I1.1 If $|x+\sqrt{5}|+|y-\sqrt{5}|+|z|=0$, determine $\alpha=x^{2}+y^{2}+z^{2}$.
I1.2 If $\beta$ is the sum of all digits of the product $\underbrace{11111 \cdots 11}_{\alpha 1_{1}^{\prime} s} \times \underbrace{99999 \ldots 99}_{\alpha 9 \text { s's }}$, determine the value of $\beta$.
I1.3 Suppose that the real function $f(x)$ satisfies $f(x y)=f(x) f(y)$ for all real numbers $x$ and $y$, and $f(1)<1$. Determine the value of $\gamma=f(\beta)+100-\beta$.

I1.4 If $n$ is a positive integer and $f(n)=2^{n}+2^{n-1}+2^{n-2}+\ldots+2^{2}+2^{1}+1$, determine the value of $\delta=f(\gamma)$.
Sum of non-negative terms $=0 \Rightarrow$ each term $=0$ at the same time $x+\sqrt{5}=0$ and $y-\sqrt{5}=0$ and $z=0$
$x=-\sqrt{5}$ and $y=\sqrt{5}$ and $z=0$
$$
\alpha=x^{2}+y^{2}+z^{2}=5+5+0=10
$$

Observe the patterns $11 \times 99=1089 ; 111 \times 999=110889$.
Deductively, $\underbrace{11111 \cdots 11}_{101^{\prime} \mathrm{s}} \times \underbrace{99999 \cdots 99}_{109_{\mathrm{s}}}=\underbrace{11111 \cdots 11}_{91^{\prime} \mathrm{s}} \underbrace{88888 \cdots 889}_{9 g_{\mathrm{s}}}$
$\beta=$ the sum of all digits $=9+9 \times 8+9=90$
$f(1)=f(1) f(1)$
$\Rightarrow f(1)[f(1)-1]=0$
$\Rightarrow f(1)=0$ or 1 (rejected)
$\because f(1)=0$
$f(x)=f(1 \times x)=f(1) f(x)=0$ for all real values of $x$.
$\gamma=f(\beta)+100-\beta=0+100-90=10$
$f(n)=2^{n+1}-1$ (sum to $n$ terms of a G.S. $a=1, r=2$, no. of terms $=n+1$ )
$\delta=f(10)=2^{11}-1=2047$