Topic Question Set
Q 1
:
Assertion (A): The HCF of two numbers is 16 and their product is 3072. Then their LCM = 162.
Reason (R): If a and b are two positive integers, then their HCF × LCM = a × b.
Reason (R): If a and b are two positive integers, then their HCF × LCM = a × b.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
(4)
As we know that HCF(a, b) × LCM(a, b) = a × b.
But 16 × 162 = 2592, i.e., 2592 ≠ 3072
Q 2
:
Assertion (A): If LCM(p, q) = 30 and HCF(p, q) = 5, then p × q = 150.
Reason (R): LCM of (a, b) × HCF of (a, b) = a × b.
Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true
(1)
We know that LCM(p, q) × HCF(p, q) = p × q.
⇒ 30 × 5 = p × q ⇒ 150 = p × q.
Q 3
:
Assertion (A): HCF(306, 657) = 9, and LCM(306, 657) = 22338.
Reason (R): If a and b are two positive integers and HCF(a, b) = 9, then LCM(a, b) = (a + b)/2.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.
(3)
We have, HCF(306, 657) = 9
⇒ LCM(306, 657) = (306 × 657) / 9 = 22338.
Q 4
:
Assertion (A): 6? can end with the digit 0 for any natural number n.
Reason (R): Any positive integer ending with the digits 0 or 5 is divisible by 5 and so its prime factorisation must contain the prime 5.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.
(4)
As we know that any number whose prime factors consist of 2? × 5?, where m and n are natural numbers, can end with the digit 0, but 6? has only prime factors as 2 and 3. So 6? cannot end with the digit 0.
Q 5
:
Assertion (A): The HCF of two numbers is 4 and their product is 38784. Then their LCM is 9696.
Reason (R): LCM(a, b) × HCF(a, b) = a × b.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false
A is false, but R is true
(1)
For Assertion (A): Given, HCF of two numbers = 4 and product of these numbers = 38784.
We know that HCF × LCM = Product of numbers.
⇒ LCM = 38784 ÷ 4 = 9696.
Q 6
:
Assertion (A): 2 is a rational number.
Reason (R): The square root of all positive integers are irrationals.
Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true
(3)
Clearly, 2 is a rational number but square root of all positive integers are not always irrational.
It may be rational or irrational number.
For example, √2 is an irrational number but √4 is not an irrational number because √4 = 2 which is rational.
So, Assertion (A) is true but Reason (R) is false
Q 7
:
Assertion (A): √5 is an irrational number.
Reason (R): If m is an odd number greater than 1, then √m is irrational.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true
(3)
Clearly, √5 is irrational so Assertion (A) is true.
Let m = 9 ⇒ √m = √9 = 3 which is rational.
So, for m is an odd number greater than 1, √m need not be irrational always.
∴ Assertion (A) is true but Reason (R) is false.
Q 8
:
Assertion (A): 15/1600 is a terminating decimal fraction.
Reason (R): If x = p/q is a rational number, such that the prime factorisation of q is of the form
where n, m are non-negative integers, then x = p/q is a terminating decimal fraction.
where n, m are non-negative integers, then x = p/q is a terminating decimal fraction.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true
(1)
For Assertion (A)
We have, 15/1600 = 15 / (2 × 2 × 2 × 2 × 2 × 5 × 5) = 15 / (2? × 5²)
⇒ The prime factorisation of denominator 1600 of 15/1600 is of the form 2?5?, where n, m are non-negative integers.
So, 15/1600 is a terminating decimal fraction.
Hence, Assertion (A) is true.
Also Reason (R) is true because it is the statement of a known theorem which says
“If x = p/q is a rational number, such that the prime factorisation of q is of the form 2?5?, where n, m are non-negative integers,
then x = p/q is a terminating decimal fraction.”
Also, it helps to prove Assertion (A) and so Reason (R) is the correct explanation of Assertion (A).
Q 9
:
Assertion (A): (3 + √5) is an irrational number.
Reason (R): The sum or difference of a rational and irrational number is an irrational.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false
A is false, but R is true
(1)
In (3 + √5), obviously 3 is rational number and √5 is an irrational number.
Also we know that sum or difference of a rational and irrational number is irrational number.
⇒ (3 + √5) is irrational number.
Hence, assertion (A) is true.
Also Reason (R) which says “The sum or difference of a rational and irrational number is irrational” is true.
Also, it helps to prove Assertion (A) and so Reason (R) is the correct explanation of Assertion (A).
