SPS SPS FM Pure (SPS FM Pure) 2023 February

Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]

Solve the equation \(2z - 5iz^* = 12\). [4]

In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\). \includegraphics{figure_4} This region is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [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]

Prove by induction that for all positive integers \(n\) $$f(n) = 3^{2n+4} - 2^{2n}$$ is divisible by 5 [6]

In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]

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]

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]

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]

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]

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]

$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$

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]
3. 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]

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]

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]