3.57: Trapezium vectors; log AP and trig GP

0606/13/O/N/24 — Question 9 · 17 marks

3.57 diagram

(a)

In trapezium

OABC

,

\overrightarrow{OA}=4\mathbf{a}

,

\overrightarrow{OC}=\mathbf{c}

,

\overrightarrow{CB}=2\mathbf{a}

. Find

\overrightarrow{AB}

.

[1]

(b)

D

on

AB

with

AD:DB=2:1

. Find

\overrightarrow{OD}

.

[2]

(c)

Find

\overrightarrow{OX}

in terms of

\mathbf{a}

,

\mathbf{c}

and

n

.

[1]

(d)

Find

\overrightarrow{AX}

in terms of

\mathbf{a}

,

\mathbf{c}

and

m

.

[2]

(e)

Hence find

m

and

n

.

[4]

(a)

An AP has first three terms

\log_{x}3

,

\log_{x}81

,

\log_{x}2187

. Find

S_{n}

as

\log_{x}(3^{k})

with

k

in terms of

n

.

[3]

(b)

A GP has terms

1

,

\tfrac23\tan\theta

,

\tfrac49\tan^{2}\theta

for

0<\theta<\tfrac{\pi}{2}

. Find

\theta

for which

S_{\infty}

exists.

[4]