Question

Which of the following statements has the truth value \(T\)?
A. Cube roots of unity are in Geometric Progression and their sum is \(1\) 
B. \[ 4 + 7 > 10 \iff 2 + 8 < 10 \] 
C. \[ \exists x \in \mathbb{N} \text{ such that } x^2 - 3x + 2 = 0 \text{ and } \exists n \in \mathbb{N} \text{ such that } n \text{ is an odd number} \] 
D. \[ 3+i \text{ is a complex number } \; \text{or} \; \sqrt{2}+\sqrt{3}=\sqrt{5} \]

• A. only A
• B. B, C and D
• C. both A and C
• D. both C and D

Hint:

In logic questions, never select an option directly. First find the truth value of each statement one by one.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Evaluate truth values of each logical statement based on mathematical facts.
Step 2: Key Formula or Approach:
A: sum of cube roots \(= 0\). B: \(T \iff F\) is False. C: \(x^2-3x+2=0\) has real natural roots. D: \(P \vee Q\) is true if one is true.
Step 3: Detailed Explanation:
A: Roots sum to \(0\), not \(1\). (F)
B: \(11 \gt 10 \iff 10 \lt 10 \implies T \iff F \equiv F\). (F)
C: Roots are \(1, 2 \in \mathbb{N}\). Odd numbers exist in \(\mathbb{N}\). (T)
D: Complex number statement is true. (T)
Step 4: Final Answer:
Statements C and D are True.