I4.1 The average of $p, q, r$ is 12 . The average of $p, q, r, t, 2 t$ is 15 . Find $t$.
I4.2 $k$ is a real number such that $k^{4}+\frac{1}{k^{4}}=t+1$, and $s=k^{2}+\frac{1}{k^{2}}$. Find $s$.
I4.3 $M$ and $N$ are the points $(1,2)$ and $(11,7)$ respectively. $P(a, b)$ is a point on $M N$ such that $M P: P N=1: s$. Find $a$.

I4.4 If the curve $y=a x^{2}+12 x+c$ touches the $x$-axis, find $c$.
$\begin{array}{l}p+q+r=36 \\ p+q+r+t+2 t=75 \\ 3 t=75-36=39 \\ t=13 \\ k^{4}+\frac{1}{k^{4}}=14 \\ k^{4}+2+\frac{1}{k^{4}}=16 \\ \left(k^{2}+\frac{1}{k^{2}}\right)^{2}=16 \\ \Rightarrow s=k^{2}+\frac{1}{k^{2}}=4 \\ M P: P N=1: 4 \\ a=\frac{4+11}{1+4}=3 \\ y=3 x^{2}+12 x+c \\ \Delta=12^{2}-4(3) c=0 \\ \Rightarrow c=12\end{array}$