If d > 0 and 0 , which of the following must be true? (c > 0 c/d (c^2 +...

1. Translate the problem requirements: We need to understand what the compound inequality \(0 < 1 - \mathrm{c}/\mathrm{d} < 1\) tells us about the relationship between c and d, then determine which of the three statements (I, II, III) must always be true given our constraints.
2. Break down the compound inequality: Split the inequality into two separate conditions and analyze what each part tells us about c and d individually.
3. Test each statement systematically: For each of the three given statements, determine whether it must be true, could be false, or is indeterminate based on our constraints.
4. Verify with strategic examples: Use concrete values that satisfy our constraints to confirm our reasoning and eliminate any statements that can be false.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given in plain English. We know that d is positive (\(\mathrm{d} > 0\)), and we have a compound inequality: \(0 < 1 - \mathrm{c}/\mathrm{d} < 1\).

Think of this as saying: "The expression \((1 - \mathrm{c}/\mathrm{d})\) is somewhere between 0 and 1." This means \((1 - \mathrm{c}/\mathrm{d})\) is a positive fraction less than 1.

Our job is to figure out which of the three statements (I, II, III) must ALWAYS be true whenever these conditions are met. Remember, "must be true" means it can never be false under our given constraints.

Process Skill: TRANSLATE - Converting the inequality language into clear mathematical understanding

2. Break down the compound inequality

Let's split our compound inequality \(0 < 1 - \mathrm{c}/\mathrm{d} < 1\) into two separate pieces and see what each tells us:

From the left side: \(0 < 1 - \mathrm{c}/\mathrm{d}\)

In everyday terms: "1 minus c/d is positive." This means \(1 > \mathrm{c}/\mathrm{d}\), or equivalently, \(\mathrm{c}/\mathrm{d} < 1\).

Mathematically: \(0 < 1 - \mathrm{c}/\mathrm{d} \rightarrow 1 > \mathrm{c}/\mathrm{d} \rightarrow \mathrm{c}/\mathrm{d} < 1\)

From the right side: \(1 - \mathrm{c}/\mathrm{d} < 1\)

In everyday terms: "1 minus c/d is less than 1." This means \(-\mathrm{c}/\mathrm{d} < 0\), or equivalently, \(\mathrm{c}/\mathrm{d} > 0\).

Since we know \(\mathrm{d} > 0\), this tells us that \(\mathrm{c} > 0\).

Mathematically: \(1 - \mathrm{c}/\mathrm{d} < 1 \rightarrow -\mathrm{c}/\mathrm{d} < 0 \rightarrow \mathrm{c}/\mathrm{d} > 0 \rightarrow \mathrm{c} > 0\) (since \(\mathrm{d} > 0\))

So our constraints give us: \(\mathrm{c} > 0\) and \(0 < \mathrm{c}/\mathrm{d} < 1\)

Process Skill: INFER - Drawing non-obvious conclusions from the compound inequality

3. Test each statement systematically

Now let's check each statement to see if it must always be true:

Statement I: \(\mathrm{c} > 0\)

From our analysis above, we discovered that \(\mathrm{c} > 0\) must be true. When \(1 - \mathrm{c}/\mathrm{d} < 1\), we get \(\mathrm{c}/\mathrm{d} > 0\), and since \(\mathrm{d} > 0\), this forces \(\mathrm{c} > 0\). ✓ MUST BE TRUE

Statement II: \(\mathrm{c}/\mathrm{d} < 1\)

From our analysis above, we discovered that \(\mathrm{c}/\mathrm{d} < 1\) must be true. When \(0 < 1 - \mathrm{c}/\mathrm{d}\), we get \(\mathrm{c}/\mathrm{d} < 1\). ✓ MUST BE TRUE

Statement III: \(\mathrm{c}^2 + \mathrm{d}^2 > 1\)

This statement is about the magnitude of c and d. Let's see if we can find values that satisfy our constraints but make this false.

