Challenging +1.8 This question requires establishing a quadratic equation from the GP property (ratio equality), solving a trigonometric quadratic with domain restrictions, verifying convergence conditions, and finding an infinite sum. While each individual step uses standard techniques, the combination of algebraic manipulation with trigonometric identities, the need to handle multiple cases carefully, and the requirement to express the final answer in a specific surd form elevates this significantly above routine GP questions. It's harder than typical A-level questions but not at the extreme difficulty of proof-heavy AEA problems.
15. The first three terms of a geometric series where \(\theta\) is a constant are
$$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$
a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\)
Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)
15. The first three terms of a geometric series where $\theta$ is a constant are
$$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$
a. Show that $4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0$
Given that $\theta$ lies in the interval $90 ^ { \circ } < \theta < 180 ^ { \circ }$,\\
b. Find the value of $\theta$.\\
c. Hence prove that this series is convergent.\\
d. Find $S _ { \infty }$, in the form $a ( 1 - \sqrt { 3 } )$\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q15 [9]}}