Find \(x \), \(y \) and \(z \) for \(\neg q \Rightarrow \neg p\).

\( \begin{array}{|c|c|c|c|c|} \hline
p & q & \neg p & \neg q & \neg q \Rightarrow \neg p \\ \hline
\text{F} & \text{T} & x & y & z \\ \hline
\end{array} \)

Determine the validity of this argument:

It is windy and Amy does not get cool.   Therefore,  Amy does not feel happy.

Find  \(x\), \(y\) and \(z\)  if  \(\left( {\begin{array}{*{20}{c}}
{x + 4}\\
{y-1}\\
{z-4}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{2-x}\\
{5-y}\\
{ -2-z}
\end{array}} \right)\)

Find the coordinates \(\left( {a,b,c} \right)\) of the point where the line with parametric equations \(x = 2 + 5t,{\rm{ }}y = -3-4t\) and \(z = 3 + t\) meets the \(XOY\) plane.

A line has a vector equation \(\left( {\begin{array}{*{20}{c}}
x\\
y
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
3\\
6
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
5\\
{ – 4}
\end{array}} \right)\). Find the shortest distance from \(P\left( {2,3} \right)\) to the line, correcting to three significant figures.

State whether two lines are parallel, intersect coincident, skew or perpendicular.

Line \(a\): \(x = 1 + s\),  \(y = – 2s\)  and   \(z = 4 + 4s\)

Line \(b\): \(x = 7 + 2t\),  \(y = 2 + 3t\)  and   \(z = t\)

Find \(a\),   if  \(\int_0^3 {x\sqrt {x + 1} } dx = \dfrac{a}{{15}}\).

Evaluate  \(\int_0^1 {\dfrac{{{x^2} + 4x}}{{\sqrt[3]{{{x^3} + 6{x^2}}}}}} dx\).

Given \(U = \left\{ {x|2 < x \le 8,x \in N} \right\}\) and the following pair of propositions,   find the truth set of \( p \veebar q \).

•    \(p:\) \(x \) is even.
•    \(q:\) \(x \) is devisible by 3.

Given  \(\vec a = \left( {\begin{array}{*{20}{c}}
3\\
5\\
{ – 1}
\end{array}} \right),\vec b = \left( {\begin{array}{*{20}{c}}
1\\
3\\
{ – 4}
\end{array}} \right)\)  and  \(\vec c = \left( {\begin{array}{*{20}{c}}
3\\
– 2\\
7
\end{array}} \right)\),   find   \(\left| {\vec a – \vec b + 3\vec c} \right|\),   correcting to two significant figures.