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

Correct Answer: \( \boxed{\text{Both C and D}} \)

Solution:  

A. Cube roots of unity are: \[ 1,\ \omega,\ \omega^2 \] where \[ \omega = \frac{-1+\sqrt{3}i}{2}, \quad \omega^2 = \frac{-1-\sqrt{3}i}{2} \] Sum of cube roots of unity: \[ 1+\omega+\omega^2 = 0 \neq 1 \] Therefore, statement A is false. 

B. Let \[ p : 4+7>10 \] Since, \[ 11>10 \] So, \(p\) is true. 

Let \[ q : 2+8<10 \] Since, \[ 10<10 \] which is false. Therefore, \[ p \leftrightarrow q = T \leftrightarrow F = F \] Hence, statement B is false. 

C. Check: \[ \exists x\in \mathbb{N} \text{ such that } x^2-3x+2=0 \] Factorizing: \[ (x-1)(x-2)=0 \] Thus, \[ x=1 \text{ or } x=2 \] Both belong to \( \mathbb{N} \), so first statement is true. 

Also, \[ \exists n\in\mathbb{N} \text{ such that } n \text{ is odd} \] Example: \[ n=1 \] which is true. Therefore, \[ T \land T = T \] Hence, statement C is true. 

D. Consider: \[ 3+i \] It is clearly a complex number, so this statement is true. 

Therefore, \[ T \lor F = T \] Hence, statement D is true. 

Therefore, statements having truth value \(T\) are: \[ \boxed{\text{C and D}} \]