Assertion Reason Questions For Class 9 Maths Chapter 3 The World Of Numbers

Assertion and reason questions for Class 9 Maths Chapter 3, The World of Numbers, are available in this Maths article. These questions are very helpful for understanding the chapter and solving number-related problems with ease. This article helps students revise important concepts such as integers, rational numbers, irrational numbers, real numbers, decimal expansions, and the number system hierarchy. Our subject experts have provided detailed solutions based on the CBSE syllabus and the NCERT textbook. This study material helps students build conceptual clarity, practise important question types, and prepare well for examinations. A free downloadable PDF is also available for quick revision and practice.

Assertion and Reason Questions on The World of Numbers for Class 9

Directions: In each question below, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:

(a) Both A and R are true, and R is the correct explanation of A.

(b) Both A and R are true, but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

Here’s a more natural, humanised version of your content while keeping the meaning the same:

Q1
Assertion (A): Every integer is a rational number. For example, −7 is a rational number.
Reason (R): Every integer m can be written in the form m/1, where the denominator is 1, a non-zero integer.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Any integer 'm' can be expressed as 'm/1', so every integer fits the definition of a rational number. The reason clearly explains why the assertion is true. Hence, option (a) is correct.

Q2
Assertion (A): √2​ is an irrational number.
Reason (R): A number is irrational if it cannot be written in the form p/q, where p and q are integers and q≠0.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: √2 = 1.41421356… is a non-terminating, non-repeating decimal, so it cannot be expressed as p/q. The reason gives the correct definition of an irrational number and directly explains the assertion. Therefore, option (a) is correct.

Q3
Assertion (A): √5​ is an irrational number.
Reason (R): The square roots of all positive integers are irrational numbers.
Answer: (c) A is true; R is false.
Explanation: The assertion is correct because √5 ≈ 2.2360… is irrational. However, the reason is false because this is too broad a statement. For example, √4 = 2, √9 = 3, √16 = 4 are all rational. Only square roots of non-perfect-square integers are irrational.

Q4
Assertion (A): 0.271 is a terminating decimal and can be written as 271/1000, which is of the form p/q, where q ≠ 0.
Reason (R): A terminating or non-terminating decimal expansion can be expressed as a rational number.
Answer: (c) A is true; R is false.
Explanation: The assertion is true because 0.271=271/1000, which is a valid rational number. But the reason is false because not every non-terminating decimal is rational. Only non-terminating recurring decimals are rational numbers.

Q5
Assertion (A): A rational number between 1/3 and 1/2 is 5/12.
Reason (R): There is exactly one rational number between any two rational numbers.
Answer: (c) A is true; R is false.
Explanation: The assertion is correct because the midpoint of 1/3 and 1/2 is (1/3+1/2)/2=5/12, and this lies between them. But the reason is false because there are infinitely many rational numbers between any two rational numbers.

Q6
Assertion (A): 2 + 6 is an irrational number.
Reason (R): The sum of a rational number and an irrational number is always irrational.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Since 2 is rational and 66​ is irrational, their sum is irrational. The reason states the general rule correctly and explains the assertion. So option (a) is correct.

Q7
Assertion (A): The sum of (3+5) and (4−5​) is an irrational number.
Reason (R): The sum of two irrational numbers is always irrational.
Answer: (d) A is false; R is false.
Explanation: The sum is (3+5)+(4−5)=7, which is rational. So the assertion is false. The reason is also false because two irrational numbers can add up to a rational number. Hence, option (d) is correct.

Q8
Assertion (A): 5−√2 = 5−1.414…= 3.585… is an irrational number.
Reason (R): The difference of a rational number and an irrational number is irrational.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: 5 is rational and √2 is irrational, so their difference is irrational. The reason states the correct general rule and explains the assertion. Therefore, option (a) is correct.

Q9
Assertion (A): 78÷74=74.
Reason (R): For any real number a>0 and rational numbers p, q,  ap÷aq=a(p−q).
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Using the exponent rule, 78÷74=7(8−4)=74. The reason gives the exact rule used here, so it correctly explains the assertion.

Q10
Assertion (A):   113×114=1112
Reason (R): For any real number a>0 and rational numbers p, q,  ap×aq=a(p+q).
Answer: (d) A is false; R is true.
Explanation: Using the rule in the reason,  113×114=11(3+4)=117, not  1112. The mistake in the assertion is that the exponents were added incorrectly. The reason is correct, so option (d) is right.

Q11
Assertion (A): The rationalising factor of  (3 + 2√5) is (3 − 2√5).
Reason (R): If the product of two irrational numbers is rational, each one is called the rationalising factor of the other.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation:  (3 + 2√5)(3 − 2√5) = 9 − 4×5 = 9 − 20 = −11, which is rational. So each one is the rationalising factor of the other. The reason gives the correct definition and explains the assertion well.

Q12
Assertion (A): 2, √3, and √5​ are examples of irrational numbers.
Reason (R): An irrational number can be expressed in the form p/q, where p and q are integers and q≠0.
Answer: (c) A is true; R is false.
Explanation: The assertion is true because all three are irrational numbers. But the reason is false because it says the opposite of the actual definition. An irrational number cannot be written as p/qp/q.

Q13
Assertion (A): If √2 = 1.414 and √3 = 1.732, then √5 = √2 + √3
Reason (R): The square root of a positive real number always exists.
Answer: (d) A is false; R is true.
Explanation: 2 + √3 = 1.414 + 1.732 = 3.146, but √5 = 2.236. So the assertion is false. The reason is true on its own, so option (d) is correct.

Q14
Assertion (A): A rational number lying between 1/4 and 1/2 is 3/8.
Reason (R): A rational number lying between two rational numbers aa and bb can be found using the mean formula (a+b)/2.
Answer: (a) Both A and R are true; R is the correct explanation of A.
Explanation: Using the mean formula, (1/4+1/2)/2=(1/4+2/4)/2=(3/4)/2=3/8. This value lies between 1/4 and 1/2. The reason correctly explains the method used to get the assertion.

Q15
Assertion (A): The product of √2 and √8​ is a rational number.
Reason (R): The product of two irrational numbers is always irrational.
Answer: (c) A is true; R is false.
Explanation: √2 × √8 = √16 = 4, which is rational. So the assertion is true. But the reason is false because the product of two irrational numbers can also be rational. Hence, option (c) is correct.

 

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