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SPS SPS SM Pure 2023 June Q4
6 marks Moderate -0.8
Relative to a fixed origin \(O\), • the point \(A\) has position vector \(5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) • the point \(B\) has position vector \(7\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) • the point \(C\) has position vector \(4\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\)
  1. Find \(|\vec{AB}|\) giving your answer as a simplified surd. [2] Given that \(ABCD\) is a parallelogram,
  2. find the position vector of the point \(D\). [2] The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC : CE = 2 : 1\)
  3. Find the coordinates of the point \(E\). [2]
SPS SPS SM Pure 2023 June Q5
5 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = e^{\frac{1}{5}x^2}\) for \(x \geq 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis, and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = e^{\frac{1}{5}x^2}\)
\(x\)00.511.52
\(y\)1\(e^{0.05}\)\(e^{0.2}\)\(e^{0.45}\)\(e^{0.8}\)
  1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 2 decimal places. [3]
  2. Use your answer to part (a) to deduce an estimate for
    1. \(\int_0^2 \left( 4 + e^{\frac{1}{5}x^2} \right) dx\)
    2. \(\int_1^3 e^{\frac{1}{5}(x-1)^2} dx\) giving your answers to 2 decimal places. [2]
SPS SPS SM Pure 2023 June Q6
5 marks Moderate -0.3
\includegraphics{figure_2} The shape \(AOCBA\), shown in Figure 2, consists of a sector \(AOB\) of a circle centre \(O\) joined to a triangle \(BOC\). The points \(A\), \(O\) and \(C\) lie on a straight line with \(AO = 7.5\) cm and \(OC = 8.5\) cm. The size of angle \(AOB\) is 1.2 radians. Find, in cm, the perimeter of the shape \(AOCBA\), giving your answer to one decimal place. [5]
SPS SPS SM Pure 2023 June Q7
6 marks Moderate -0.3
A ball is released from rest from a height of 5 m and bounces repeatedly on horizontal ground. After hitting the ground for the first time, the ball rises to a maximum height of 3 m. In a model for the motion of the ball • the maximum height after each bounce is 60% of the previous maximum height • the motion takes place in a vertical line
  1. Using the model
    1. show that the maximum height after the 3rd bounce is 1.08 m,
    2. find the total distance the ball travels from release to when the ball hits the ground for the 5th time.
    [3] According to the model, after the ball is released, there is a limit, \(D\) metres, to the total distance the ball will travel.
  2. Find the value of \(D\) [2] With reference to the model,
  3. give a reason why, in reality, the ball will not travel \(D\) metres in total. [1]
SPS SPS SM Pure 2023 June Q8
7 marks Moderate -0.8
\includegraphics{figure_4} A circle with centre \((9, -6)\) touches the \(x\)-axis as shown in Figure 4.
  1. Write down an equation for the circle. [3] A line \(l\) is parallel to the \(x\)-axis. The line \(l\) cuts the circle at points \(P\) and \(Q\). Given that the distance \(PQ\) is 8
  2. find the two possible equations for \(l\). [4]
SPS SPS SM Pure 2023 June Q9
4 marks Standard +0.3
A curve has equation $$y = 4x^2 - 5x$$ The curve passes through the point \(P(2, 6)\) Use differentiation from first principles to find the value of the gradient of the curve at \(P\). [4]
SPS SPS SM Pure 2023 June Q10
5 marks Moderate -0.3
\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,
  1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
  2. Determine whether, according to the model, Lee can safely inspect this defect. [2]
SPS SPS SM Pure 2023 June Q11
10 marks Standard +0.3
The function \(f\) is defined by $$f(x) = \frac{12x}{3x + 4} \quad x \in \mathbb{R}, x \geq 0$$
  1. Find the range of \(f\). [2]
  2. Find \(f^{-1}\). [3]
  3. Show, for \(x \in \mathbb{R}, x \geq 0\), that $$ff(x) = \frac{9x}{3x + 1}$$ [3]
  4. Show that \(ff(x) = \frac{7}{2}\) has no solutions. [2]
SPS SPS SM Pure 2023 June Q12
6 marks Standard +0.3
  1. Solve, for \(-180° \leq x < 180°\), the equation $$3 \sin^2 x + \sin x + 8 = 9 \cos^2 x$$ giving your answers to 2 decimal places. [4]
  2. Hence find the smallest positive solution of the equation $$3\sin^2(2\theta - 30°) + \sin(2\theta - 30°) + 8 = 9 \cos^2(2\theta - 30°)$$ giving your answer to 2 decimal places. [2]
SPS SPS SM Pure 2023 June Q13
6 marks Moderate -0.8
A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N\) ppm (parts per million), is modelled by the equation $$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$ where \(t\) hours is the time after the treatment was applied. Use the equation of the model to answer parts (a) and (b).
  1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3] For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
  2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. [3]
SPS SPS SM Pure 2023 June Q14
6 marks Standard +0.3
  1. Prove that the sum of the squares of 2 consecutive odd integers is always 2 more than a multiple of 8 [3]
  2. Use proof by contradiction to show that \(\log_2 5\) is irrational. [3]
SPS SPS SM Pure 2023 June Q15
6 marks Moderate -0.5
The resting metabolic rate, \(R\) ml of oxygen consumed per hour, of a particular species of mammal is modelled by the formula, $$R = aM^b$$ where • \(M\) grams is the mass of the mammal • \(a\) and \(b\) are constants
  1. Show that this relationship can be written in the form $$\log_{10} R = b \log_{10} M + \log_{10} a$$ [2] \includegraphics{figure_3} A student gathers data for \(R\) and \(M\) and plots a graph of \(\log_{10} R\) against \(\log_{10} M\) The graph is a straight line passing through points \((0.7, 1.2)\) and \((1.8, 1.9)\) as shown in Figure 3.
  2. Using this information, find a complete equation for the model. Write your answer in the form $$R = aM^b$$ giving the value of each of \(a\) and \(b\) to 3 significant figures. [3]
  3. With reference to the model, interpret the value of the constant \(a\) [1]
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]