Determine the equation of the line that passes through the point \(( 1,3 )\) and is perpendicular to the line with equation \(3 x + 6 y - 5 = 0\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be determined.
In a triangle \(A B C , A B = 9 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(A C = 4 \mathrm {~cm}\).
Show that \(\cos C A B = \frac { 2 } { 3 }\).
Hence find the exact value of \(\sin C A B\).
Find the exact area of triangle \(A B C\). [0pt]
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Given the function \(f ( x ) = 3 x ^ { 3 } - 7 x - 1\), defined for all real values of \(x\), prove from first principles that \(f ^ { \prime } ( x ) = 9 x ^ { 2 } - 7\). [0pt]
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The cubic polynomial \(2 x ^ { 3 } - k x ^ { 2 } + 4 x + k\), where \(k\) is a constant, is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( 2 ) = 16\).
Show that \(k = 3\).
For the remainder of the question, you should use this value of \(k\).
Use the factor theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
Hence show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. [0pt]
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