✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 1: Real Numbers
Assertion-Reason Question Bank
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Assertion (A)
The exponent of \(2\) in the prime factorisation of \(144\) is \(4\).
Reason (R)
Prime factorisation of \(144\) is \(2^4 \times 3^2\).
Correct Option: (A)
\(144 = 16 \times 9 = 2^4 \times 3^2\). The exponent of \(2\) is \(4\). Reason (R) provides the prime factorisation proving Assertion (A).
\(144 = 16 \times 9 = 2^4 \times 3^2\). The exponent of \(2\) is \(4\). Reason (R) provides the prime factorisation proving Assertion (A).
Correct Answer: (A)
\(144 = 2^4 \times 3^2\). Exponent of \(2\) is \(4\). Reason correctly explains Assertion.
\(144 = 2^4 \times 3^2\). Exponent of \(2\) is \(4\). Reason correctly explains Assertion.
Assertion (A)
\(\text{LCM}(a, b) \ge \text{HCF}(a, b)\) for any two positive integers \(a\) and \(b\).
Reason (R)
\(\text{HCF}(a, b)\) divides \(\text{LCM}(a, b)\) completely.
Correct Option: (B)
Both statements are true. \(\text{HCF}\) is a factor of \(\text{LCM}\), hence \(\text{LCM} \ge \text{HCF}\). But R is a property of HCF and LCM, making both true without R fully explaining why LCM is greater than or equal to HCF in magnitude comparison.
Both statements are true. \(\text{HCF}\) is a factor of \(\text{LCM}\), hence \(\text{LCM} \ge \text{HCF}\). But R is a property of HCF and LCM, making both true without R fully explaining why LCM is greater than or equal to HCF in magnitude comparison.
Correct Answer: (B)
Both statements are true, but R is not the direct explanation of A.
Both statements are true, but R is not the direct explanation of A.
Assertion (A)
If \(\text{HCF}(a, 8) = 4\) and \(\text{LCM}(a, 8) = 24\), then \(a = 12\).
Reason (R)
\(a = \frac{\text{HCF} \times \text{LCM}}{b} = \frac{4 \times 24}{8} = 12\).
Correct Option: (A)
Using \(a \times b = \text{HCF} \times \text{LCM}\), \(a \times 8 = 4 \times 24 \implies a = 96/8 = 12\). Reason explains Assertion.
Using \(a \times b = \text{HCF} \times \text{LCM}\), \(a \times 8 = 4 \times 24 \implies a = 96/8 = 12\). Reason explains Assertion.
Correct Answer: (A)
Using \(a \times b = \text{HCF} \times \text{LCM}\), we get \(a = 12\). Reason explains Assertion.
Using \(a \times b = \text{HCF} \times \text{LCM}\), we get \(a = 12\). Reason explains Assertion.
Assertion (A)
For any two positive integers \(a\) and \(b\), \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\).
Reason (R)
For \(a = 6\) and \(b = 20\), \(\text{HCF}(6, 20) = 2\) and \(\text{LCM}(6, 20) = 60\), giving \(2 \times 60 = 120 = 6 \times 20\).
Correct Option: (A)
The formula \(\text{HCF} \times \text{LCM} = a \times b\) holds true for any two positive integers. Reason (R) correctly demonstrates this identity with \(a=6, b=20\).
The formula \(\text{HCF} \times \text{LCM} = a \times b\) holds true for any two positive integers. Reason (R) correctly demonstrates this identity with \(a=6, b=20\).
Correct Answer: (A)
Identity \(\text{HCF} \times \text{LCM} = a \times b\) is verified by Reason.
Identity \(\text{HCF} \times \text{LCM} = a \times b\) is verified by Reason.
Assertion (A)
\(7 \times 11 \times 13 + 13\) is a composite number.
Reason (R)
\(7 \times 11 \times 13 + 13 = 13 \times (77 + 1) = 13 \times 78\), which has factors other than \(1\) and itself.
