8
q - 1
q ^ { 2 } - 7
\end{array} \right)$$
where \(q\) is a constant.
- Find the values of \(q\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
- Find the angle between \(\mathbf { u }\) and \(\mathbf { v }\) when \(q = 0\).
6 - The first and second terms of a geometric progression are \(p\) and \(2 p\) respectively, where \(p\) is a positive constant. The sum of the first \(n\) terms is greater than \(1000 p\). Show that \(2 ^ { n } > 1001\). [2]
- In another case, \(p\) and \(2 p\) are the first and second terms respectively of an arithmetic progression. The \(n\)th term is 336 and the sum of the first \(n\) terms is 7224 . Write down two equations in \(n\) and \(p\) and hence find the values of \(n\) and \(p\).
7 (a) Solve the equation \(3 \sin ^ { 2 } 2 \theta + 8 \cos 2 \theta = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
(b)
\includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-13_540_750_251_735}
The diagram shows part of the graph of \(y = a + \tan b x\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left( - \frac { 1 } { 6 } \pi , 0 \right)\) and the \(y\)-axis at \(( 0 , \sqrt { } 3 )\). Find the values of \(a\) and \(b\).
8 - Express \(x ^ { 2 } - 4 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7\) for \(x < k\), where \(k\) is a constant. - State the largest value of \(k\) for which f is a decreasing function.
The value of \(k\) is now given to be 1 . - Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
- The function g is defined by \(\mathrm { g } ( x ) = \frac { 2 } { x - 1 }\) for \(x > 1\). Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf.