Questions — SPS SPS SM (125 questions)

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SPS SPS SM 2021 November Q3
5 marks Standard +0.3
In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]
SPS SPS SM 2021 November Q4
5 marks Standard +0.3
Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]
SPS SPS SM 2021 November Q5
4 marks Moderate -0.3
  1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
  2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]
SPS SPS SM 2021 November Q6
7 marks Challenging +1.2
  1. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
  2. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]
SPS SPS SM 2021 November Q7
5 marks Standard +0.3
The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}
  1. Find the intersections of this graph with the \(x\) axis. [1]
  2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]
SPS SPS SM 2021 November Q8
11 marks Standard +0.3
  1. Express \(2\sqrt{3} \cos 2x - 6 \sin 2x\) in the form \(R\cos(2x + \alpha)\) where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) [3]
  2. Hence
    1. Solve the equation \(2\sqrt{3} \cos 2x - 6 \sin 2x = 6\) for \(0 \leq x \leq 2\pi\) Giving your answers in terms of \(\pi\). [3]
  3. It can be shown that \(y = 9 \sin 2x + 4 \cos 2x\) can be written as \(y = \sqrt{97} \sin(2x + 24.0°)\)
    1. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2x + 4 \cos 2x\) to the curve \(y = \sin x\) [3]
    2. Find the exact maximum and minimum values of the function; $$f(x) = \frac{1}{11 - 9 \sin 2x - 4 \cos 2x}$$ [2]
SPS SPS SM 2021 November Q9
7 marks Moderate -0.3
    1. Show that \(\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x\) [1]
    2. Hence find \(\int 2\cos^2 4x \, dx\) [3]
  2. Find \(\int \sin^3 x \, dx\) [3]
SPS SPS SM 2021 November Q10
7 marks Standard +0.3
  1. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\) [3]
  2. A curve is defined parametrically by the equations; $$x = \cos \theta \qquad y = \left(\frac{\sin \theta}{2}\right)\left(\sin \frac{\theta}{2}\right)$$ Show that the cartesian equation of the curve can be written as \(y^2 = \frac{1}{8}(1-x)^2(1+x)\) [4]
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 SM 2023 October Q1
3 marks Easy -1.2
In this question you must show detailed reasoning. Find the smallest positive integers \(m\) and \(n\) such that \(\left(\frac{64}{49}\right)^{-\frac{3}{2}} = \frac{m}{n}\) [3]
SPS SPS SM 2023 October Q2
4 marks Easy -1.2
In this question you must show detailed reasoning. Express \(\frac{8 + \sqrt{7}}{2 + \sqrt{7}}\) in the form \(a + b\sqrt{7}\), where \(a\) and \(b\) are integers. [4]
SPS SPS SM 2023 October Q3
5 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\). Given that • \(C\) has equation \(y = f(x)\) where \(f(x)\) is a quadratic expression in \(x\) • \(C\) cuts the \(x\)-axis at \(0\) and \(6\) • \(l\) cuts the \(y\)-axis at \(60\) and intersects \(C\) at the point \((10, 80)\) use inequalities to define the region \(R\) shown shaded in Figure 3. [5]
SPS SPS SM 2023 October Q4
6 marks Moderate -0.3
In this question you must show detailed reasoning. A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form using set notation. [6]
SPS SPS SM 2023 October Q5
9 marks Moderate -0.3
A sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 8 \quad \text{and} \quad u_{n+1} = u_n + 3.$$
  1. Show that \(u_5 = 20\). [1]
  2. The \(n\)th term of the sequence can be written in the form \(u_n = pn + q\). State the values of \(p\) and \(q\). [2]
  3. State what type of sequence it is. [1]
  4. Find the value of \(N\) such that \(\sum_{n=1}^{2N} u_n - \sum_{n=1}^{N} u_n = 1256\). [5]
SPS SPS SM 2023 October Q6
8 marks Standard +0.3
In part (ii) of this question you must show detailed reasoning.
  1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
  2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]
SPS SPS SM 2023 October Q7
5 marks Easy -1.3
  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]