Questions — SPS (686 questions)

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SPS SPS SM 2020 June Q1
6 marks Moderate -0.8
A curve has equation $$y = 2x^3 - 2x^2 - 2x + 8$$
  1. Find \(\frac{dy}{dx}\) [2]
  2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation. [4]
SPS SPS SM 2020 June Q2
9 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that - \(AFE\) is a straight line - \(AF = FE = 10.7\)m - \(BF = FD = 9.2\)m - angle \(BFD = 1.82\) radians find
  1. the perimeter of the stage, in metres, to one decimal place, [5]
  2. the area of the stage, in m², to one decimal place. [4]
SPS SPS SM 2020 June Q3
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 is a sketch showing the line \(l_1\) with equation \(y = 2x - 1\) and the point \(A\) with coordinates \((-2, 3)\). The line \(l_2\) passes through \(A\) and is perpendicular to \(l_1\)
  1. Find the equation of \(l_2\) writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [3]
The point \(B\) and the point \(C\) lie on \(l_1\) such that \(ABC\) is an isosceles triangle with \(AB = AC = 2\sqrt{13}\)
  1. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5x^2 - 12x - 32 = 0$$ [4]
Given that \(B\) lies in the 3rd quadrant
  1. find, using algebra and showing your working, the coordinates of \(B\). [4]
SPS SPS SM 2020 June Q4
4 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{g}(x)\). The curve has a single turning point, a minimum, at the point \(M(4, -1.5)\). The curve crosses the \(x\)-axis at two points, \(P(2, 0)\) and \(Q(7, 0)\). The curve crosses the \(y\)-axis at a single point \(R(0, 5)\).
  1. State the coordinates of the turning point on the curve with equation \(y = 2\text{g}(x)\). [1]
  2. State the largest root of the equation $$\text{g}(x + 1) = 0$$ [1]
  3. State the range of values of \(x\) for which \(\text{g}'(x) \leqslant 0\) [1]
Given that the equation \(\text{g}(x) + k = 0\), where \(k\) is a constant, has no real roots,
  1. state the range of possible values for \(k\). [1]
SPS SPS SM 2020 June Q5
4 marks Easy -1.2
Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of $$\left(1 + \frac{3}{4}x\right)^6$$ simplifying each term. [4]
SPS SPS SM 2020 June Q6
3 marks Moderate -0.8
A company which makes batteries for electric cars has a 10-year plan for growth. • In year 1 the company will make 2600 batteries • In year 10 the company aims to make 12000 batteries In order to calculate the number of batteries it will need to make each year, from year 2 to year 9, the company considers the following model: *the number of batteries made will increase by the same percentage each year* Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10. [3]
SPS SPS SM 2020 June Q7
9 marks Moderate -0.3
  1. Solve, for \(-90° \leqslant \theta < 270°\), the equation, $$\sin(2\theta + 10°) = -0.6$$ giving your answers to one decimal place. [5]
    1. A student's attempt at the question "Solve, for \(-90° < x < 90°\), the equation \(7\tan x = 8\sin x\)" is set out below. \begin{align} 7\tan x &= 8\sin x
      7 \times \frac{\sin x}{\cos x} &= 8\sin x
      7\sin x &= 8\sin x \cos x
      7 &= 8\cos x
      \cos x &= \frac{7}{8}
      x &= 29.0° \text{ (to 3 sf)} \end{align} Identify two mistakes made by this student, giving a brief explanation of each mistake. [2]
    2. Find the smallest positive solution to the equation $$7\tan(4\alpha + 199°) = 8\sin(4\alpha + 199°)$$ [2]
SPS SPS SM 2020 June Q8
4 marks Challenging +1.2
Prove by contradiction that there are no positive integers \(a\) and \(b\) with \(a\) odd such that $$a + 2b = \sqrt{8ab}$$ [4]
SPS SPS SM 2020 June Q9
4 marks Moderate -0.5
\includegraphics{figure_1} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = ab^t$$ where \(a\) and \(b\) are constants. The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log_{10} P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model, find a complete equation for the model giving the value of \(b\) to 4 significant figures. [4]
SPS SPS SM 2020 June Q10
8 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3x - 2\sqrt{x}\), \(x \geqslant 0\) and the line \(l\) with equation \(y = 8x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\). [5]
  2. \includegraphics{figure_4} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\). [3]
SPS SPS SM 2020 June Q11
9 marks Standard +0.3
  1. Sketch the curve with equation $$y = k - \frac{1}{2x}$$ where \(k\) is a positive constant State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. [3]
The straight line \(l\) has equation \(y = 2x + 3\) Given that \(l\) cuts the curve in two distinct places,
  1. find the range of values of \(k\), writing your answer in set notation. [6]
SPS SPS SM 2020 June Q12
8 marks Standard +0.3
\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 $$y = 3 \times 2^{2x}$$ The point \(P\left(a, 96\sqrt{2}\right)\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3 \times 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is $$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]
SPS SPS SM 2020 June Q13
6 marks Standard +0.3
\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\). *(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]
SPS SPS ASFM 2020 May Q1
5 marks Easy -1.3
You are given that \(z = 3 - 4\mathrm{i}\).
