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SPS SPS SM 2021 February Q8
7 marks Standard +0.3
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.
Type of defectColourFabricSewingSizing
Probability0.250.300.400.05
Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
  1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
  2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]
SPS SPS SM 2022 October Q1
4 marks Easy -1.2
  1. Sketch the curve \(y = 3^{-x}\) [2]
  2. Solve the inequality \(3^{-x} < 27\) [2]
SPS SPS SM 2022 October Q2
6 marks Easy -1.2
  1. Complete the square for \(1 - 4x - x^2\) [3]
  2. Sketch the curve \(y = 1 - 4x - x^2\), including the coordinates of any maximum or minimum points and the y intercept only. [3]
SPS SPS SM 2022 October Q3
7 marks Moderate -0.8
A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
  1. the number of houses built in 1986, the first year of the building programme, [5]
  2. the total number of houses built in the 25 years of the programme. [2]
SPS SPS SM 2022 October Q4
8 marks Standard +0.3
  1. Find the positive value of \(x\) such that $$\log_x 64 = 2$$ [2]
  2. Solve for \(x\) $$\log_2(11 - 6x) = 2\log_2(x - 1) + 3$$ [6]
SPS SPS SM 2022 October Q5
11 marks Moderate -0.3
The first term of a geometric series is 120. The sum to infinity of the series is 480.
  1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
  2. Find, to 2 decimal places, the difference between the 5th and 6th term. [2]
  3. Calculate the sum of the first 7 terms. [2]
The sum of the first \(n\) terms of the series is greater than 300.
  1. Calculate the smallest possible value of \(n\). [4]
SPS SPS SM 2022 October Q6
6 marks Easy -1.2
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
  2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]
SPS SPS SM 2022 October Q7
7 marks Standard +0.8
A sequence is defined by $$u_1 = 3$$ $$u_{n+1} = 2 - \frac{4}{u_n}, \quad n \geq 1$$ Find the exact values of
  1. \(u_2\), \(u_3\) and \(u_4\) [3]
  2. \(u_{61}\) [1]
  3. \(\sum_{i=1}^{99} u_i\) [3]
SPS SPS SM 2022 October Q8
8 marks Standard +0.3
The equation \(k(3x^2 + 8x + 9) = 2 - 6x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11k^2 - 30k - 9 > 0$$ [4]
  2. Find the range of possible values for \(k\). [4]
SPS SPS SM 2022 October Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]
SPS SPS FM 2022 February Q1
4 marks Easy -1.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & 1 \\ 0 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}\).
  1. Find \(\mathbf{A} + 3\mathbf{B}\). [2]
  2. Show that \(\mathbf{A} - \mathbf{B} = k\mathbf{I}\), where \(\mathbf{I}\) is the identity matrix and \(k\) is a constant whose value should be stated. [2]
SPS SPS FM 2022 February Q2
8 marks Moderate -0.8
The complex numbers \(3 - 2i\) and \(2 + i\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + iy\) and showing clearly how you obtain these answers,
  1. \(2z - 3w\), [2]
  2. \((iz)^2\), [3]
  3. \(\frac{z}{w}\). [3]
SPS SPS FM 2022 February Q3
8 marks Moderate -0.3
The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_3}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
  2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]
SPS SPS FM 2022 February Q4
4 marks Moderate -0.8
The transformation \(S\) is a shear parallel to the \(x\)-axis in which the image of the point \((1, 1)\) is the point \((0, 1)\).
  1. Draw a diagram showing the image of the unit square under \(S\). [2]
  2. Write down the matrix that represents \(S\). [2]
SPS SPS FM 2022 February Q5
11 marks Moderate -0.8
  1. Sketch the curve \(y = \left(\frac{1}{2}\right)^x\), and state the coordinates of any point where the curve crosses an axis. [3]
  2. Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve \(y = \left(\frac{1}{2}\right)^x\), the axes, and the line \(x = 2\). [4]
  3. The point \(P\) on the curve \(y = \left(\frac{1}{2}\right)^x\) has \(y\)-coordinate equal to \(\frac{1}{6}\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac{\log_{10} 3}{\log_{10} 2}.$$ [4]
SPS SPS FM 2022 February Q6
7 marks Moderate -0.8
In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
  2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]
SPS SPS FM 2022 February Q7
8 marks Moderate -0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\).
  1. Find \(\mathbf{A}^2\) and \(\mathbf{A}^3\). [3]
  2. Hence suggest a suitable form for the matrix \(\mathbf{A}^n\). [1]
  3. Use induction to prove that your answer to part (ii) is correct. [4]
SPS SPS FM 2022 February Q8
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
  2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]
SPS SPS FM 2022 February Q9
8 marks Standard +0.3
The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$
  1. Find the angle between \(AB\) and \(AC\). [6]
  2. Find the area of triangle \(ABC\). [2]
SPS SPS SM 2022 February Q1
6 marks Easy -1.3
  1. Evaluate \(27^{-\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1-\sqrt{5}}{3+\sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]
SPS SPS SM 2022 February Q2
8 marks Moderate -0.3
  1. Solve the equation \(x^4 - 10x^2 + 25 = 0\). [4]
  2. Given that \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\), find \(\frac{dy}{dx}\). [2]
  3. Hence find the number of stationary points on the curve \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\). [2]
SPS SPS SM 2022 February Q3
8 marks Moderate -0.3
Solve each of the following equations, for \(0° \leqslant x \leqslant 180°\).
  1. \(2\sin^2 x = 1 + \cos x\). [4]
  2. \(\sin 2x = -\cos 2x\). [4]
SPS SPS SM 2022 February Q4
8 marks Easy -1.3
  1. By expanding the brackets, show that \((x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12\). [3]
  2. Sketch the curve \(y = x^3 - 6x^2 + 5x + 12\), giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve \(y = -x^3 + 6x^2 - 5x - 12\). Label this curve \(C_2\). [2]
SPS SPS SM 2022 February Q5
6 marks Easy -1.2
The gradient of a curve is given by \(\frac{dy}{dx} = 2x^{-\frac{1}{2}}\), and the curve passes through the point \((4, 5)\). Find the equation of the curve. [6]
SPS SPS SM 2022 February Q6
8 marks Moderate -0.3
The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_6}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
  2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]