Questions — SPS SPS SM Pure (97 questions)

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SPS SPS SM Pure 2022 June Q8
8 marks Moderate -0.3
The function \(f(x)\) is such that \(f(x) = -x^3 + 2x^2 + kx - 10\) The graph of \(y = f(x)\) crosses the \(x\)-axis at the points with coordinates \((a, 0)\), \((2, 0)\) and \((b, 0)\) where \(a < b\)
  1. Show that \(k = 5\) [1 mark]
  2. Find the exact value of \(a\) and the exact value of \(b\) [3 marks]
  3. The functions \(g(x)\) and \(h(x)\) are such that $$g(x) = x^3 + 2x^2 - 5x - 10$$ $$h(x) = -8x^3 + 8x^2 + 10x - 10$$
    1. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = g(x)\) Fully justify your answer. [2 marks]
    2. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = h(x)\) Fully justify your answer. [2 marks]
SPS SPS SM Pure 2022 June Q9
5 marks Standard +0.3
A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u_n\) Find the exact value of \(\sum_{n=5}^{\infty} u_n\) [5 marks]
SPS SPS SM Pure 2022 June Q10
6 marks Moderate -0.8
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leq x \leq 2\pi\). State the values of the intercepts with the coordinate axes. [2 marks]
    1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that \(\cos \theta = \frac{1}{2}\). [2 marks]
    2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval \(0 \leq x \leq \pi\) [2 marks]
SPS SPS SM Pure 2022 June Q11
7 marks Standard +0.3
A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where \(p\) and \(q\) are constants. It is given that \(u_2 = 500\) and \(u_4 = 356\)
  1. Find the two possible values of \(u_3\) [5 marks]
  2. When \(u_n\) is a decreasing sequence, the limit of \(u_n\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\). [2 marks]
SPS SPS SM Pure 2022 June Q12
5 marks Moderate -0.8
A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$
  1. Find \(\frac{dy}{dx}\). [2 marks]
  2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]
SPS SPS SM Pure 2022 June Q13
4 marks Moderate -0.3
$$\frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \equiv A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}$$ Find the values of the constants \(A\), \(B\) and \(C\). [4]
SPS SPS SM Pure 2022 June Q14
6 marks Moderate -0.3
A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)
  1. Sketch a graph to show the region R. Shade the region R.
  2. Find the area of R [6 marks]
SPS SPS SM Pure 2022 June Q15
6 marks Standard +0.8
  1. Prove that $$n - 1 \text{ is divisible by } 3 \Rightarrow n^3 - 1 \text{ is divisible by } 9$$ [3 marks]
  2. Show that if \(\log_2 3 = \frac{p}{q}\), then $$2^p = 3^q.$$ Use proof by contradiction to prove that \(\log_2 3\) is irrational. [3 marks]
SPS SPS SM Pure 2022 June Q16
7 marks Standard +0.8
\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]
SPS SPS SM Pure 2022 June Q17
4 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = 2x^2 - x\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = -0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).
  1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). [1]
The estimate for R is found to be 2.58. Using this value, and showing your working,
  1. estimate the value of \(\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx\). [3]
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]