Correct Option: (A)
Factoring out \(13\) gives \(13 \times 78\), showing it has more than two factors, hence it is composite. Reason explains Assertion.
Factoring out \(13\) gives \(13 \times 78\), showing it has more than two factors, hence it is composite. Reason explains Assertion.
Correct Answer: (A)
Composite numbers have factors other than 1 and itself. Reason shows the factorisation.
Composite numbers have factors other than 1 and itself. Reason shows the factorisation.
Assertion (A)
The total number of prime factors of \(210\) is \(4\).
Reason (R)
Prime factorisation of \(210\) is \(2 \times 3 \times 5 \times 7 \times 11\).
Correct Option: (C)
Assertion (A) is true (\(210 = 2 \times 3 \times 5 \times 7\), total 4 factors). Reason (R) includes an extra \(11\) (\(210 \times 11 = 2310\)), so R is false.
Assertion (A) is true (\(210 = 2 \times 3 \times 5 \times 7\), total 4 factors). Reason (R) includes an extra \(11\) (\(210 \times 11 = 2310\)), so R is false.
Correct Answer: (C)
Assertion true (4 prime factors); Reason false (incorrect prime factorisation).
Assertion true (4 prime factors); Reason false (incorrect prime factorisation).
Assertion (A)
The decimal expansion of an irrational number is non-terminating non-repeating.
Reason (R)
\(\sqrt{5}\) is an irrational number.
Correct Option: (B)
Both A and R are true statements, but R is an example of an irrational number, not the definition of its decimal expansion.
Both A and R are true statements, but R is an example of an irrational number, not the definition of its decimal expansion.
Correct Answer: (B)
Both true, but R is an example rather than an explanation for A.
Both true, but R is an example rather than an explanation for A.
Assertion (A)
\(\text{HCF}(a, b) \times (a \times b) = \text{LCM}(a, b)\).
Reason (R)
\(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\) for any two positive integers \(a, b\).
Correct Option: (D)
Assertion (A) is false (the formula states \(\text{HCF} \times \text{LCM} = a \times b\)). Reason (R) is the true formula.
Assertion (A) is false (the formula states \(\text{HCF} \times \text{LCM} = a \times b\)). Reason (R) is the true formula.
Correct Answer: (D)
Assertion false (formula is incorrect); Reason true.
Assertion false (formula is incorrect); Reason true.
Assertion (A)
The product of three consecutive natural numbers is always divisible by \(24\).
Reason (R)
The product of three consecutive natural numbers is always divisible by \(6\).
Correct Option: (D)
Assertion (A) is false (e.g., \(1 \times 2 \times 3 = 6\), not divisible by \(24\)). Reason (R) is true (\(n(n+1)(n+2)\) is always divisible by \(3! = 6\)).
Assertion (A) is false (e.g., \(1 \times 2 \times 3 = 6\), not divisible by \(24\)). Reason (R) is true (\(n(n+1)(n+2)\) is always divisible by \(3! = 6\)).
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
\(\sqrt{3}\) is an irrational number.
Reason (R)
If \(p\) is a prime number, then \(\sqrt{p}\) is always an irrational number.
Correct Option: (A)
Since \(3\) is a prime number, \(\sqrt{3}\) is irrational according to the theorem that \(\sqrt{p}\) is irrational for any prime \(p\). Reason explains Assertion.
Since \(3\) is a prime number, \(\sqrt{3}\) is irrational according to the theorem that \(\sqrt{p}\) is irrational for any prime \(p\). Reason explains Assertion.
Correct Answer: (A)
Square root of any prime number is irrational. Reason explains Assertion.
Square root of any prime number is irrational. Reason explains Assertion.
Assertion (A)
The HCF of \(a^2 b^3\) and \(a^3 b^2\) is \(a^2 b^2\).
Reason (R)
LCM of \(a^2 b^3\) and \(a^3 b^2\) is \(a^6 b^6\).
