Find the vectors \(\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }\) and \(\overrightarrow { C D }\).
Find
the length of a side of the square base of \(P\),
the cosine of the angle between one of the slanting edges of \(P\) and its base,
the height of \(P\),
the position vector of \(E\).
A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
Find the position vector of the other vertex of this octahedron.
6. (i) A curve with equation \(y = \mathrm { f } ( x )\) has \(\mathrm { f } ( x ) \geqslant 0\) for \(x \geqslant a\) and
$$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$
where \(a\) and \(b\) are constants with \(b > a\).
Use integration by substitution to show that for the positive constants \(r\) and \(h\)
$$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$
(ii)
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Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }\)
This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \(( 0 , p )\).