3.22: Particle relative motion; trig AP and log GP

0606/12/M/J/24 — Question 9 · 16 marks

(a)
Particle PP moves with speed 14.514.5 parallel to (20−21)\begin{pmatrix}20\\-21\end{pmatrix}. Find the velocity vector of PP.
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(b)
Initially PP has position (35)\begin{pmatrix}3\\5\end{pmatrix}. Write the position vector of PP at time tt.
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(c)
QQ has position (13)+(5−7.5)t\begin{pmatrix}1\\3\end{pmatrix}+\begin{pmatrix}5\\-7.5\end{pmatrix}t. Find the distance PQPQ in terms of tt, simplified.
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(d)
Hence show that PP and QQ never collide.
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(a)(i)
An AP has first three terms 32sin⁡x\dfrac{3}{2}\sin x, 52sin⁡x\dfrac{5}{2}\sin x, 72sin⁡x\dfrac{7}{2}\sin x. Show Sn=n(n+2)2sin⁡xS_{n}=\dfrac{n(n+2)}{2}\sin x.
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(a)(ii)
Given x=2π3x=\dfrac{2\pi}{3}, find the exact sum of the first 2020 terms.
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(b)(i)
A GP has first three terms ln⁡(y2)\ln(y^{2}), ln⁡(4y2)\ln(4y^{2}), ln⁡(16y4)\ln(16y^{4}). Find the nnth term.
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(b)(ii)
Find SnS_{n} in its simplest form.
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(c)
A GP has terms (1−w4)2\bigl(1-\tfrac{w}{4}\bigr)^{2}, (1−w4)4\bigl(1-\tfrac{w}{4}\bigr)^{4}, (1−w4)6\bigl(1-\tfrac{w}{4}\bigr)^{6}. Find ww for which S∞S_{\infty} exists.
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