Correct Option: (C)
Assertion (A) is true (take minimum powers: \(a^2 b^2\)). Reason (R) is false because LCM takes maximum powers: \(a^3 b^3\), not \(a^6 b^6\).
Assertion (A) is true (take minimum powers: \(a^2 b^2\)). Reason (R) is false because LCM takes maximum powers: \(a^3 b^3\), not \(a^6 b^6\).
Correct Answer: (C)
Assertion true; Reason false (LCM is \(a^3 b^3\)).
Assertion true; Reason false (LCM is \(a^3 b^3\)).
Assertion (A)
The HCF of any two consecutive even numbers is \(1\).
Reason (R)
Two consecutive even numbers can be written as \(2n\) and \(2n+2\), both having common factor \(2\).
Correct Option: (D)
Assertion (A) is false (HCF of consecutive even numbers is \(2\), e.g., \(\text{HCF}(4,6)=2\)). Reason (R) is true.
Assertion (A) is false (HCF of consecutive even numbers is \(2\), e.g., \(\text{HCF}(4,6)=2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false (HCF is 2); Reason true.
Assertion false (HCF is 2); Reason true.
Assertion (A)
\(\sqrt{2}\) is an irrational number.
Reason (R)
The HCF of any two distinct prime numbers is always \(1\).
Correct Option: (B)
Assertion (A) is true (\(\sqrt{2}\) is irrational). Reason (R) is also true (HCF of prime numbers is \(1\)). However, R does not explain why \(\sqrt{2}\) is irrational.
Assertion (A) is true (\(\sqrt{2}\) is irrational). Reason (R) is also true (HCF of prime numbers is \(1\)). However, R does not explain why \(\sqrt{2}\) is irrational.
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
The sum of two irrational numbers is always an irrational number.
Reason (R)
\((2 + \sqrt{3}) + (2 - \sqrt{3}) = 4\), which is a rational number.
Correct Option: (D)
Assertion (A) is false (sum can be rational). Reason (R) gives a correct counter-example proving it rational.
Assertion (A) is false (sum can be rational). Reason (R) gives a correct counter-example proving it rational.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
For any positive integer \(n\), \(n^2 - n\) is always an odd number.
Reason (R)
\(n^2 - n = n(n-1)\), which is the product of two consecutive integers and thus always even.
Correct Option: (D)
Assertion (A) is false (\(n^2-n\) is always even). Reason (R) is true.
Assertion (A) is false (\(n^2-n\) is always even). Reason (R) is true.
Correct Answer: (D)
Assertion false (it is always even); Reason true.
Assertion false (it is always even); Reason true.
Assertion (A)
\(2 + \sqrt{3}\) is an irrational number.
Reason (R)
The product of a non-zero rational number and an irrational number is always a rational number.
Correct Option: (C)
Assertion (A) is true (rational + irrational = irrational). Reason (R) claims product of rational and irrational is rational, which is false (it is irrational).
Assertion (A) is true (rational + irrational = irrational). Reason (R) claims product of rational and irrational is rational, which is false (it is irrational).
Correct Answer: (C)
Assertion true; Reason false (product of rational and irrational is irrational).
Assertion true; Reason false (product of rational and irrational is irrational).
Assertion (A)
If \(n\) is any natural number, \(6^n - 5^n\) always ends with \(1\).
Reason (R)
\(6^n\) always ends with \(6\) and \(5^n\) always ends with \(0\).
Correct Option: (C)
Assertion (A) is true (\(6^n\) ends in \(6\), \(5^n\) ends in \(5\), \(6 - 5 = 1\)). Reason (R) is false because \(5^n\) ends in \(5\), not \(0\).
Assertion (A) is true (\(6^n\) ends in \(6\), \(5^n\) ends in \(5\), \(6 - 5 = 1\)). Reason (R) is false because \(5^n\) ends in \(5\), not \(0\).
Correct Answer: (C)
Assertion true; Reason false (\(5^n\) ends in 5).
Assertion true; Reason false (\(5^n\) ends in 5).
