SPS SPS FM (SPS FM) 2021 November

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [3 marks]

The equation of a curve in polar coordinates is $$r = 11 + 9 \sec \theta.$$ Show that a cartesian equation of the curve is $$(x - 9)\sqrt{x^2 + y^2} = 11x.$$ [3 marks]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|.$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6 marks]

Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]

Use a trigonometrical substitution to show that $$\int_0^2 \frac{1}{(16 - x^2)^{\frac{3}{2}}} dx = \frac{1}{16\sqrt{3}}$$ [4 marks]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]

The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).

1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]

In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).

1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]

1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4 marks]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9 marks]