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SPS SPS FM Statistics 2021 June Q5
7 marks Moderate -0.3
Eleven students in a class sit a Mathematics exam and their average score is 67% with a standard deviation of 12%. One student from the class is absent and sits the paper later, achieving a score of 85%.
  1. Find the mean score for the whole class and the standard deviation for the whole class. [5]
  2. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier. [2]
SPS SPS FM Statistics 2021 June Q6
6 marks Standard +0.3
Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]
SPS SPS FM Statistics 2021 June Q7
12 marks Standard +0.8
A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).
  1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
  2. Find \(E(X)\). [3]
  3. Find \(P(X > E(X)|X > 2)\) [4]
SPS SPS FM Pure 2021 June Q1
2 marks Moderate -0.8
A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]
SPS SPS FM Pure 2021 June Q2
6 marks Moderate -0.3
The equation \(x^3 - 3x + 1 = 0\) has three real roots.
  1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
  2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
  3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]
SPS SPS FM Pure 2021 June Q3
4 marks Moderate -0.3
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of \(P\). [2]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]
SPS SPS FM Pure 2021 June Q4
8 marks Standard +0.8
Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]
SPS SPS FM Pure 2021 June Q5
5 marks Standard +0.3
\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]
SPS SPS FM Pure 2021 June Q6
5 marks Moderate -0.3
  1. Use the binomial expansion, in ascending powers of \(x\), to show that $$\sqrt{4-x} = 2 - \frac{1}{4}x + kx^2 + ...$$ where \(k\) is a rational constant to be found. [4] A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt{3}\).
  2. State, giving a reason, if the expansion is valid for this value of \(x\). [1]
SPS SPS FM Pure 2021 June Q7
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
SPS SPS FM Pure 2021 June Q8
6 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\) $$f(n) = 2^{n+2} + 3^{2n+1}$$ is divisible by 7 [6]
SPS SPS FM Pure 2021 June Q9
6 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$
  1. State the range of \(f\) [1]
  2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
  3. state the set of possible values for \(k\). [2]
SPS SPS FM Pure 2021 June Q10
8 marks Challenging +1.8
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]
SPS SPS FM Pure 2021 June Q11
7 marks Standard +0.8
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x} dx\) Fully justify your answer. [6 marks]
SPS SPS FM Pure 2021 June Q13
8 marks Standard +0.8
$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,
  1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
  2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]
SPS SPS FM Pure 2021 June Q14
6 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.
  1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
  2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]
SPS SPS FM Pure 2021 June Q15
7 marks Standard +0.3
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = -\frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
SPS SPS FM Pure 2021 June Q16
7 marks Challenging +1.8
Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]
SPS SPS SM Pure 2021 June Q1
5 marks Moderate -0.8
A curve has equation $$y = 2x^3 - 4x + 5$$ Find the equation of the tangent to the curve at the point \(P(2, 13)\). Write your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers to be found. Solutions relying on calculator technology are not acceptable. [5]
SPS SPS SM Pure 2021 June Q2
4 marks Easy -1.2
Given that the point \(A\) has position vector \(3\mathbf{i} - 7\mathbf{j}\) and the point \(B\) has position vector \(8\mathbf{i} + 3\mathbf{j}\),
  1. find the vector \(\overrightarrow{AB}\) [2]
  2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]
SPS SPS SM Pure 2021 June Q3
5 marks Moderate -0.8
\includegraphics{figure_1} The shape \(ABCDOA\), as shown in Figure 1, consists of a sector \(COD\) of a circle centre \(O\) joined to a sector \(AOB\) of a different circle, also centre \(O\). Given that arc length \(CD = 3\) cm, \(\angle COD = 0.4\) radians and \(AOD\) is a straight line of length 12 cm,
  1. find the length of \(OD\), [2]
  2. find the area of the shaded sector \(AOB\). [3]
SPS SPS SM Pure 2021 June Q4
5 marks Moderate -0.3
The function \(\mathbf{f}\) is defined by $$\mathbf{f}(x) = \frac{3x - 7}{x - 2} \quad x \in \mathbb{R}, x \neq 2$$
  1. Find \(\mathbf{f}^{-1}(7)\) [2]
  2. Show that \(\mathbf{f}(x) = \frac{ax + b}{x - 3}\) where \(a\) and \(b\) are integers to be found. [3]
SPS SPS SM Pure 2021 June Q5
6 marks Moderate -0.8
A car has six forward gears. The fastest speed of the car • in 1st gear is 28 km h⁻¹ • in 6th gear is 115 km h⁻¹ Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,
  1. find the fastest speed of the car in 3rd gear. [3]
Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
  1. find the fastest speed of the car in 5th gear. [3]
SPS SPS SM Pure 2021 June Q6
6 marks Moderate -0.8
  1. Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$(1 + kx)^{10}$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. [3]
Given that in the expansion of \((1 + kx)^{10}\) the coefficient \(x^3\) is 3 times the coefficient of \(x\),
  1. find the possible values of \(k\). [3]
SPS SPS SM Pure 2021 June Q7
8 marks Standard +0.3
Given that \(k\) is a positive constant and \(\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4\)
  1. show that \(3k + 5\sqrt{k} - 12 = 0\) [4]
  2. Hence, using algebra, find any values of \(k\) such that $$\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$$ [4]