Questions — SPS (686 questions)

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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]
SPS SPS ASFM Mechanics 2021 May Q6
Moderate -0.3
At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30°\) to the horizontal. The distance travelled down the chute by each brick is \(8\) m. A brick of mass \(3\) kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5\) m s\(^{-1}\).
  1. Find the potential energy lost by the brick in moving down the chute.
(2)
  1. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
(5)
  1. Hence find the coefficient of friction between the brick and the chute.
(3) Another brick of mass \(3\) kg slides down the chute. This brick is given an initial speed of \(2\) m s\(^{-1}\) at the top of the chute.
  1. Find the speed of this brick when it reaches the bottom of the chute.
(5)
SPS SPS ASFM Statistics 2021 May Q1
10 marks Moderate -0.8
  1. The complex number \(3 + 2i\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w^*\). Find
    1. the modulus of \(w\), [1]
    2. the argument of \(w^*\), giving your answer in radians, correct to 2 decimal places. [3]
  2. Find the complex number \(u\) given that \(u + 2u^* = 3 + 2i\). [4]
  3. Sketch, on an Argand diagram, the locus given by \(|z + 1| = |z|\). [2]
SPS SPS ASFM Statistics 2021 May Q2
11 marks Standard +0.3
  1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]
Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\).
  1. Find the point of intersection of \(l_1\) and \(l_2\). [4]
  2. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]
SPS SPS ASFM Statistics 2021 May Q3
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 Statistics 2021 May Q4
8 marks Moderate -0.3
Every time a spinner is spun, the probability that it shows the number 4 is 0.2, independently of all other spins.
  1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required. [2]
  2. Calculate the probability that the number of spins required is between 3 and 10 inclusive. [2]
  3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\). [4]
SPS SPS ASFM Statistics 2021 May Q5
8 marks Moderate -0.3
Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.
  1. Arlosh calculates Spearman's rank correlation coefficient \(r_s\) for the critics' ratings. He calculates that \(\Sigma d^2 = 72\). Show that this value must be incorrect. [2]
  2. Arlosh checks his working with Sarah, whose answer \(r_s = \frac{39}{35}\) is correct. Find the correct value of \(\Sigma d^2\). [2]
  3. Carry out an appropriate two-tailed significance test of the value of \(r_s\) at the 5% significance level, stating your hypotheses clearly. [4]
SPS SPS ASFM Statistics 2021 May Q6
9 marks Moderate -0.8
A spinner has edges numbered 1, 2, 3, 4 and 5. When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table.
Score, \(N\)12345
Probability0.30.20.2\(x\)\(y\)
It is known that E\((N) = 2.55\).
  1. Find Var\((N)\). [7]
  2. Find E\((3N + 2)\). [1]
  3. Find Var\((3N + 2)\). [1]
SPS SPS ASFM Statistics 2021 May Q7
Moderate -0.3
A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
  1. Find the probability of exactly 4 faults in a 15 metre length of cloth. (2)
  2. Find the probability of more than 10 faults in 60 metres of cloth. (3)
A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
  1. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. (4)
The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.
  1. Find the retailer's expected profit. (4)
SPS SPS FM Pure 2021 May Q1
5 marks Moderate -0.3
Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.
  1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
  2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]
SPS SPS FM Pure 2021 May Q2
9 marks Standard +0.3
The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_2}
  1. The greatest value of \(r\) on the curve occurs at the point \(P\).
    1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
    2. Find the value of \(r\) at the point \(P\). [1]
    3. Mark the point \(P\) on a copy of the graph. [1]
  2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).
  1. Find \(\mathbf{A}^4\). [1]
  2. Describe the transformation that \(\mathbf{A}\) represents. [2]
The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).
  1. Write down the matrix \(\mathbf{B}\). [1]
The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).
  1. Find the coordinates of \(P'\). [1]
SPS SPS FM Pure 2021 May Q4
3 marks Easy -1.2
Using the formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\), show that \(\sum_{r=1}^{10} r(3r - 2) = 1045\). [3]