- The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
- Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
- It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
$$z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80 = 0$$
where \(p\) and \(r\) are real numbers.
- Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
- Find the value of \(p\) and the value of \(r\).
[0pt]
[BLANK PAGE]