Questions — SPS (686 questions)

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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]
SPS SPS SM 2022 February Q7
10 marks Moderate -0.3
The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}
  1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
  2. Find the area of the shaded segment. [3]
  3. Find the perimeter of the shaded segment. [4]
SPS SPS SM 2022 February Q8
9 marks Moderate -0.8
The diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}
  1. Evaluate \(f(-3)\). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
SPS SPS FM Pure 2022 June Q1
7 marks Standard +0.3
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int_0^9 \frac{1}{\sqrt{x}} dx\); [3 marks]
    2. \(\int_0^9 \frac{1}{x\sqrt{x}} dx\). [3 marks]
  2. Explain briefly why the integrals in part (a) are improper integrals. [1 mark]
SPS SPS FM Pure 2022 June Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = f(x + 1)\), [2]
  2. \(y = f(|x|)\). [2]
Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find
  1. the value of \(a\) and the value of \(b\), [2]
  2. the value of \(x\) for which \(f(x) = 5x\). [3]
SPS SPS FM Pure 2022 June Q3
8 marks Standard +0.3
  1. Show on an Argand diagram the locus of points given by $$|z - 10 - 12i| = 8$$ [2] Set \(A\) is defined by $$A = \left\{z : 0 \leq \arg(z - 10 - 10i) \leq \frac{\pi}{2}\right\} \cap \{z : |z - 10 - 12i| < 8\}$$
  2. Shade the region defined by \(A\) on your Argand diagram. [2]
  3. Determine the area of the region defined by \(A\). [4]
SPS SPS FM Pure 2022 June Q4
8 marks Standard +0.3
The curve with equation \(y = f(x)\) where $$f(x) = x^2 + \ln(2x^2 - 4x + 5)$$ has a single turning point at \(x = \alpha\)
  1. Show that \(\alpha\) is a solution of the equation $$2x^3 - 4x^2 + 7x - 2 = 0$$ [4]
The iterative formula $$x_{n+1} = \frac{1}{7}(2 + 4x_n^2 - 2x_n^3)$$ is used to find an approximate value for \(\alpha\). Starting with \(x_1 = 0.3\)
  1. calculate, giving each answer to 4 decimal places,
    1. the value of \(x_2\)
    2. the value of \(x_4\)
    [2]
Using a suitable interval and a suitable function that should be stated,
  1. show that \(\alpha\) is 0.341 to 3 decimal places. [2]
SPS SPS FM Pure 2022 June Q5
4 marks Standard +0.3
The triangle \(T\) has vertices at the points \((1, k)\), \((3,0)\) and \((11,0)\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T'\) by the matrix $$\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix}$$ Given that the area of triangle \(T'\) is 364 square units, find the value of \(k\). [4]
SPS SPS FM Pure 2022 June Q6
7 marks Moderate -0.3
The complex number \(w\) is given by $$w = 10 - 5i$$
  1. Find \(|w|\). [1]
  2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [1]
The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + i)(z + 3i) = w$$
  1. Use algebra to find \(z\), giving your answer in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
Given that $$\arg(\lambda + 9i + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,
  1. find the value of \(\lambda\). [1]
SPS SPS FM Pure 2022 June Q7
7 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = xe^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis. Use integration by parts to find an exact value for the volume of the solid generated. [7]
SPS SPS FM Pure 2022 June Q8
5 marks Standard +0.8
With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\). The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\). Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]
SPS SPS FM Pure 2022 June Q9
6 marks Standard +0.3
Prove by induction that for \(n \in \mathbb{Z}^+\) $$f(n) = 4^{n+1} + 5^{2n-1}$$ is divisible by 21 [6]
SPS SPS FM Pure 2022 June Q10
8 marks Standard +0.8
The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}
  1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
  2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]