Test: Irrational Numbers - Class 9 MCQ

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers. Pi is a non-terminating, non-repeating decimal.

The quotient of a non zero rational number with an irrational number will be irrational.

The real numbers are closed under addition and multiplication, meaning that performing these operations on any two real numbers results in another real number. They also satisfy associative, commutative, and distributive properties for these operations.

A. 'It refers to closure under addition and/or multiplication, which are true.

B. Associative law holds for multiplication, which is true.

C. Distributive law holds, which is also true.

D. The phrase 'closed with respect to commutative law of addition' incorrectly applies the concept of closure to a property (commutativity) rather than an operation. Closure pertains to operations like addition or multiplication, not laws describing their behavior.

Answer. D

Solution. Thus, the false statement is in Option D because it misapplies the term 'closure.'

The expression  involves the square roots of non-perfect squares (12, 10, and 2), which are irrational numbers.

When you add or subtract irrational numbers like these, the result is generally also an irrational number (unless there is some specific cancellation, which is not the case here).

Thus, the correct answer is:

Option 3: an irrational number

(a + b)² = a²  +  b²  + 2ab
= (√5 + √6)² 
= (√5)² + (√6)² + 2(√5)(√6)

= 5 + 6 +2√30

= 11 + 2√30

There are infinitely many irrational numbers between 5/7 and 7/9.
5/7 = 0.71428571428 and 7/9 = 0.77777777777
so we have to find a number between these two numbers
(0.7507500075000.... is also an irrational number between 5/7 and 7/9;but it is not the only one;there are many others)

The irrational number among the options is:

√13 is classified as an irrational number because:

• It cannot be expressed as a fraction of two integers.
• Its decimal representation is non-repeating and non-terminating.

In contrast, the other options (9, 36, and 25) are all rational numbers because:

• They can be expressed as fractions (e.g., 9 = 9/1).
• They have finite decimal representations.

(√5 + √7)² = (√5)² + (√7)² + 2(√5)(√7)

= 5 + 7 + 2√35

= 12 + 2√35

Solution:

• Let the other number be x.


• According to the given information, 4*x = -20/9.


• So, x = -20/9 / 4 = -20/9 * 1/4 = -5/9.


 

2.31312345…as it has a non terminating non repeating decimal form.

A rational number between 2 and 3 =  2 + 3  / 2 = 2.5

Circles are all similar, and "the circumference divided by the diameter" produces the same value regardless of their radius. This value is the ratio of the circumference of a circle to its diameter and is called π (Pi). This constant appears in the calculation of the area of a circle and is a type of an irrational number known as a transcendental number that can be expressed neither by a fraction nor by any radical sign such as a square root, nor their combination. 

i) √23

√23 = √23/1 = p/q, but p is not an integer.

Hence √23 is an irrational number.

ii) √256

√256 = 16/1 = p/q, where p and q are integers and q ≠ 0.

Hence √225 is a rational number.

iii) 0.3796

0.3796 is a rational number because it is a terminating decimal number.

iv) 7.478478...

7.478478... is a rational number as it is a non-terminating recurring decimal i.e, the block of numbers 478 is repeating.

v) 1.101001000100001 . . . .

It is an irrational number because it is a non-terminating and non-recurring decimal.

The number π  is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.

The decimal expansion 0.080080008000080000080000008….. is a. Non-terminating, non-recurring. Non-terminating, recurring.

Infinitely many. You can say 5.1 is a rational number lying between 5 and 6. 5.01, 5.001, 5.0001, 5.00001 and so forth.

∠A = 50° ∠C = 35°
so  ∠A +∠B + ∠C = 180°
      ∠B = 180 - ∠B -∠C
      ∠B = 180 - 50° - 35°
      ∠B = 180 - 85°
      ∠B = 95°
By similarity ∠B = ∠E
so ∠E = 95°

he correct answer is a.

The co prime numbers are 2,3 . Co means near prime means which cannot have a square root.

The real numbers consist of both rational and irrational numbers. so the other three options are not true as the first one is not true because not only every real number is a rational number but the real number is rational and irrational both. same goes for the second one and the last option is not true because one half and the other half can be irrational or rational both.