Questions — SPS (686 questions)

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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]
SPS SPS FM 2023 February Q5
9 marks Standard +0.3
  1. Expand \(\sqrt{1 + 2x}\) in ascending powers of x, up to and including the term in \(x^3\). [4]
  2. Hence expand \(\frac{\sqrt{1 + 2x}}{1 + 9x^2}\) in ascending powers of x, up to and including the term in \(x^3\). [3]
  3. Determine the range of values of x for which the expansion in part (b) is valid. [2]
SPS SPS FM 2023 February Q6
7 marks Standard +0.3
  1. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. Find the probability that all the men are next to each other. [3]
  2. Find the probability that no two men are next to one another. [4]
SPS SPS FM 2023 February Q7
8 marks Standard +0.8
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ P is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of P. [3]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]
Q is a point on \(l_1\) which is 12 metres away from P. R is the point on \(l_2\) such that QR is perpendicular to \(l_1\).
  1. Determine the length QR. [2]
SPS SPS FM 2023 February Q8
5 marks Standard +0.3
In this question you must show detailed reasoning. The equation f(x) = 0, where f(x) = \(x^4 + 2x^3 + 2x^2 + 26x + 169\), has a root x = 2 + 3i.
  1. Express f(x) as a product of two quadratic factors. [4]
  2. Hence write down all the roots of the equation f(x) = 0. [1]
SPS SPS FM 2023 February Q9
3 marks Standard +0.8
O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. The arc BC is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram). A funnel can be modelled by rotating OABCD by \(2\pi\) radians about the y-axis. \includegraphics{figure_9} Find the volume of the funnel according to the model. [3]
SPS SPS FM 2023 February Q10
7 marks Challenging +1.2
A transformation is equivalent to a shear parallel to the x-axis followed by a shear parallel to the y-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of s the matrices which represent each of the shears. [7]
SPS SPS FM 2023 February Q11
6 marks Challenging +1.2
Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and Re (z) \(\geq\) 9. [6]
SPS SPS FM Pure 2023 June Q1
5 marks Easy -1.2
You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).
  1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
  2. State the range of \(gf(x)\). [1]
  3. Solve the inequality \(|3x - 1| > 1\). [3]
SPS SPS FM Pure 2023 June Q2
6 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
  2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]
SPS SPS FM Pure 2023 June Q3
6 marks Standard +0.3
You are given that \(f(x) = \ln(2x - 5) + 2x^2 - 30\), for \(x > 2.5\).
  1. Show that \(f(x) = 0\) has a root \(\alpha\) in the interval \([3.5, 4]\). [2]
A student takes 4 as the first approximation to \(\alpha\). Given \(f(4) = 3.099\) and \(f'(4) = 16.67\) to 4 significant figures,
  1. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures. [2]
  2. Show that \(\alpha\) is the only root of \(f(x) = 0\). [2]
SPS SPS FM Pure 2023 June Q4
7 marks Standard +0.3
You are given that \(M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}\).
  1. Show that \(M\) is non-singular. [2]
The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(M\). Given that the area of hexagon \(R\) is 5 square units,
  1. find the area of hexagon \(S\). [1]
The matrix \(M\) represents an enlargement, with centre \((0,0)\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \((0,0)\).
  1. Find the value of \(k\). [2]
  2. Find the value of \(\theta\). [2]
SPS SPS FM Pure 2023 June Q5
5 marks Standard +0.3
\includegraphics{figure_5} Figure 1 shows a sketch of a triangle \(ABC\). Given \(\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{BC} = \mathbf{i} - 9\mathbf{j} + 3\mathbf{k}\), show that \(\angle BAC = 105.9°\) to one decimal place. [5]