Questions — SPS SPS FM (161 questions)

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SPS SPS FM 2024 October Q6
7 marks Standard +0.8
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 FM 2024 October Q7
9 marks Standard +0.8
The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}
  1. Determine the coordinates of \(B\). [6]
  2. Hence determine the exact value of \(\tan\alpha\). [3]
SPS SPS FM 2024 October Q8
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]
SPS SPS FM 2024 October Q9
9 marks Standard +0.8
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]
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]
SPS SPS FM 2025 February Q7
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 \(\text{Re}(z) \geq 9\). [6]
SPS SPS FM 2025 February Q8
4 marks Moderate -0.3
A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).
  1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
  2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]
SPS SPS FM 2025 February Q9
9 marks Standard +0.8
\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
  2. Use the model to find the volume of water that the bottle can contain. [7]
SPS SPS FM 2025 October Q1
3 marks Easy -1.2
Determine the equation of the line that passes through the point \((1, 3)\) and is perpendicular to the line with equation \(3x + 6y - 5 = 0\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
SPS SPS FM 2025 October Q2
6 marks Moderate -0.8
In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.
  1. Show that \(\cos CAB = \frac{2}{3}\). [2]
  2. Hence find the exact value of \(\sin CAB\). [2]
  3. Find the exact area of triangle \(ABC\). [2]
SPS SPS FM 2025 October Q3
4 marks Moderate -0.5
Given the function \(f(x) = 3x^3 - 7x - 1\), defined for all real values of \(x\), prove from first principles that \(f'(x) = 9x^2 - 7\). [4]
SPS SPS FM 2025 October Q4
8 marks Moderate -0.3
The cubic polynomial \(2x^3 - kx^2 + 4x + k\), where \(k\) is a constant, is denoted by f(x). It is given that f'(2) = 16.
  1. Show that \(k = 3\). [3]
For the remainder of the question, you should use this value of \(k\).
  1. Use the factor theorem to show that \((2x + 1)\) is a factor of f(x). [2]
  2. Hence show that the equation f(x) = 0 has only one real root. [3]
SPS SPS FM 2025 October Q5
4 marks Standard +0.3
In this question you must show detailed reasoning. Consider the expansion of \(\left(\frac{x^2}{2} + \frac{a}{x}\right)^6\). The constant term is 960. Find the possible values of \(a\). [4]
SPS SPS FM 2025 October Q6
6 marks Moderate -0.3
The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$
  1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
  2. Find the nature of the stationary point, justifying your answer. [2]
SPS SPS FM 2025 October Q7
7 marks Standard +0.8
The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]
SPS SPS FM 2025 October Q8
4 marks Standard +0.8
A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]
SPS SPS FM 2025 October Q9
7 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)
  1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
  2. Find the equation of C. You may leave your answer in factorised form. [3]
The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,
  1. find the range of possible values for \(k\). [2]
SPS SPS FM 2025 October Q10
4 marks Moderate -0.8
The graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis followed by a translation.
    1. State the scale factor of the stretch. [1]
    2. Give full details of the translation. [2]
Alternatively the graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.
  1. State the scale factor of the stretch parallel to the \(y\)-axis. [1]
SPS SPS FM 2025 October Q11
8 marks Standard +0.3
The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$
  1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
  2. Find the range of the function fg. [2]
  3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]
SPS SPS FM 2025 October Q12
6 marks Standard +0.3
Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]