(a) The graph of \(y = 2px^{2} - p^{2}x - 14\) passes through the point (3, 10). Find the values of p. (b) Two lines, \(3y - 2x = 21\) and \(4y + 5x = 5\) i...

(a) Since the graph passes through \((3,10)\), substitute \(x=3\) and \(y=10\) into \(y=2px^2-p^2x-14\):

\[10=2p(3)^2-p^2(3)-14.\]

\[10=18p-3p^2-14\]

\[3p^2-18p+24=0\]

\[p^2-6p+8=0\]

\[(p-2)(p-4)=0.\]

Therefore,

\[\boxed{p=2\text{ or }p=4}.\]

For these two values, the corresponding curves are \(y=4x^2-4x-14\) and \(y=8x^2-16x-14\). Both pass through \((3,10)\), as shown below.

(b) The equations of the two lines are

\[3y-2x=21 \quad \text{and} \quad 4y+5x=5.\]

Multiply the first equation by 5 and the second equation by 2:

\[15y-10x=105\]

\[8y+10x=10.\]

Adding the equations gives

\[23y=115\]

\[y=5.\]

Substitute \(y=5\) into \(3y-2x=21\):

\[3(5)-2x=21\]

\[15-2x=21\]

\[-2x=6\]

\[x=-3.\]

Hence, the coordinates of \(Q\) are

\[\boxed{Q=(-3,5)}.\]