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SPS SPS SM Pure 2021 June Q8
5 marks Moderate -0.3
  1. Given that \(\mathbf{f}(x) = x^2 - 4x + 2\), find \(\mathbf{f}(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
  2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]
SPS SPS SM Pure 2021 June Q9
7 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows part of the curve with equation \(y = 3\cos x^2\). The point \(P(c, d)\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.
  1. State the value of \(c\) and the value of \(d\). [1]
  2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3\cos x^2\) to the curve with equation
    1. \(y = 3\cos\left(\frac{x^2}{4}\right)\)
    2. \(y = 3\cos(x - 36)^2\)
    [2]
  3. Solve, for \(450° \leq \theta < 720°\), $$3\cos\theta = 8\tan\theta$$ giving your solution to one decimal place. [4]
SPS SPS SM Pure 2021 June Q10
10 marks Moderate -0.3
$$g(x) = 2x^3 + x^2 - 41x - 70$$
  1. Use the factor theorem to show that \(g(x)\) is divisible by \((x - 5)\). [2]
  2. Hence, showing all your working, write \(g(x)\) as a product of three linear factors. [4]
The finite region \(R\) is bounded by the curve with equation \(y = g(x)\) and the \(x\)-axis, and lies below the \(x\)-axis.
  1. Find, using algebraic integration, the exact value of the area of \(R\). [4]
SPS SPS SM Pure 2021 June Q11
9 marks Standard +0.3
  1. A circle \(C_1\) has equation $$x^2 + y^2 + 18x - 2y + 30 = 0$$ The line \(l\) is the tangent to \(C_1\) at the point \(P(-5, 7)\). Find an equation of \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found. [5]
  2. A different circle \(C_2\) has equation $$x^2 + y^2 - 8x + 12y + k = 0$$ where \(k\) is a constant. Given that \(C_2\) lies entirely in the 4th quadrant, find the range of possible values for \(k\). [4]
SPS SPS SM Pure 2021 June Q12
5 marks Standard +0.3
Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]
SPS SPS SM Pure 2021 June Q13
3 marks Moderate -0.8
  1. Prove that for all positive values of \(x\) and \(y\) $$\sqrt{xy} \leq \frac{x + y}{2}$$ [2]
  2. Prove by counter example that this is not true when \(x\) and \(y\) are both negative. [1]
SPS SPS SM Pure 2021 June Q14
13 marks Moderate -0.3
\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log_{10} P\) for the population over a period of 100 years. The line \(l\) meets the vertical axis at \((0, 5)\) as shown. The gradient of \(l\) is \(\frac{1}{200}\).
  1. Write down an equation for \(l\). [2]
  2. Find the value of \(a\) and the value of \(b\). [4]
  3. With reference to the model interpret
    1. the value of the constant \(a\),
    2. the value of the constant \(b\).
    [2]
  4. Find
    1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
    2. the number of years it takes the population to reach 200000, according to the model.
    [3]
  5. State two reasons why this may not be a realistic population model. [2]
SPS SPS SM Pure 2021 June Q15
9 marks Standard +0.8
A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)
  1. find \(g(x)\), [7]
  2. prove that the stationary point at \((2, 9)\) is a maximum. [2]
SPS SPS FM 2020 September Q1
3 marks Moderate -0.3
Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by $$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$ where \(p\) and \(q\) are constants. Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM 2020 September Q2
4 marks Challenging +1.2
A sequence of numbers \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 3$$ $$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$
  1. Find \(\sum_{r=1}^{100} a_r\) [3]
  2. Hence find \(\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r\) [1]
SPS SPS FM 2020 September Q3
4 marks Moderate -0.3
Using algebraic integration and making your method clear, find the exact value of $$\int_1^5 \frac{4x + 9}{x + 3} \, dx = a + \ln b$$ where \(a\) and \(b\) are constants to be found [4]
SPS SPS FM 2020 September Q4
5 marks Moderate -0.8
  1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]
Adele uses \(\theta = 5°\) to test the approximation in part (a). Adele's working is shown below.
Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\)
Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\)
    1. Identify the mistake made by Adele in her working.
    2. Show that \(8 - 5\theta^2\) can be used to give a good approximation to \(1 + 4\cos \theta + 3\cos^2 \theta\) for an angle of size \(5°\) [2]
SPS SPS FM 2020 September Q5
7 marks Standard +0.3
\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
  1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
  2. Hence, using algebraic integration, find the exact area of \(R\). [3]
SPS SPS FM 2020 September Q6
5 marks Standard +0.8
A sequence is defined by \(U_n = 2^{n+1} + 9 \times 13^n\) for positive integer values of \(n\). Prove by induction that \(U_n\) is divisible by 11. [5]
SPS SPS FM 2020 September Q7
5 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]
SPS SPS FM 2020 September Q8
8 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.
  1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
  2. find, showing your working, the integer value of \(k\). [5]
SPS SPS FM 2020 September Q9
7 marks Standard +0.8
Show that $$\int_0^{\pi/2} \frac{\sin 2\theta}{1 + \cos \theta} \, d\theta = 2 - 2\ln 2$$ [7]
SPS SPS FM 2020 September Q10
5 marks Challenging +1.2
A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]
SPS SPS FM 2020 September Q11
4 marks Standard +0.3
Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\) [4]
SPS SPS FM 2020 September Q12
9 marks Standard +0.3
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
SPS SPS ASFM Mechanics 2021 May Q1
7 marks Challenging +1.3
In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
  2. Given that \(\alpha^3\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^3\), \(\beta^3\) and \(\gamma^3\). [4]
SPS SPS ASFM Mechanics 2021 May Q2
6 marks Moderate -0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), $$f(n) = 2^{2n-1} + 3^{2n-1} \text{ is divisible by 5}.$$ [6]
SPS SPS ASFM Mechanics 2021 May Q3
13 marks Challenging +1.2
The \(2 \times 2\) matrix \(\mathbf{A}\) represents a transformation \(T\) which has the following properties. • The image of the point \((0, 1)\) is the point \((3, 4)\). • An object shape whose area is \(7\) is transformed to an image shape whose area is \(35\). • \(T\) has a line of invariant points.
  1. Find a possible matrix for \(\mathbf{A}\). [8]
The transformation \(S\) is represented by the matrix \(\mathbf{B}\) where \(\mathbf{B} = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}\).
  1. Find the equation of the line of invariant points of \(S\). [2]
  2. Show that any line of the form \(y = x + c\) is an invariant line of \(S\). [3]
SPS SPS ASFM Mechanics 2021 May Q4
14 marks Standard +0.8
\includegraphics{figure_4} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length \(1\) m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length \(1\) m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius \(0.6\) m with constant angular velocity \(\omega\) rad s\(^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.
  1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
  2. Show that
    1. \(T_1 = \frac{4g}{5}(m_1 + m_2)\). [2]
    2. \(\omega^2 = \frac{4g(m_1 + 2m_2)}{4m_1}\). [3]
  3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2]
In a different case, where \(m_1 = 2.5\) and \(m_2 = 2.8\), \(P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
  1. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega\) rad s\(^{-1}\) to zero. [5]
SPS SPS ASFM Mechanics 2021 May Q5
10 marks Standard +0.3
A car of mass \(1250\) kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v\) m s\(^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.54\) m s\(^{-2}\). At a point \(B\) on the road the car's speed is \(20\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.3\) m s\(^{-2}\).
  1. Find the values of \(k\) and \(P\). [7]
The power is increased to \(15\) kW.
  1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]