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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]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Prove by induction that, for all positive integers \(n\), \(7^n + 3^{n-1}\) is a multiple of \(4\). [5]
SPS SPS FM Pure 2021 May Q6
8 marks Standard +0.3
\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.
  1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]
A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),
  1. determine the value of \(a\) and the value of \(k\). [3]
Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,
  1. determine this value of \(k\). [3]
SPS SPS FM Pure 2021 May Q7
9 marks Standard +0.3
Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),
  1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]
Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),
  1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]
SPS SPS FM Pure 2021 May Q8
8 marks Challenging +1.3
  1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
  2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM Pure 2021 May Q9
6 marks Standard +0.8
  1. Using the Maclaurin series for \(\ln(1 + x)\), find the first four terms in the series expansion for \(\ln(1 + 3x^2)\). [2]
  2. Find the range of \(x\) for which the expansion is valid. [1]
  3. Find the exact value of the series $$\frac{3^1}{2 \times 2^2} - \frac{3^2}{3 \times 2^4} + \frac{3^3}{4 \times 2^6} - \frac{3^4}{5 \times 2^8} + \ldots$$ [3]
SPS SPS FM Pure 2021 May Q10
7 marks Standard +0.8
A particular radioactive substance decays over time. A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac{dx}{dt} + \frac{1}{10}x = e^{-0.1t}\cos t.$$
  1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). [3]
Initially there was \(10\) g of the substance.
  1. Find the particular solution of the differential equation. [2]
  2. Find to \(6\) significant figures the amount of substance that would be predicted by the model at
    1. \(6\) hours, [1]
    2. \(6.25\) hours. [1]
SPS SPS FM Pure 2021 May Q1
7 marks Standard +0.3
In this question you must show detailed reasoning.
  1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
  2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]
SPS SPS FM Pure 2021 May Q2
8 marks Standard +0.8
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
In this question you must show detailed reasoning. Show that $$\int_5^{\infty} (x - 1)^{-\frac{3}{2}} dx = 1$$ [5]
SPS SPS FM Pure 2021 May Q4
6 marks Standard +0.3
You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.
  1. State the plane of reflection of \(R\). [1]
  2. Determine the value of \(a\). [3]
  3. With reference to \(R\) explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3 \times 3\) identity matrix. [2]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Express \(\frac{5x^2+x+12}{x^3+4x}\) in partial fractions. [5]
SPS SPS FM Pure 2021 May Q6
6 marks Challenging +1.2
A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]
SPS SPS FM Pure 2021 May Q7
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p + x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form