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SPS SPS FM 2020 December Q2
4 marks Moderate -0.3
Let \(a, b\) satisfy \(0 < a < b\).
  1. Find, in terms of \(a\) and \(b\), the value of $$\int_a^b \frac{81}{x^4} dx$$ [2]
  2. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]
SPS SPS FM 2020 December Q3
4 marks Moderate -0.3
  1. Sketch the graph of \(y = |3x - 1|\). [1]
  2. Hence, solve \(5x + 3 < |3x - 1|\). [3]
SPS SPS FM 2020 December Q4
6 marks Moderate -0.8
The following diagram shows the curve \(y = a \sin(b(x + c)) + d\), where \(a, b, c\) and \(d\) are all positive constants and \(x\) is measured in radians. The curve has a maximum point at \((1, 3.5)\) and a minimum point at \((2, 0.5)\). \includegraphics{figure_4}
  1. Write down the value of \(a\) and the value of \(d\). [2]
  2. Find the value of \(b\). [2]
  3. Find the smallest possible value of \(c\), given that \(c > 0\). [2]
SPS SPS FM 2020 December Q5
4 marks Moderate -0.8
The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$
  1. Write down the matrix representing A. [1]
  2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]
SPS SPS FM 2020 December Q6
4 marks Challenging +1.2
Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]
SPS SPS FM 2020 December Q7
7 marks Standard +0.3
The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}
  1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
  2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]
SPS SPS FM 2020 December Q8
5 marks Standard +0.3
  1. The \(2 \times 2\) matrix A is given by $$\mathbf{A} = \begin{pmatrix} 7 & 3 \\ 2 & 1 \end{pmatrix}.$$ The \(2 \times 2\) matrix B satisfies $$\mathbf{BA}^2 = \mathbf{A}.$$ Find the matrix B. [3]
  2. The \(2 \times 2\) matrix C is given by $$\mathbf{C} = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix}.$$ By considering \(\mathbf{C}^2\), show that the matrices \(\mathbf{I} - \mathbf{C}\) and \(\mathbf{I} + \mathbf{C}\) are inverse to each other. [2]
SPS SPS FM 2020 December Q9
5 marks Moderate -0.3
Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]
SPS SPS FM 2020 December Q10
4 marks Challenging +1.2
The \(2 \times 2\) matrix M is defined by $$\mathbf{M} = \begin{pmatrix} 0 & 0.25 \\ 0.36 & 0 \end{pmatrix}$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by M. [4]
SPS SPS FM 2020 December Q11
6 marks Standard +0.3
In the triangle \(PQR\), \(PQ = 6\), \(PR = k\), \(P\hat{Q}R = 30°\).
  1. For the case \(k = 4\), find the two possible values of \(QR\) exactly. [3]
  2. Determine the value(s) of \(k\) for which the conditions above define a unique triangle. [3]
SPS SPS FM 2020 December Q12
7 marks Standard +0.3
Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.
  1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]
The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
  1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
  2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]
SPS SPS FM 2020 December Q13
5 marks Standard +0.3
A series is given by $$\sum_{r=1}^k 9^{r-1}$$
  1. Write down a formula for the sum of this series. [1]
  2. Prove by induction that \(P(n) = 9^n - 8n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1. [4]
SPS SPS FM 2020 October Q1
7 marks Moderate -0.8
  1. Find the binomial expansion of \((2 + x)^5\), simplifying the terms. [4]
  2. Hence find the coefficient of \(y^3\) in the expansion of \((2 + 3y + y^2)^5\). [3]
SPS SPS FM 2020 October Q2
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z}{y})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS FM 2020 October Q3
8 marks Standard +0.3
  1. Give full details of a sequence of two transformations needed to transform the graph \(y = |x|\) to the graph of \(y = |2(x + 3)|\). [3]
  2. Solve \(|x| > |2(x + 3)|\), giving your answer in set notation. [5]
SPS SPS FM 2020 October Q4
5 marks Standard +0.3
Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^n r(3r + 1) = n(n + 1)^2\). [5]
SPS SPS FM 2020 October Q5
6 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows triangle \(ABC\), with \(AB = x\) cm, \(AC = (x + 2)\) cm, \(BC = 2\sqrt{7}\) cm and angle \(CAB = 60°\).
  1. Find the value of \(x\). [4]
  2. Find the area of triangle \(ABC\), giving your answer in an exact form as simply as possible. [2]
SPS SPS FM 2020 October Q6
5 marks Moderate -0.3
Prove by contradiction that \(\sqrt{7}\) is irrational. [5]
SPS SPS FM 2020 October Q7
7 marks Moderate -0.3
A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).
  1. Find \(\frac{dy}{dx}\). [1]
  2. Hence sketch the gradient function for the curve. [4]
  3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]
SPS SPS FM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS FM 2020 October Q9
8 marks Challenging +1.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]
SPS SPS SM 2020 October Q1
2 marks Easy -1.8
Simplify fully the following expressions:
  1. \(\frac{7y^{13}}{35y^7}\) [1]
  2. \(6x^{-2} \div x^{-5}\) [1]
SPS SPS SM 2020 October Q2
3 marks Easy -1.8
A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).
  1. State what type of sequence this is. [1]
  2. Find \(u_{17}\). [2]
SPS SPS SM 2020 October Q3
6 marks Moderate -0.3
  1. Write \(3x^2 - 6x + 1\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [2]
  2. Solve \(3x^2 - 6x + 1 \leq 0\), giving your answer in set notation. [4]
SPS SPS SM 2020 October Q4
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
  2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]