Questions — SPS (686 questions)

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SPS SPS FM Pure 2023 February Q4
8 marks Standard +0.3
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
SPS SPS FM Pure 2023 February Q5
6 marks Standard +0.3
Prove by induction that for all positive integers \(n\) $$f(n) = 3^{2n+4} - 2^{2n}$$ is divisible by 5 [6]
SPS SPS FM Pure 2023 February Q6
4 marks Standard +0.8
In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]
SPS SPS FM Pure 2023 February Q7
10 marks Challenging +1.3
  1. Prove that $$\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad -k < x < k$$ stating the value of the constant \(k\). [5]
  2. Hence, or otherwise, solve the equation $$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$ [5]
SPS SPS FM Pure 2023 February Q8
6 marks Challenging +1.8
The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]
SPS SPS FM Pure 2023 February Q9
7 marks Standard +0.8
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation \(|z - 3| = 2\) [1]
\includegraphics{figure_9}
  1. There is a unique complex number \(w\) that satisfies both \(|w - 3| = 2\) and \(\arg(w + 1) = \alpha\) where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
    1. Find the value of \(\alpha\). [2]
    2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4]
SPS SPS FM Pure 2023 February Q10
10 marks Challenging +1.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$ [8]
  2. Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f(x)\). [2]
SPS SPS FM Pure 2023 February Q11
9 marks Challenging +1.2
In an Argand diagram, the points \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2i\).
  1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [6]
The points \(D\), \(E\) and \(F\) are the midpoints of the sides of triangle \(ABC\).
  1. Find the exact area of triangle \(DEF\). [3]
SPS SPS FM Pure 2023 February Q12
11 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$
  1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
    3x - 6y + 4z &= 1
    3x + 2y - z &= 0 \end{align} [5]
    1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]
SPS SPS FM Pure 2023 February Q13
11 marks Challenging +1.8
In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}\) for \(0 \leqslant \theta \leqslant \pi\). \includegraphics{figure_13}
  1. Find the exact area enclosed by the curve. [4]
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}} e^{\frac{1}{6}}\). [7]
SPS SPS FM Pure 2023 February Q14
7 marks Challenging +1.3
  1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln\left(\frac{1}{2} + \cos x\right)\). [4]
  2. By considering the root of the equation \(\ln\left(\frac{1}{2} + \cos x\right) = 0\) deduce that \(\pi \approx 3\sqrt{3 \ln\left(\frac{3}{2}\right)}\). [3]
SPS SPS FM 2024 October Q1
8 marks Moderate -0.8
    1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
    2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
  2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]
SPS SPS FM 2024 October Q2
5 marks Easy -1.2
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.8
  1. Find and simplify the first three terms in the expansion of \((2-5x)^5\) in ascending powers of \(x\). [3]
  2. In the expansion of \((1+ax)^2(2-5x)^5\), the coefficient of \(x\) is 48. Find the value of \(a\). [3]
SPS SPS FM 2024 October Q4
11 marks Moderate -0.3
The functions f and g are defined for all real values of \(x\) by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).
  1. Find the range of f. [3]
  2. Give a reason why f has no inverse. [1]
  3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
  4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]
SPS SPS FM 2024 October Q5
5 marks Moderate -0.3
In this question you must show detailed reasoning Find the equation of the normal to the curve \(y = \frac{x^2-32}{\sqrt{x}}\) at the point on the curve where \(x = 4\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM 2024 October Q6
6 marks Standard +0.8
Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]
SPS SPS FM 2024 October Q7
7 marks Standard +0.8
In this question you must show detailed reasoning. A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]
SPS SPS FM 2024 October Q8
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
SPS SPS FM 2024 October Q9
6 marks Standard +0.8
  1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
  2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.
[6 marks]
SPS SPS SM 2023 October Q1
3 marks Easy -1.2
In this question you must show detailed reasoning. Find the smallest positive integers \(m\) and \(n\) such that \(\left(\frac{64}{49}\right)^{-\frac{3}{2}} = \frac{m}{n}\) [3]
SPS SPS SM 2023 October Q2
4 marks Easy -1.2
In this question you must show detailed reasoning. Express \(\frac{8 + \sqrt{7}}{2 + \sqrt{7}}\) in the form \(a + b\sqrt{7}\), where \(a\) and \(b\) are integers. [4]
SPS SPS SM 2023 October Q3
5 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\). Given that • \(C\) has equation \(y = f(x)\) where \(f(x)\) is a quadratic expression in \(x\) • \(C\) cuts the \(x\)-axis at \(0\) and \(6\) • \(l\) cuts the \(y\)-axis at \(60\) and intersects \(C\) at the point \((10, 80)\) use inequalities to define the region \(R\) shown shaded in Figure 3. [5]
SPS SPS SM 2023 October Q4
6 marks Moderate -0.3
In this question you must show detailed reasoning. A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form using set notation. [6]
SPS SPS SM 2023 October Q5
9 marks Moderate -0.3
A sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 8 \quad \text{and} \quad u_{n+1} = u_n + 3.$$
  1. Show that \(u_5 = 20\). [1]
  2. The \(n\)th term of the sequence can be written in the form \(u_n = pn + q\). State the values of \(p\) and \(q\). [2]
  3. State what type of sequence it is. [1]
  4. Find the value of \(N\) such that \(\sum_{n=1}^{2N} u_n - \sum_{n=1}^{N} u_n = 1256\). [5]