  1. Find
    [3] On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w^*\) is represented by the point \(B\).
  2. Describe the geometrical relationship between the points \(A\) and \(B\). [2]
SPS SPS ASFM 2020 May Q2
10 marks Standard +0.3
The position vector of point \(A\) is \(\mathbf{a} = -9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\). The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf{a}\).
  1. Determine the shortest distance between the origin, \(O\), and \(l\). [2] \(l\) is also perpendicular to the vector \(\mathbf{b}\) where \(\mathbf{b} = -2\mathbf{i} + \mathbf{j} + \mathbf{k}\).
  2. Find a vector which is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). [1]
  3. Write down an equation of \(l\) in vector form. [1] \(P\) is a point on \(l\) such that \(PA = 2OA\).
  4. Find angle \(POA\) giving your answer to 3 significant figures. [3] \(C\) is a point whose position vector, \(\mathbf{c}\), is given by \(\mathbf{c} = p\mathbf{a}\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf{r} = \mathbf{c} + \mu\mathbf{b}\). The point with position vector \(9\mathbf{i} + 8\mathbf{j} - 12\mathbf{k}\) lies on \(m\).
  5. Find the value of \(p\). [3]
SPS SPS ASFM 2020 May Q3
14 marks Standard +0.3
In this question you must show detailed reasoning. You are given that \(f(z) = 4z^4 - 12z^3 + 41z^2 - 128z + 185\) and that \(2 + \mathrm{i}\) is a root of the equation \(f(z) = 0\).
  1. Express \(f(z)\) as the product of two quadratic factors with integer coefficients. [5]
  2. Solve \(f(z) = 0\). [3] Two loci on an Argand diagram are defined by \(C_1 = \{z:|z| = r_1\}\) and \(C_2 = \{z:|z| = r_2\}\) where \(r_1 > r_2\). You are given that two of the points representing the roots of \(f(z) = 0\) are on \(C_1\) and two are on \(C_2\). \(R\) is the region on the Argand diagram between \(C_1\) and \(C_2\).
  3. Find the exact area of \(R\). [4]
  4. \(\omega\) is the sum of all the roots of \(f(z) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\). [2]
SPS SPS ASFM 2020 May Q4
9 marks Standard +0.8
In this question you must show detailed reasoning. You are given that \(\alpha\), \(\beta\) and \(\gamma\) are the roots of the equation \(5x^3 - 2x^2 + 3x + 1 = 0\).
  1. Find the value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\). [5]
  2. Find a cubic equation whose roots are \(\alpha^2\), \(\beta^2\) and \(\gamma^2\) giving your answer in the form \(ax^3 + bx^2 + cx + d = 0\) where \(a\), \(b\), \(c\) and \(d\) are integers. [4]
SPS SPS ASFM 2020 May Q5
5 marks Challenging +1.2
A transformation T is represented by the matrix \(\mathbf{T}\) where \(\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q'\). Find the smallest possible value of the area of \(Q'\). [5]
SPS SPS ASFM 2020 May Q6
6 marks Challenging +1.2
In this question you must show detailed reasoning. M is the matrix \(\begin{pmatrix} 1 & 6 \\ 0 & 2 \end{pmatrix}\). Prove that \(\mathbf{M}^n = \begin{pmatrix} 1 & 3(2^{n+1} - 2) \\ 0 & 2^n \end{pmatrix}\), for any positive integer \(n\). [6]
SPS SPS ASFM 2020 May Q7
6 marks Standard +0.3
\includegraphics{figure_7} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(OB\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm{ms}^{-1}\) (see diagram).
  1. Find the speed of \(B\) at the instant when \(OB\) makes an angle of 0.8 radians with the downward vertical through \(O\). [3]
  2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\). [3]
SPS SPS ASFM 2020 May Q8
14 marks Standard +0.3
\includegraphics{figure_8} 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 \(\omega \mathrm{rad s}^{-1}\). 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. \(T_1 = \frac{49}{4}(m_1 + m_2)\), [2]
    2. \(\omega^2 = \frac{49(m_1 + 2m_2)}{4m_1}\). [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.
  4. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm{rad s}^{-1}\) to zero. [5]
SPS SPS ASFM 2020 May Q9
10 marks Challenging +1.8
Three particles, \(P\), \(Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(km\) respectively, where \(k\) is a constant. Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly. The coefficient of restitution between each pair of particles is \(e\).
  1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. [9] Given that the mass of \(P\) is \(km\) and that there is a second collision,
  2. write down, in terms of \(u\), \(k\) and \(e\), the speed of \(Q\) after this second collision. [1]
SPS SPS ASFM 2020 May Q10
6 marks Standard +0.3
On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution Po(120).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10-minute period. [2]
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received. [3]
  3. State a necessary assumption for the validity of your calculation in part (b). [1]
SPS SPS ASFM 2020 May Q11
7 marks Standard +0.3
The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other. [3]
  2. Find the probability that no two men are next to one another. [4]
SPS SPS ASFM 2020 May Q12
7 marks Challenging +1.8
Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than 1%. Determine the smallest possible value of \(n\). [7]