CBSE Class 10 Mathematics Real Numbers Worksheet Set 02

Question. If two positive integers a and b are written as a = x³y² and b = xy³, where x and y are prime numbers, then the HCF (a, b) is:
(a) xy
(b) xy²
(c) x³y³
(d) x²y²

Answer: B

Question. Find the greatest number of 5 digits, that will give us remainder of 5, when divided by 8 and 9 respectively.
(a) 99921
(b) 99931
(c) 99941
(d) 99951

Answer: C

Question. The ratio between the LCM and HCF of 5, 15, 20 is:
(a) 9 : 1
(b) 4 : 3
(c) 11 : 1
(d) 12 : 1

Answer: D

Question. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again for the first time?
(a) 12.20 pm
(b) 12.12 pm
(c) 12.11 pm
(d) none of these

Answer: D

Question. The HCF of 2472, 1284 and a third number N is 12. If their LCM is 2³ × 3² × 5 × 103 × 107, then the number N is :
(a) 2² × 3² × 7
(b) 2² × 3³ × 103
(c) 2² × 3² × 5
(d) 2⁴ × 3² × 11

Answer: C

Question. Two natural numbers whose difference is 66 and the least common multiple is 360, are:
(a) 120 and 54
(b) 90 and 24
(c) 180 and 114
(d) 130 and 64

Answer: B

Question. HCF of 5² × 3² and 3⁵ × 5³ is:
(a) 5³ × 3⁵
(b) 5 × 3³
(c) 5³ × 3²
(d) 5² × 3²

Answer: D

Question. The HCF and the LCM of 12, 21, 15 respectively are
(a) 3, 140
(b) 12, 420
(c) 3, 420
(d) 420, 3

Answer: C

Question. In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
Assertion (A): For no value of n, where n is a natural number, the number 6n ends with the digit zero.
Reason (R): For a number to end with digit zero, its prime factors should have 2 and 5.
(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.

Answer: A

Question. In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
Assertion (A): If LCM of two numbers is 2475 and their product is 12375, then their HCF is 5.
Reason (R): HCF (a, b) × LCM (a, b) = a × b.
(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.

Answer: A

FILL IN THE BLANK

Question. H.C.F. of 6, 72 and 120 is ..........
Answer: 6

Question. 156 as a product of its prime factors ..........
Answer: \( 2^2 \times 3 \times 13 \)

Question. If \( a = bq + r \), least value of \( r \) is ..........
Answer: Zero

Question. If every positive even integer is of the form \( 2q \), then every positive odd integer is of the form .........., where \( q \) is some integer.
Answer: \( 2q + 1 \)

Question. The exponent of 2 in the prime factorisation of 144, is ..........
Answer: 4

Question. \( \sqrt{2}, \sqrt{3}, \sqrt{7} \), etc. are .......... numbers.
Answer: Irrational

Question. \( 7\sqrt{5} \) is a/an .......... number.
Answer: irrational

Question. An algorithm which is used to find HCF of two positive numbers is ..........
Answer: Euclid’s division algorithm

Question. \( 6 + \sqrt{2} \) is a/an .......... number.
Answer: irrational

Question. Every point on the number line corresponds to a .......... number.
Answer: Real

Question. A .......... is a proven statement used for proving another statement.
Answer: lemma

Question. The product of three numbers is .......... to the product of their HCF and LCM.
Answer: Not equal

Question. L.C.M. of 96 and 404 is ..........
Answer: 9696

Question. If \( p \) is a prime number and it divides \( a^2 \) then it also divides .........., where \( a \) is a positive integer.
Answer: \( a \)

Question. \( \frac{35}{50} \) is a .......... decimal expansion.
Answer: terminating

Question. An .......... is a series of well defined steps which gives a procedure for solving a type of problem.
Answer: algorithm

Question. Every real number is either a .......... number or an .......... number.
Answer: Rational, irrational

Question. Euclid’s Division Lemma is a restatement of ..........
Answer: Long division process

Question. \( \frac{1}{\sqrt{2}} \) is a/an .......... number.
Answer: irrational

Question. Numbers having non-terminating, non-repeating decimal expansion are known as ..........
Answer: Irrational numbers

Question. \( \sqrt{5} \) is a/an .......... number.
Answer: irrational

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true but R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : \( 6^n \) ends with the digit zero, where \( n \) is natural number.
Reason (R) : Any number ends with digit zero, if its prime factor is of the form \( 2^m \times 5^n \), where \( m, n \) are natural numbers.
Answer: Solution : \( 6^n = (2 \times 3)^n = 2^n \times 3^n \), Its prime factors do not contain \( 5^n \) i.e., it is not of the form of the form \( 2^m \times 5^n \), where \( m, n \) are natural numbers. Hence, cannot end with 0.
Therefore assertion is false but reason is true.
Hence, option (d) is correct.

Question. Assertion (A) : \( \sqrt{a} \) is an irrational number, where \( a \) is a prime number.
Reason (R) : Square root of any prime number is an irrational number.
Answer: Solution : Square root of every prime number is an irrational number. So, both A and R are true and R is the correct explanation of A.
Hence, option (a) is correct.

Question. Assertion (A) : For any two positive integers \( a \) and \( b \), \( \text{HCF } (a, b) \times \text{LCM } (a, b) = a \times b \)
Reason (R) : The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.
Answer: Solution : We have,
\( \text{LCM } (a, b) \times \text{HCF } (a, b) = a \times b \)
\( \text{LCM } \times 5 = 150 \)
\( \therefore \text{LCM } = \frac{150}{5} = 30 \)

\( \implies \text{LCM } = 30 \)
Therefore, A is true but R is false
Hence, option (c) is correct.

Question. Assertion (A) : A number \( N \) when divided by 15 gives the reminder 2. Then the remainder is same when \( N \) is divided by 5.
Reason (R) : \( \sqrt{3} \) is an irrational number.
Answer: Solution : 5 is a factor of 15 so when a number \( N \) is divided by 15 or 5 it gives the remainder 2.
Also, \( \sqrt{3} \) is an irrational number.
Both A and R are true but R is not the correct explanation for A.
Hence, option (b) is correct.

Please click on below link to download CBSE Class 10 Mathematics Real Numbers Worksheet Set B