Real Numbers (वास्तविक संख्या) MCQ Test — Class 9 Maharashtra SSC

Solution:

Real numbers are composed of both rational and irrational numbers. Therefore, every irrational number is a real number.

Option B is false because rational numbers are also real numbers.

Option C is false because fractions like 1/2 are rational but not integers.

Option D is false because negative integers (e.g., -3) are integers but not whole numbers.

Solution:

A rational number can be expressed as p/q, where p and q are integers and q ≠ 0. Its decimal expansion is either terminating or non-terminating repeating.

The decimal expansion of 7/8 is 0.875, which is terminating. Thus, 7/8 is a rational number. Ravi's conclusion for 7/8 is correct.

The decimal expansion of 1/3 is 0.333..., which is non-terminating repeating. According to the definition, this also represents a rational number. Ravi's conclusion that 1/3 is irrational is incorrect.

Therefore, 7/8 is rational, and 1/3 is also rational.

Solution:

In the right-angled triangle OAB, by Pythagoras theorem, OB² = OA² + AB².

Given OA = 2 units and AB = 1 unit, OB² = 2² + 1² = 4 + 1 = 5. So, OB = √5 units.

To represent √5 on the number line, we need to transfer the length OB from the origin (O) to the number line.

Therefore, the next logical step is to draw an arc with center O (the origin) and radius OB (which is √5) to intersect the number line.

Solution:

A. The sum of a rational number and an irrational number is always irrational. For example, 2 + √3 is irrational. This statement is ALWAYS TRUE.

B. The product x × y is not always rational. If x is a non-zero rational number, x × y is irrational. If x = 0, then 0 × y = 0, which is rational. So, this statement is FALSE.

C. The product y × y (or y²) is not always irrational. For example, √2 × √2 = 2, which is rational. So, this statement is FALSE.

D. The quotient x / y is not always rational. If x is a non-zero rational number, x / y is irrational. If x = 0, then 0 / y = 0, which is rational. So, this statement is FALSE.

Solution:

The property of rational numbers is that they are 'dense' on the number line.

This means that between any two distinct rational numbers, no matter how close they are, there exists another rational number.

This process can be repeated infinitely, leading to infinitely many rational numbers between any two distinct rational numbers.

Solution:

A rational number has a decimal expansion that is either terminating or non-terminating repeating.

The given number, 0.1010010001..., is non-terminating. While it has a 'pattern' (increasing number of zeros), it does not have a repeating block of digits that occurs periodically. For example, '01' does not repeat consistently.

Therefore, it is a non-terminating, non-repeating decimal, which means it is an irrational number.

The error in the student's reasoning is confusing a general 'pattern' with a specific 'repeating block' required for rational numbers.

Solution:

The set of real numbers (ℝ) is defined as the union of the set of rational numbers (ℚ) and the set of irrational numbers (𝕀).

Option A describes rational numbers, but excludes irrational numbers.

Option B is incomplete, as it excludes zero and is not the primary definition.

Option C is the definition of rational numbers, excluding irrational numbers.

Solution:

Rational numbers have decimal expansions that are either terminating or non-terminating repeating.

Integers and whole numbers are subsets of rational numbers.

Numbers whose decimal expansions are non-terminating and non-repeating are, by definition, irrational numbers.

Solution:

The number line is a continuous line where every point corresponds to a unique real number.

Rational numbers, integers, and natural numbers are subsets of real numbers, and while they can be represented, they do not fill the number line completely (there are 'gaps' representing irrational numbers if only considering rational numbers).

Therefore, every point on the number line represents a unique real number.

Solution:

Convert the given options to decimal approximations:

A. √2 ≈ 1.414... (This is less than 1.5)

B. √2.3. To check this, consider 1.5² = 2.25 and 1.6² = 2.56. Since 2.25 < 2.3 < 2.56, it follows that √2.25 < √2.3 < √2.56, which means 1.5 < √2.3 < 1.6. (Specifically, √2.3 ≈ 1.516...)

C. 3/2 = 1.5 (This is not strictly between 1.5 and 1.6)

D. 7/4 = 1.75 (This is greater than 1.6)

Therefore, √2.3 is the number that lies between 1.5 and 1.6.