Questions — SPS SPS SM Pure (97 questions)

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SPS SPS SM Pure 2023 June Q16
8 marks Standard +0.3
\includegraphics{figure_5} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H\) m, is modelled by the equation $$H = 0.8 + k \cos(30t - 70)°$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).
  1. Find the time of day at which the height of the sea is at its maximum. [2] Given that the maximum height of the sea relative to the path is 2 m,
    1. find a complete equation for the model,
    2. state the minimum height of the sea relative to the path.
    [2] It is safe to use the path when the sea is 10 centimetres or more below the path.
  3. Find the times between which it is safe to use the path. (Solutions relying entirely on calculator technology are not acceptable.) [4]
SPS SPS SM Pure 2023 June Q17
5 marks Standard +0.8
\includegraphics{figure_7} Figure 7 shows the curves with equations $$y = kx^2 \quad x \geq 0$$ $$y = \sqrt{kx} \quad x \geq 0$$ where \(k\) is a positive constant. The finite region \(R\), shown shaded in Figure 7, is bounded by the two curves. Show that, for all values of \(k\), the area of \(R\) is \(\frac{1}{3}\) [5]
SPS SPS SM Pure 2023 June Q18
6 marks Moderate -0.8
Given that \(p\) is a positive constant,
  1. show that $$\sum_{n=1}^{11} \ln(p^n) = k \ln p$$ where \(k\) is a constant to be found, [2]
  2. show that $$\sum_{n=1}^{11} \ln(8p^n) = 33\ln(2p^2)$$ [2]
  3. Hence find the set of values of \(p\) for which $$\sum_{n=1}^{11} \ln(8p^n) < 0$$ giving your answer in set notation. [2]
SPS SPS SM Pure 2023 October Q1
8 marks Moderate -0.8
In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.
  1. Differentiate with respect to \(x\)
    1. \(x^2 e^{3x + 2}\), [4]
    2. \(\frac{\cos(2x^4)}{3x}\). [4]
SPS SPS SM Pure 2023 October Q2
11 marks Standard +0.3
  1. The curve \(C\) has equation $$y = \frac{x}{9 + x^2}.$$ Use calculus to find the coordinates of the turning points of \(C\). [6]
  2. Given that $$y = (1 + e^{2x})^{\frac{3}{2}},$$ find the value of \(\frac{dy}{dx}\) at \(x = \frac{1}{2} \ln 3\). [5]
SPS SPS SM Pure 2023 October Q3
12 marks Moderate -0.3
  1. Given that \(\cos A = \frac{3}{4}\), where \(270° < A < 360°\), find the exact value of \(\sin 2A\). [5]
    1. Show that \(\cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right) = \cos 2x\). [3] Given that $$y = 3\sin^2 x + \cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right),$$
    2. show that \(\frac{dy}{dx} = \sin 2x\). [4]
SPS SPS SM Pure 2023 October Q4
12 marks Moderate -0.3
$$f(x) = 12 \cos x - 4 \sin x.$$ Given that \(f(x) = R \cos(x + \alpha)\), where \(R \geq 0\) and \(0 \leq \alpha \leq 90°\),
  1. find the value of \(R\) and the value of \(\alpha\). [4]
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leq x < 360°\), giving your answers to one decimal place. [5]
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\). [1]
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. [2]
SPS SPS SM Pure 2023 October Q5
8 marks Standard +0.3
The curve \(C\) has equation $$y = \frac{3 + \sin 2x}{2 + \cos 2x}$$
  1. Show that $$\frac{dy}{dx} = \frac{6\sin 2x + 4\cos 2x + 2}{(2 + \cos 2x)^2}$$ [4]
  2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac{\pi}{2}\). Write your answer in the form \(y = ax + b\), where \(a\) and \(b\) are exact constants. [4]
SPS SPS SM Pure 2023 September Q1
6 marks Moderate -0.8
In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.
  1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left(1+\frac{x}{2}\right)^7\), giving each coefficient in exact simplified form. [3]
  2. Hence determine the coefficient of \(x\) in the expansion of $$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]
SPS SPS SM Pure 2023 September Q2
6 marks Moderate -0.8
\includegraphics{figure_2} The figure above shows a triangle with vertices at \(A(2,6)\), \(B(11,6)\) and \(C(p,q)\).
