Questions — SPS (686 questions)

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SPS SPS FM Pure 2023 June Q6
5 marks Standard +0.3
A spherical balloon is inflated so that its volume increases at a rate of \(10\text{ cm}^3\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm. [For a sphere of radius \(r\), surface area \(= 4\pi r^2\) and volume \(= \frac{4}{3}\pi r^3\)]. [5]
SPS SPS FM Pure 2023 June Q7
6 marks Challenging +1.2
Prove that for all \(n \in \mathbb{N}\) $$\begin{pmatrix} 3 & 4i \\ i & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & 4ni \\ ni & 1-2n \end{pmatrix}$$ [6]
SPS SPS FM Pure 2023 June Q8
7 marks Challenging +1.2
  1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]
The complex number \(w\) satisfies \(|w - 4i| = 3\).
  1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]
SPS SPS FM Pure 2023 June Q9
8 marks Standard +0.3
  1. Use the binomial expansion to show that \((1 - 2x)^{-\frac{1}{4}} \approx 1 + x + \frac{5}{8}x^2\) for sufficiently small values of \(x\). [2]
  2. For what values of \(x\) is the expansion valid? [1]
  3. Find the expansion of \(\sqrt{\frac{1+2x}{1-2x}}\) in ascending powers of \(x\) as far as the term in \(x^2\). [3]
  4. Use \(x = \frac{1}{20}\) in your answer to part (iii) to find an approximate value for \(\sqrt{11}\). [2]
SPS SPS FM Pure 2023 June Q10
6 marks Standard +0.3
The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number. Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]
SPS SPS FM Pure 2023 June Q11
7 marks Challenging +1.2
In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]
SPS SPS FM Pure 2023 June Q12
7 marks Challenging +1.8
\includegraphics{figure_12} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\). [7]
SPS SPS FM Pure 2023 June Q13
10 marks Challenging +1.2
  1. Solve the differential equation $$\frac{dy}{dx} = y(1 + y)(1 - x),$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\), where \(f\) is a function to be determined. [7]
  2. By considering the sign of \(\frac{dy}{dx}\) near \((1, 1)\), or otherwise, show that this point is a maximum point on the curve \(y = f(x)\). [3]
SPS SPS FM Pure 2023 June Q14
7 marks Challenging +1.8
A curve \(C\) has equation $$x^3 + y^3 = 3xy + 48$$ Prove that \(C\) has two stationary points and find their coordinates. [7]
SPS SPS FM Pure 2023 June Q15
8 marks Challenging +1.2
In this question you must use detailed reasoning.
  1. Show that \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1+\sin 2x}{-\cos 2x} dx = \ln(\sqrt{a} + b)\), where \(a\) and \(b\) are integers to be determined. [6]
  2. Show that \(\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1+\sin 2x}{-\cos 2x} dx\) is undefined, explaining your reasoning clearly. [2]
SPS SPS SM Pure 2023 June Q1
3 marks Easy -1.8
Find $$\int (x^4 - 6x^2 + 7) dx$$ giving your answer in simplest form. [3]
SPS SPS SM Pure 2023 June Q2
3 marks Moderate -0.8
Curve C has equation $$y = x^3 - 7x^2 + 5x + 4$$ The point \(P(2, -6)\) lies on \(C\) Find the equation of the tangent to \(C\) at \(P\) Give your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [3]
SPS SPS SM Pure 2023 June Q3
3 marks Standard +0.3
Express in partial fractions, $$\frac{9x^2}{(x-1)^2(2x+1)}$$ [3]
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]