Questions — SPS SPS FM (161 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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SPS SPS FM 2021 November Q6
7 marks Challenging +1.8
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]
SPS SPS FM 2021 November Q7
7 marks Challenging +1.3
The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]
SPS SPS FM 2021 November Q8
11 marks Standard +0.3
In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).
  1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
  2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]
SPS SPS FM 2021 November Q9
7 marks Standard +0.3
  1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
  2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]
SPS SPS FM 2021 November Q10
13 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4 marks]
  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9 marks]
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]
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]