Evaluate the mathematical validity of the three theoretical propositions listed below, whi… — State PSC Exam

Evaluate the mathematical validity of the three theoretical propositions listed below, which pertain to the fundamental properties of real numbers:

I. For two distinct irrational numbers $x$ and $y$ , it is mathematically possible for both their sum $(x + y)$ and their product $(x \cdot y)$ to evaluate to rational numbers simultaneously.
II. If $N$ represents any natural number, its principal square root $\sqrt{N}$ must be either a natural number or an irrational number; it can never equate to a non-integer rational fraction.
III. The integer $0$ is formally categorized as a positive even number, and the constant $\pi$ is categorized as a rational number because it represents the exact ratio of a circle's circumference to its diameter.

Which of the given propositions is/are correct?

Only I and IIAnswer
B
Only I and III
C
Only II
D
I, II, and III

Answer

The correct answer is the option stating that only propositions I and II are correct.

Proposition I is mathematically accurate because using conjugate irrational numbers yields rational sums and products. Proposition II is accurate because the square root of a natural number cannot exist as a non-integer fraction; it must either be an integer or strictly irrational. Proposition III contains two definitive errors: zero is neither positive nor negative, and pi is an irrational number because it cannot be formed by the ratio of two integers.

Step-by-Step Solution

Analyze the validity of Proposition I regarding the closure of irrational numbers.

Determined to be True. If

x = 5 + \sqrt{3}

and

y = 5 - \sqrt{3}

(both irrational), their sum is

10

(rational) and their product is

25 - 3 = 22

(rational). Thus, it is mathematically possible.

To test the understanding that the set of irrational numbers is not closed under addition or multiplication.

Analyze the validity of Proposition II regarding the square roots of natural numbers.

Determined to be True. A rational fraction in its simplest form

p/q

(where

q > 1

) squared is

p^2/q^2

. This can never simplify to a whole number

N

. Therefore,

\sqrt{N}

is either a whole integer (if

N

is a perfect square) or strictly irrational.

To verify the structural classification of square roots and rational fractions.

Analyze the validity of Proposition III regarding the classification of zero and pi.

Determined to be False. First, while zero (

0

) is an even integer, it separates positive and negative numbers and is definitively neither positive nor negative. Second, while

\pi = C/d

, a rational number requires the ratio of two integers. In any true circle, circumference and diameter cannot both be integers simultaneously. Thus,

\pi

is irrational.

To identify specific, common misconceptions regarding zero's sign parity and the rational geometric definition of pi.

Key Concept

Classification and structural properties of real numbers, including the irrationality of pi, the neutrality of zero, and arithmetic operations on irrational numbers.

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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