Let's evaluate each of the given statements one by one based on the conditions provided: integer [tex]\( k \)[/tex] is negative and integer [tex]\( c \)[/tex] is the square of a positive integer less than 10.
Step-by-Step Solution:
1. Analyze Statement A: [tex]\( k^2 \geq c \)[/tex]
Here, we need to determine whether the square of [tex]\( k \)[/tex] (which is negative) is greater than or equal to [tex]\( c \)[/tex] (which is a positive integer less than 100).
Given the result, [tex]\( k^2 \geq c \)[/tex] is False.
2. Analyze Statement B: [tex]\( c - k \leq 0 \)[/tex]
We need to determine whether [tex]\( c - k \leq 0 \)[/tex]. Since [tex]\( k \)[/tex] is negative and [tex]\( c \)[/tex] is positive, subtracting [tex]\( k \)[/tex] (a negative number) from [tex]\( c \)[/tex] (a positive number) essentially adds to [tex]\( c \)[/tex]:
[tex]\[
c - k = c + |k| > c
\][/tex]
Given the result, [tex]\( c - k \leq 0 \)[/tex] is False.
3. Analyze Statement C: [tex]\( (k c)^2 \geq \sqrt{c} \)[/tex]
This statement examines whether the square of the product of [tex]\( k \)[/tex] and [tex]\( c \)[/tex] is greater than or equal to the square root of [tex]\( c \)[/tex]:
[tex]\[
(k c)^2 = (k^2 \cdot c^2) \geq \sqrt{c}
\][/tex]
Given the result, [tex]\( (k c)^2 \geq \sqrt{c} \)[/tex] is True.
4. Analyze Statement D: [tex]\( k^2 \leq c^2 \)[/tex]
This statement evaluates if the square of [tex]\( k \)[/tex] is less than or equal to the square of [tex]\( c \)[/tex]:
[tex]\[
k^2 \leq c^2
\][/tex]
Given the result, [tex]\( k^2 \leq c^2 \)[/tex] is True.
Therefore, based on the analysis:
- Statement A is false.
- Statement B is false.
- Statement C is true.
- Statement D is true.
Conclusion:
The statements that must be true are:
- [tex]\( (k c)^2 \geq \sqrt{c} \)[/tex]
- [tex]\( k^2 \leq c^2 \)[/tex]