Assertion (A)
If \(p\) is prime, \(\sqrt{p}\) cannot be written in \(m/n\) form where \(m, n \in \mathbb{Z}, n \ne 0\).
Reason (R)
The square root of a non-perfect square natural number is always irrational.
Correct Option: (B)
Both A and R are true mathematical facts regarding irrationality of roots, but R is a broader statement.
Both A and R are true mathematical facts regarding irrationality of roots, but R is a broader statement.
Correct Answer: (B)
Both true, but R is not the specific explanation of A.
Both true, but R is not the specific explanation of A.
Assertion (A)
The LCM of \(15\) and \(25\) is \(75\).
Reason (R)
Every composite number can be uniquely expressed as a product of prime numbers.
Correct Option: (B)
Assertion (A) is true (\(15 = 3 \times 5, 25 = 5^2 \implies \text{LCM} = 3 \times 5^2 = 75\)). Reason (R) is the Fundamental Theorem of Arithmetic (true), but it does not directly explain the specific calculation for \(15\) and \(25\).
Assertion (A) is true (\(15 = 3 \times 5, 25 = 5^2 \implies \text{LCM} = 3 \times 5^2 = 75\)). Reason (R) is the Fundamental Theorem of Arithmetic (true), but it does not directly explain the specific calculation for \(15\) and \(25\).
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
\(3 \sqrt{2}\) is an irrational number.
Reason (R)
The square of an irrational number is always a rational number.
Correct Option: (C)
Assertion (A) is true (\(3\sqrt{2}\) is irrational). Reason (R) is false because the square of an irrational like \((\sqrt[4]{2})^2 = \sqrt{2}\) is still irrational.
Assertion (A) is true (\(3\sqrt{2}\) is irrational). Reason (R) is false because the square of an irrational like \((\sqrt[4]{2})^2 = \sqrt{2}\) is still irrational.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The number \(3^n\) can never end with the digit \(0\) for any natural number \(n\).
Reason (R)
For a number to end with digit \(0\), its prime factorisation must contain prime factor \(3\).
Correct Option: (C)
Assertion (A) is true (\(3^n\) lacks prime factor \(2\) and \(5\)). Reason (R) is false because a number ending in \(0\) must contain factors \(2\) and \(5\), not \(3\).
Assertion (A) is true (\(3^n\) lacks prime factor \(2\) and \(5\)). Reason (R) is false because a number ending in \(0\) must contain factors \(2\) and \(5\), not \(3\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The number \(\frac{23}{2^3 \times 5^2}\) is a rational number.
Reason (R)
All rational numbers have terminating decimal expansions.
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because many rational numbers (like \(1/3\)) have non-terminating repeating decimal expansions.
Assertion (A) is true. Reason (R) is false because many rational numbers (like \(1/3\)) have non-terminating repeating decimal expansions.
Correct Answer: (C)
Assertion true; Reason false (not all rationals have terminating expansions).
Assertion true; Reason false (not all rationals have terminating expansions).
Assertion (A)
\(\sqrt{4}\) is an irrational number.
Reason (R)
\(\sqrt{4} = 2\), which is a rational number.
Correct Option: (D)
Assertion (A) is false (\(\sqrt{4}=2\) is rational). Reason (R) is true.
Assertion (A) is false (\(\sqrt{4}=2\) is rational). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
\(\text{HCF}(p, q) = 1\) if \(p\) and \(q\) are co-prime numbers.
Reason (R)
\(17\) and \(23\) are prime numbers.
Correct Option: (B)
Assertion (A) is true by definition of co-primes. Reason (R) is true (\(17\) and \(23\) are primes), but it is just a specific pair, not the general explanation of co-prime definition.
Assertion (A) is true by definition of co-primes. Reason (R) is true (\(17\) and \(23\) are primes), but it is just a specific pair, not the general explanation of co-prime definition.
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
\(\pi\) is a rational number because \(\pi = 22/7\).