  1. Given that the point \(D(6,2)\) is the midpoint of \(AC\), determine the value of \(p\) and the value of \(q\). [2]
The straight line \(l\) passes through \(D\) and is perpendicular to \(AC\). The point \(E\) is the intersection of \(l\) and \(AB\).
  1. Find the coordinates of \(E\). [4]
SPS SPS SM Pure 2023 September Q3
7 marks Moderate -0.8
$$x^2 + y^2 - 2x - 2y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).
  1. Determine the value of \(r\) and the coordinates of \(C\). [3]
The point \(P(4,2)\) lies on the circle.
  1. Show that an equation of the tangent to the circle at \(P\) is [4] $$y = 14 - 3x.$$
SPS SPS SM Pure 2023 September Q4
8 marks Moderate -0.8
$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$ $$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$
  1. Find and simplify an expression for the composite function \(gf(x)\). [2]
  2. State the domain and range of \(gf(x)\). [2]
  3. Solve the equation $$gf(x) = 27.$$ [3]
The equation \(gf(x) = k\), where \(k\) is a constant, has solutions.
  1. State the range of the possible values of \(k\). [1]
SPS SPS SM Pure 2023 September Q5
7 marks Moderate -0.8
Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(4\mathbf{i} + 2\mathbf{j}\), \(3\mathbf{i} + 4\mathbf{j}\) and \(-\mathbf{i} + 12\mathbf{j}\), respectively.
  1. Find the magnitude of the vector \(\overrightarrow{OC}\) [2]
  2. Find the angle that the vector \(\overrightarrow{OB}\) makes with the vector \(\mathbf{j}\) to the nearest degree [2]
  3. Show that the points \(A\), \(B\) and \(C\) are collinear [3]
SPS SPS SM Pure 2023 September Q6
8 marks Moderate -0.8
Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by 10% compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.
  1. Show that the volume of the liquid in the container at the end of the second month is 202.5 litres. [1]
  2. Find the volume of the liquid in the container at the end of the twelfth month. [2]
At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.
  1. Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months. [5]
SPS SPS SM Pure 2023 September Q7
5 marks Moderate -0.3
\includegraphics{figure_7} The figure above shows a circular sector \(OAB\) whose centre is at \(O\). The radius of the sector is 60 cm. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively, so that \(|OC| = |OD| = 24\) cm. Given that the length of the arc \(AB\) is 48 cm, find the area of the shaded region \(ABDC\), correct to the nearest cm\(^2\). [5 marks]
SPS SPS SM Pure 2023 September Q8
9 marks Moderate -0.3
A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$
  1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
  2. Sketch in separate diagrams the graph of \(\ldots\)
    1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
    2. \(\ldots y = (3+x)(4-x)^2\). [2]
    3. \(\ldots y = (2-x)(5+x)^2\). [2]
    Each of the sketches must include any points where the graph meets the coordinate axes.
SPS SPS SM Pure 2023 September Q9
6 marks Challenging +1.2
Solve the following trigonometric equation in the range given. $$4\tan^2\theta\cos\theta = 15, \quad 0 \leq \theta < 360°.$$ [6 marks]
SPS SPS SM Pure 2023 September Q10
12 marks Standard +0.3
\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.
  1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
  2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]
SPS SPS SM Pure 2023 September Q11
4 marks Moderate -0.3
It is given that $$f(x) = x^2 - kx + (k+3),$$ where \(k\) is a constant. If the equation \(f(x) = 0\) has real roots find the range of the possible values of \(k\). [4]
SPS SPS SM Pure 2023 September Q12
8 marks Standard +0.3
\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$
  1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]
The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).
  1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]
SPS SPS SM Pure 2023 September Q13
6 marks Standard +0.8
Prove or disprove each of the following statements:
  1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
  2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
  3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]
SPS SPS SM Pure 2023 September Q14
8 marks Standard +0.3
\includegraphics{figure_14} The diagram above shows the curve with equation $$y = (x-4)^2, \quad x \in \mathbb{R},$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(AB\) and \(BC\). [8]