We need: \(\mathrm{d} > 0\), \(\mathrm{c} > 0\), and \(0 < \mathrm{c}/\mathrm{d} < 1\)

Let's try small positive values: \(\mathrm{c} = 0.1\) and \(\mathrm{d} = 0.2\)

Check our constraints: \(\mathrm{d} = 0.2 > 0\) ✓, \(\mathrm{c}/\mathrm{d} = 0.1/0.2 = 0.5\), so \(0 < 0.5 < 1\) ✓

Check statement III: \(\mathrm{c}^2 + \mathrm{d}^2 = (0.1)^2 + (0.2)^2 = 0.01 + 0.04 = 0.05 < 1\)

So statement III can be false! ✗ NOT ALWAYS TRUE

Process Skill: APPLY CONSTRAINTS - Using specific examples to test whether statements must be true

4. Verify with strategic examples

Let's double-check our work with another example to make sure our reasoning is solid:

Example: \(\mathrm{c} = 0.3\), \(\mathrm{d} = 0.4\)

Check constraints:

• \(\mathrm{d} = 0.4 > 0\) ✓

• \(\mathrm{c}/\mathrm{d} = 0.3/0.4 = 0.75\), so \(1 - \mathrm{c}/\mathrm{d} = 1 - 0.75 = 0.25\)

• Indeed, \(0 < 0.25 < 1\) ✓

Check statements:

• I: \(\mathrm{c} = 0.3 > 0\) ✓

• II: \(\mathrm{c}/\mathrm{d} = 0.75 < 1\) ✓

• III: \(\mathrm{c}^2 + \mathrm{d}^2 = (0.3)^2 + (0.4)^2 = 0.09 + 0.16 = 0.25 < 1\) ✗

This confirms our analysis. Statements I and II must always be true, but statement III can be false.

5. Final Answer

Based on our systematic analysis:

• Statement I (\(\mathrm{c} > 0\)) must be true

• Statement II (\(\mathrm{c}/\mathrm{d} < 1\)) must be true

• Statement III (\(\mathrm{c}^2 + \mathrm{d}^2 > 1\)) is not always true

Therefore, only statements I and II must be true.

The answer is C) I and II only

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the compound inequality structure

Students often struggle with compound inequalities like \(0 < 1 - \mathrm{c}/\mathrm{d} < 1\) and may try to solve it as a single inequality rather than breaking it into two separate conditions. They might attempt to manipulate the entire expression at once instead of systematically analyzing what each part (\(0 < 1 - \mathrm{c}/\mathrm{d}\) and \(1 - \mathrm{c}/\mathrm{d} < 1\)) tells us independently.

Faltering Point 2: Confusing "must be true" with "could be true"

Students frequently misunderstand that they need to find statements that are ALWAYS true under the given constraints, not just statements that are sometimes true. This leads them to incorrectly validate statements that work for some examples but fail for others, rather than proving the statements hold universally.

Faltering Point 3: Overlooking the significance of the constraint \(\mathrm{d} > 0\)

Students may not fully appreciate how the condition \(\mathrm{d} > 0\) is crucial for making inferences about the sign of c. They might try to analyze c/d ratios without properly considering that since d is positive, the sign of c/d directly reflects the sign of c.

Errors while executing the approach

Faltering Point 1: Algebraic manipulation errors when isolating variables

When working with inequalities like \(1 - \mathrm{c}/\mathrm{d} < 1\), students often make sign errors or forget to flip inequality signs appropriately. For example, they might incorrectly go from \(1 - \mathrm{c}/\mathrm{d} < 1\) to \(\mathrm{c}/\mathrm{d} < 0\) instead of the correct \(\mathrm{c}/\mathrm{d} > 0\), especially when dealing with negative terms.

Faltering Point 2: Choosing inappropriate counterexamples

Students may select test values that don't actually satisfy the given constraints, leading to invalid conclusions. For instance, they might choose \(\mathrm{c} = 0.5\), \(\mathrm{d} = 0.2\) (which gives \(\mathrm{c}/\mathrm{d} = 2.5 > 1\), violating our constraint) and then base their conclusions on this invalid example.

Faltering Point 3: Incomplete verification of constraints

When testing specific examples, students often check only one part of the compound inequality or forget to verify that \(\mathrm{d} > 0\), leading to conclusions based on examples that don't actually meet all the given conditions.

Errors while selecting the answer

Faltering Point 1: Misreading Roman numeral answer choices

Students frequently misinterpret which combination of statements corresponds to which answer choice. They might correctly determine that statements I and II are true but accidentally select "A) I only" or "B) II only" instead of "C) I and II only" due to careless reading of the options.

Faltering Point 2: Including statements that are "usually true" rather than "must be true"

Students may include Statement III in their final answer because they tested a few examples where \(\mathrm{c}^2 + \mathrm{d}^2 > 1\) happened to be true, without finding a proper counterexample that shows it can be false. They mistake "frequently true" for "always true."