SPS SPS FM (SPS FM) 2020 September

Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by $$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$ where \(p\) and \(q\) are constants. Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\). [3]

A sequence of numbers \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 3$$ $$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$

1. Find \(\sum_{r=1}^{100} a_r\) [3]
2. Hence find \(\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r\) [1]

Using algebraic integration and making your method clear, find the exact value of $$\int_1^5 \frac{4x + 9}{x + 3} \, dx = a + \ln b$$ where \(a\) and \(b\) are constants to be found [4]

1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]

Adele uses \(\theta = 5°\) to test the approximation in part (a). Adele's working is shown below.

Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places.

Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\)

Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\)

1. 1. Identify the mistake made by Adele in her working.
   2. Show that \(8 - 5\theta^2\) can be used to give a good approximation to \(1 + 4\cos \theta + 3\cos^2 \theta\) for an angle of size \(5°\) [2]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
2. Hence, using algebraic integration, find the exact area of \(R\). [3]

A sequence is defined by \(U_n = 2^{n+1} + 9 \times 13^n\) for positive integer values of \(n\). Prove by induction that \(U_n\) is divisible by 11. [5]

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.

1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
2. find, showing your working, the integer value of \(k\). [5]

Show that $$\int_0^{\pi/2} \frac{\sin 2\theta}{1 + \cos \theta} \, d\theta = 2 - 2\ln 2$$ [7]

A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\) [4]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]