Find the remainder when \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6\) is divided by \(( 2 x + 1 )\).
Given that \(2 \sin 2 \theta = \cos 2 \theta\),
show that \(\tan 2 \theta = 0.5\).
Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }\). [0pt]
[P1 June 2001 Question 2]
(a) Using the substitution \(u = 2 ^ { x }\), show that the equation \(4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0\) can be written in the form \(u ^ { 2 } - 2 u - 15 = 0\).
Hence solve the equation \(4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0\), giving your answers to 2 d . p. [0pt]
[P2 November 2002 Question 2]
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The shape of a badge is a sector \(A B C\) of a circle with centre \(A\) and radius \(A B\), as shown in Fig 1. The triangle \(A B C\) is equilateral and has a perpendicular height 3 cm .