Questions — SPS SPS FM (161 questions)

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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 2024 October Q1
8 marks Moderate -0.8
    1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
    2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
  2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]
SPS SPS FM 2024 October Q2
5 marks Easy -1.2
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.8
  1. Find and simplify the first three terms in the expansion of \((2-5x)^5\) in ascending powers of \(x\). [3]
  2. In the expansion of \((1+ax)^2(2-5x)^5\), the coefficient of \(x\) is 48. Find the value of \(a\). [3]
SPS SPS FM 2024 October Q4
11 marks Moderate -0.3
The functions f and g are defined for all real values of \(x\) by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).
  1. Find the range of f. [3]
  2. Give a reason why f has no inverse. [1]
  3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
  4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]
SPS SPS FM 2024 October Q5
5 marks Moderate -0.3
In this question you must show detailed reasoning Find the equation of the normal to the curve \(y = \frac{x^2-32}{\sqrt{x}}\) at the point on the curve where \(x = 4\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM 2024 October Q6
6 marks Standard +0.8
Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]
SPS SPS FM 2024 October Q7
7 marks Standard +0.8
In this question you must show detailed reasoning. A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]
SPS SPS FM 2024 October Q8
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
SPS SPS FM 2024 October Q9
6 marks Standard +0.8
  1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
  2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.
[6 marks]
SPS SPS FM 2023 October Q1
4 marks Moderate -0.8
This question requires detailed reasoning. Express \(\frac{3 + \sqrt{20}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [4]
SPS SPS FM 2023 October Q2
6 marks Moderate -0.8
Solve each of the following equations, for \(0° < x < 360°\).
  1. \(\sin \frac{1}{2}x = 0.8\) [3]
  2. \(\sin x = 3 \cos x\) [3]
SPS SPS FM 2023 October Q3
6 marks Easy -1.3
  1. Sketch the curve \(y = -\frac{1}{x}\). [2]
  2. The curve \(y = -\frac{1}{x}\) is translated by 2 units parallel to the x-axis in the positive direction. State the equation of the transformed curve. [2]
  3. Describe a transformation that transforms the curve \(y = -\frac{1}{x}\) to the curve \(y = -\frac{1}{3x}\). [2]
SPS SPS FM 2023 October Q4
7 marks Moderate -0.3
In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4\sqrt{x - 3x + 1}\) at the point on the curve where x = 4. Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]
SPS SPS FM 2023 October Q5
6 marks Moderate -0.8
  1. Find the binomial expansion of \((3 + kx)^3\), simplifying the terms. [4]
  2. It is given that, in the expansion of \((3 + kx)^3\), the coefficient of \(x^2\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form. [2]
SPS SPS FM 2023 October Q6
8 marks Standard +0.3
In this question you must show detailed reasoning. The functions f and g are defined for all real values of \(x\) by $$f(x) = x^3 \text{ and } g(x) = x^2 + 2.$$
  1. Write down expressions for
    1. \(fg(x)\), [1]
    2. \(gf(x)\). [1]
  2. Hence find the values of \(x\) for which \(fg(x) - gf(x) = 24\). [6]
SPS SPS FM 2023 October Q7
6 marks Standard +0.8
The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q\sqrt{r}\) where \(p\), \(q\) and \(r\) are integers. [6]
SPS SPS FM 2023 October Q8
5 marks Standard +0.8
Prove that \(2^{3n} - 3^n\) is divisible by 5 for all integers \(n \geq 1\). [5]
SPS SPS FM 2023 October Q9
12 marks Standard +0.3
  1. \includegraphics{figure_9} The shape ABC shown in the diagram is a student's design for the sail of a small boat. The curve AC has equation \(y = 2 \log_2 x\) and the curve BC has equation \(y = \log_2\left(x - \frac{3}{2}\right) + 3\). State the x-coordinate of point A. [1]
  2. Determine the x-coordinate of point B. [3]
  3. By solving an equation involving logarithms, show that the x-coordinate of point C is 2. [4] It is given that, correct to 3 significant figures, the area of the sail is 0.656 units\(^2\).
  4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines. [4]
SPS SPS FM 2024 October Q1
6 marks Moderate -0.8
Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),
  1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
  2. Find the maximum value of \(f(x)\). [1]
  3. Explain why \(f^{-1}(x)\) does not exist. [1]
SPS SPS FM 2024 October Q2
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.3
  1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left(2 + \frac{1}{3}kx\right)^6\), where \(k\) is a constant. [3]
  2. In the expansion of \((3 - 4x)\left(2 + \frac{1}{3}kx\right)^6\), the constant term is equal to the coefficient of \(x^2\). Determine the exact value of \(k\), given that \(k\) is positive. [3]
SPS SPS FM 2024 October Q4
3 marks Moderate -0.8
The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]
SPS SPS FM 2024 October Q5
9 marks Standard +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]