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Chapter 1: Real Numbers

Overview

This page provides comprehensive Chapter 1: Real Numbers - Assertion Reason Worksheet - SJMaths. Assertion and Reason type questions for Class 10 Real Numbers. Practice for CBSE Board Exams.

Assertion-Reason Worksheet

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Directions:

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:

  • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
  • (C) Assertion (A) is true but Reason (R) is false.
  • (D) Assertion (A) is false but Reason (R) is true.
  1. Question 1:
    Assertion (A): If HCF$(a, b) = 5$ and $a \times b = 150$, then LCM$(a, b) = 30$.
    Reason (R): For any two positive integers $a$ and $b$, HCF$(a, b) \times$ LCM$(a, b) = a \times b$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Both statements are true. The formula in R is used to calculate the LCM in A ($150/5 = 30$). Thus R is the correct explanation.
  2. Question 2:
    Assertion (A): The number $6^n$ ends with the digit 0 for any natural number $n$.
    Reason (R): The prime factorization of $6^n$ is $2^n \times 3^n$, which does not contain the prime 5.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Assertion is false because for a number to end with 0, it must have 2 and 5 as factors. $6^n$ only has 2 and 3. Reason is true.
  3. Question 3:
    Assertion (A): $\sqrt{3}$ is an irrational number.
    Reason (R): The square root of any prime number is an irrational number.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    3 is a prime number, so $\sqrt{3}$ is irrational. R correctly explains A.
  4. Question 4:
    Assertion (A): The product of two consecutive positive integers is divisible by 2.
    Reason (R): For any two consecutive integers $n, n+1$, one is always even.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Since one of the two consecutive integers is even, their product must be even (divisible by 2). R is the correct explanation for A.
  5. Question 5:
    Assertion (A): $3\sqrt{2}$ is an irrational number.
    Reason (R): The product of a non-zero rational number and an irrational number is always irrational.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    3 is rational and $\sqrt{2}$ is irrational. Their product is irrational. R explains A.
  6. Question 6:
    Assertion (A): HCF of 10 and 15 is 5.
    Reason (R): LCM of 10 and 15 is 30.
    (A)
    (B)
    (C)
    (D)
    Solution: (B)
    Both statements are true independently. The LCM being 30 is not the reason for HCF being 5.
  7. Question 7:
    Assertion (A): The number $5 \times 7 \times 11 + 7$ is a composite number.
    Reason (R): A composite number has factors other than 1 and itself.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    $5 \times 7 \times 11 + 7 = 7(5 \times 11 + 1) = 7(56)$. Since it has factors 7 and 56, it is composite. R explains A.
  8. Question 8:
    Assertion (A): $\sqrt{2} + \sqrt{3}$ is an irrational number.
    Reason (R): The sum of two irrational numbers is always irrational.
    (A)
    (B)
    (C)
    (D)
    Solution: (C)
    Assertion is true. Reason is false because sum of two irrationals can be rational (e.g., $(2+\sqrt{3}) + (2-\sqrt{3}) = 4$).
  9. Question 9:
    Assertion (A): 2 is a rational number.
    Reason (R): It can be expressed in the form $p/q$ where $p, q$ are integers and $q \neq 0$ (e.g., $2/1$).
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Definition of rational number. R explains A.
  10. Question 10:
    Assertion (A): The number $4^n$ ends with the digit 0 for some natural number $n$.
    Reason (R): The prime factorization of $4^n$ is $2^{2n}$, which contains only the prime 2.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Assertion is false (needs factor 5 to end in 0). Reason is true.
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