CBSE Class 10 Maths HOTs Real Numbers Set 06

Say True (T) or False (F).

Question. LCM of two or more numbers is always divisible by their H.C.F.
Answer: T

Question. Numbers of the form \( 3m + 1 \) are always even.
Answer: F

Question. Rational number \( \frac{21}{700} \) is not terminating decimal.
Answer: F

Question. \( 7 \times 11 \times 13 + 13 \) is a prime number.
Answer: F

Question. H.C.F. of two prime numbers is always 1.
Answer: T

Question. \( (38)^n \) can end with zero.
Answer: F

Question. If 'a' and 'b' are positive integers \( (a > b) \) such that \( a = b \times q + r \) then \( b < r \le 0 \).
Answer: F

Question. Every real number is always rational.
Answer: F

Question. Product of a non zero rational with an irrational is always irrational number.
Answer: T

Question. Product of 3 consecutive natural numbers is always divisible by 6.
Answer: T

Question. \( \sqrt{12} \times \sqrt{27} \) is an irrational number.
Answer: F

Question. Reciprocal of an irrational number is always irrational number.
Answer: T

Question. There are infinitely many rational numbers between any two irrational numbers.
Answer: T

Question. Product of two prime numbers is always equal to their L.C.M.
Answer: T

Question. There is no irrational number between any two rational numbers.
Answer: F

Question. H.C.F. of two numbers can be 18 if their L.C.M. is 380.
Answer: F

Question. Prime factors of terminating decimals are in the form of \( 2^m \times 5^n \).
Answer: T

Question. If \( \sqrt{ab} \) be an irrational number then \( \sqrt{a} + \sqrt{b} \) is also irrational number.
Answer: T

Question. Decimal representation of irrational numbers is always non terminating repeating.
Answer: F

Question. L.C.M. of \( \frac{2}{7}, \frac{6}{14} \) is \( \frac{6}{7} \).
Answer: T

Fill in the Blank.

Question. ________ is only even prime number.
Answer: 2

Question. \( \frac{\text{H.C.F. of two distinct natural numbers}}{\text{L.C.M. of same distinct natural numbers}} \) is always a ________.
Answer: fractional number

Question. Euclid's division lemma is applicable to calculate only ________.
Answer: H.C.F.

Question. Every composite number can be factorized as a product of primes and this factorization is ________.
Answer: unique

Question. Sum of a rational number with an irrational is always ________.
Answer: irrational

Question. For any two positive integers 'a' and 'b', \( \text{H.C.F. } (a, b) \times \text{L.C.M } (a, b) = \) ________.
Answer: \( a \times b \)

Question. Difference of a/an ________ and a/an ________ is always irrational.
Answer: Rational and irrational

Question. ________ is neither prime nor composite.
Answer: 1

Question. Every rational number and every irrational number is always a ________.
Answer: real number

Question. L.C.M. of \( \frac{1}{2}, \frac{1}{3} \) is ________ and their H.C.F is ________.
Answer: 1 and \( \frac{1}{6} \)

MCQs with more than one correct options.

Question. There is no ________ number between \( 0.\overline{9} \) and 1
(a) Rational
(b) irrational
(c) decimal number
(d) None of the options
Answer: (a) Rational, (b) irrational, (c) decimal number

Question. Number \( 4^n \) can end with
(a) 5
(b) 4
(c) 1
(d) 6
Answer: (b) 4, (d) 6

Question. If \( a = \sqrt{12} \) and \( b = \sqrt{3} \) then \( \frac{a}{b} \) is a/an
(a) rational number
(b) irrational number
(c) Integer
(d) fraction
Answer: (a) rational number, (c) Integer

Question. Square of any positive integer is of the form
(a) \( 3m + 2 \)
(b) \( 3m + 1 \)
(c) \( 3m \)
(d) None
Answer: (b) \( 3m + 1 \), (c) \( 3m \)

Question. \( p^2 - 1 \), is always divisible by (if \( p \) is an odd positive integer)
(a) 4
(b) 5
(c) 6
(d) 8
Answer: (a) 4, (d) 8

Question. Prime factors of denominators of any rational number having terminating decimal can be of the form.
(a) \( 3^m \times 5^m \)
(b) \( 2^m \times 5^n \)
(c) \( 2^m \)
(d) \( 5^m \)
Answer: (b) \( 2^m \times 5^n \), (c) \( 2^m \), (d) \( 5^m \)

Question. Number \( \sqrt{3} + \sqrt{2} \) is not a/an
(a) whole number
(b) irrational number
(c) real number
(d) terminating decimal
Answer: (a) whole number, (d) terminating decimal

Question. Number 2.375 is
(a) rational number
(b) irrational number
(c) terminating decimal
(d) None
Answer: (a) rational number, (c) terminating decimal

Question. If H.C.F. of two numbers is 1 then two numbers can be
(a) Prime
(b) Coprime
(c) Even
(d) None of the options
Answer: (a) Prime, (b) Coprime

Question. L.C.M. of numbers 1, 2, 3 is equal to their
(a) Product
(b) division
(c) sum
(d) None
Answer: (a) Product, (c) sum