Reason (R)
\(\pi\) is an irrational number, and \(22/7\) is an approximate rational value.
Correct Option: (D)
Assertion (A) is false (\(\pi\) is irrational). Reason (R) is true (\(22/7\) is a rational approximation).
Assertion (A) is false (\(\pi\) is irrational). Reason (R) is true (\(22/7\) is a rational approximation).
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The HCF of \(12\) and \(18\) is \(6\).
Reason (R)
HCF is the product of the smallest power of each common prime factor (\(12 = 2^2 \times 3^1, 18 = 2^1 \times 3^2 \implies \text{HCF} = 2^1 \times 3^1 = 6\)).
Correct Option: (A)
\(12 = 2^2 \times 3^1\) and \(18 = 2^1 \times 3^2\). Taking the minimum power of common prime factors \(2\) and \(3\) gives \(2^1 \times 3^1 = 6\). Reason (R) gives the exact rule and calculation for Assertion (A).
\(12 = 2^2 \times 3^1\) and \(18 = 2^1 \times 3^2\). Taking the minimum power of common prime factors \(2\) and \(3\) gives \(2^1 \times 3^1 = 6\). Reason (R) gives the exact rule and calculation for Assertion (A).
Correct Answer: (A)
\(\text{HCF} = 6\). Reason explains Assertion.
\(\text{HCF} = 6\). Reason explains Assertion.
Assertion (A)
The number \(0.1010010001\dots\) is an irrational number.
Reason (R)
Rational numbers have decimal expansions that are either terminating or non-terminating repeating.
Correct Option: (B)
Assertion (A) is true (the decimal is non-terminating non-repeating, so irrational). Reason (R) is the definition of rational numbers (true). R supports classification by contrast.
Assertion (A) is true (the decimal is non-terminating non-repeating, so irrational). Reason (R) is the definition of rational numbers (true). R supports classification by contrast.
Correct Answer: (B)
Both true, but R defines rational numbers rather than directly explaining the irrationality of the given number.
Both true, but R defines rational numbers rather than directly explaining the irrationality of the given number.
Assertion (A)
The product of two consecutive positive integers is always divisible by \(2\).
Reason (R)
Out of two consecutive positive integers, exactly one is even.
Correct Option: (B)
Both statements are true. Out of \(n\) and \(n+1\), one is even, so their product is even and divisible by \(2\). R is a true property of integers.
Both statements are true. Out of \(n\) and \(n+1\), one is even, so their product is even and divisible by \(2\). R is a true property of integers.
Correct Answer: (B)
Both true, R gives property of consecutive integers.
Both true, R gives property of consecutive integers.
Assertion (A)
The HCF of two numbers is always greater than their LCM.
Reason (R)
HCF divides LCM completely without leaving any remainder for any two positive integers.
Correct Option: (D)
Assertion (A) is false because \(\text{HCF} \le \text{LCM}\). Reason (R) is true because HCF is a factor of LCM.
Assertion (A) is false because \(\text{HCF} \le \text{LCM}\). Reason (R) is true because HCF is a factor of LCM.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The number \(5^n\) cannot end with the digit \(0\) for any natural number \(n\).
Reason (R)
For a number to end with \(0\), its prime factorisation must contain both prime factors \(2\) and \(5\), whereas \(5^n\) contains only \(5\).
Correct Option: (A)
Prime factorisation of \(5^n\) contains only factor \(5\). By Fundamental Theorem of Arithmetic, prime factorisation is unique, so \(2\) can never occur in \(5^n\). Thus it cannot end in \(0\). Reason explains Assertion.
Prime factorisation of \(5^n\) contains only factor \(5\). By Fundamental Theorem of Arithmetic, prime factorisation is unique, so \(2\) can never occur in \(5^n\). Thus it cannot end in \(0\). Reason explains Assertion.
Correct Answer: (A)
Reason correctly explains why \(5^n\) cannot end in 0.
Reason correctly explains why \(5^n\) cannot end in 0.