For two conditions on the real number $x$: $$\begin{aligned} & p : | x - 1 | \leq 3 , \\ & q : | x | \leq a \end{aligned}$$ What is the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on real number $x$: $$\begin{aligned} & p : ( x - 1 ) ( x - 4 ) = 0 , \\ & q : 1 < 2 x \leq a \end{aligned}$$ Find the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$. [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

For two conditions $p$ and $q$ on real numbers $x$: $$\begin{aligned} & p : x ^ { 2 } - 4 x + 3 > 0 , \\ & q : x \leq a \end{aligned}$$ What is the minimum value of the real number $a$ such that $\sim p$ is a sufficient condition for $q$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

3. Let $x \in R$. Then ``$x > 1$'' is ``$x ^ { 3 } > 1$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

4. Let $x \in \mathbb{R}$. Then ``$1 < x < 2$'' is ``$|x - 2| < 1$'' a
(A) sufficient but not necessary condition
(B) necessary but not sufficient condition
(C) necessary and sufficient condition
(D) neither sufficient nor necessary condition

Let $x \in \mathbb{R}$. Then ``$|x - 2| < 1$'' is ``$x^2 + x - 2 > 0$'' a
(A) sufficient but not necessary condition
(B) necessary but not sufficient condition
(C) necessary and sufficient condition
(D) neither sufficient nor necessary condition