hence find the coefficient of \(x^3\) in the expansion of \(\frac{1+x^2}{(2+x)^2}\). [2]
SPS SPS FM 2023 January Q6
7 marks Standard +0.3
The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics{figure_6} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
  1. How many selections of seven cards are possible? [3]
  2. Find the probability that the seven cards include exactly one card showing the letter A. [4]
SPS SPS FM 2023 January Q7
9 marks Standard +0.3
With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations \begin{align} l_1: \quad \mathbf{r} &= (-9\mathbf{i} + 10\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})
l_2: \quad \mathbf{r} &= (3\mathbf{i} + \mathbf{j} + 17\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k}) \end{align} 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 \(A\) has position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\).
  1. Show that \(A\) lies on \(l_1\). [1]
SPS SPS FM 2023 January Q8
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM 2023 January Q9
5 marks Moderate -0.3
Please remember to show detailed reasoning in your answer \includegraphics{figure_9} The diagram shows the curve with equation \(y = (2x - 3)^2\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis. [5]
SPS SPS FM 2023 January Q10
6 marks Standard +0.3
The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\) [4]
A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).
  1. Determine, in terms of \(a\), the area of \(S'\) [2]
SPS SPS FM 2023 January Q11
10 marks Challenging +1.2
\includegraphics{figure_11} Figure 1 shows an Argand diagram. The set \(P\) of points that lie within the shaded region including its boundaries, is defined by $$P = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\}$$ where \(a\), \(b\), \(c\) and \(d\) are integers.
  1. Write down the values of \(a\), \(b\), \(c\) and \(d\). [3]
The set \(Q\) is defined by $$Q = \{z \in \mathbb{C} : a \leq |z + b + ci| \leq d\} \cap \{z \in \mathbb{C} : |z - i| \leq |z - 3i|\}$$
  1. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form. [7]
SPS SPS FM 2023 February Q1
2 marks Easy -1.8
Matrices A and B are given by \(\mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}\). Use A and B to disprove the proposition: "Matrix multiplication is commutative". [2]
SPS SPS FM 2023 February Q2
3 marks Moderate -0.8
A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]
SPS SPS FM 2023 February Q3
5 marks Standard +0.3
Express \(\frac{(x-7)(x-2)}{(x+2)(x-1)^2}\) in partial fractions. [5]
SPS SPS FM 2023 February Q4
5 marks Standard +0.3
  1. You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of k. [4]
  2. Determine whether the line with equation \(y = kx\) in part above is a line of invariant points under T. [1]