CBSE Class 10 Mathematics Real Numbers MCQs Set 13

Question. The sum of exponents of prime factors in the prime-factorisation of 196 is:
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (b) 4

Question. The total number of factors of a prime number is
(a) 1
(b) 0
(c) 2
(d) 3
Answer: (c) 2
 

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) 3, 420
 

Question. The decimal representation of \( \frac{11}{2^3 \times 5} \) will:
(a) terminate after 1 decimal place
(b) terminate after 2 decimal places
(c) terminate after 3 decimal places
(d) not terminate
Answer: (c) terminate after 3 decimal places

Question. The LCM of smallest two digit composite number and smallest composite number is:
(a) 12
(b) 4
(c) 20
(d) 44
Answer: (c) 20

Question. If two positive integers \( a \) and \( b \) are written as \( a = x^3y^2 \) and \( b = xy^3 \), where \( x \) and \( y \) are prime numbers, then the HCF (\( a, b \)) is:
(a) \( xy \)
(b) \( xy^2 \)
(c) \( x^3y^3 \)
(d) \( x^2y^2 \)
Answer: (b) \( xy^2 \)

Question. If two positive integers \( p \) and \( q \) can be expressed as \( p = ab^2 \) and \( q = a^3b \) where \( a \) and \( b \) are prime numbers, then the LCM (\( p, q \)) is:
(a) \( ab \)
(b) \( a^2b^2 \)
(c) \( a^3b^2 \)
(d) \( a^3b^3 \)
Answer: (c) \( a^3b^2 \)
 

Question. 7 × 11 × 13 × 15 + 15 is a:
(a) Composite number
(b) Whole number
(c) Prime number
(d) (a) and (b) both
Answer: (d) (a) and (b) both
 

Question. LCM of (\( 2^3 \times 3 \times 5 \)) and (\( 2^4 \times 5 \times 7 \)) is
(a) 40
(b) 560
(c) 1120
(d) 1680
Answer: (d) 1680
 

Question. 1.23451326... is
(a) an integer
(b) an irrational number
(c) a rational number
(d) None of the options
Answer: (b) an irrational number
 

Question. If the LCM of \( a \) and 18 is 36 and the HCF of \( a \) and 18 is 2, then \( a = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4
 

Question. The product of a non-zero rational and an irrational number is:
(a) always irrational
(b) always rational
(c) rational or irrational
(d) one
Answer: (a) always irrational
 

Question. The number of decimal places after which the decimal expansion of the rational number \( \frac{9}{2^4 \times 5} \) will terminate, is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4
Explanation: Number is
\( \frac{9}{2^4 \times 5} = \frac{9 \times 5^3}{2^4 \times 5^4} = \frac{1125}{10^4} = 0.1125 \)
Therefore, number terminate after 4 decimal places.

Question. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is:
(a) 10
(b) 100
(c) 504
(d) 2520
Answer: (d) 2520
Explanation: As we require the least number, the problem is based on finding the LCM.
Factors of 1 to 10 numbers are as follows:
1 = 1
2 = 1 × 2
3 = 1 × 3
4 = 1 × 2 × 2
5 = 1 × 5
6 = 1 × 2 × 3
7 = 1 × 7
8 = 1 × 2 × 2 × 2
9 = 1 × 3 × 3
10 = 1 × 2 × 5
LCM of number 1 to 10
= LCM (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
= 1 × 2 × 2 × 2 × 3 × 3 × 5 × 7
= 2520

Question. The decimal expansion of the rational number \( \frac{14587}{1250} \) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places
Answer: (d) four decimal places
 

Question. If HCF (\( a, b \)) = 45 and \( a \times b = 30375 \), then LCM (\( a, b \)) is:
(a) 1875
(b) 1350
(c) 625
(d) 675
Answer: (d) 675
 

Question. The cube of any positive integer is not of the form:
(a) \( 9q \)
(b) \( 9q + 1 \)
(c) \( 9q + 3 \)
(d) \( 9q + 8 \)
Answer: (c) \( 9q + 3 \)
 

Question. 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75, what is the HCF of (525, 3000)?
(a) 25
(b) 125
(c) 75
(d) 15
Answer: (c) 75
 

Question. If HCF of two numbers is 1, the numbers are called relatively ......... or ......... .
(a) Prime, co-prime
(b) Composite, prime
(c) Both (a) and (b)
(d) None of the options
Answer: (a) Prime, co-prime
Explanation: Prime numbers are those numbers which have only two factors i.e., 1 and itself. Example, 3, 5, 11 etc.
Co-prime numbers: Two numbers that have only 1 as a common factor.
Example, 35 and 39
35 = 1 × 5 × 7, 39 = 1 × 3 × 13
Here, common factor is 1.

Fill in the Blanks

Fill in the blanks/tables with suitable information:

Question. \( \left( \frac{2 + \sqrt{5}}{3} \right) \) is .................... number.
Answer: irrational
Explanation: As \( \sqrt{5} \) is irrational, \( 2 + \sqrt{5} \) is irrational

Question. The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, the other number is .................... .
Answer: 81
Explanation: HCF of two numbers is 27 and their LCM is 162.
Let the other number be \( x \).
Product of two numbers
= HCF × LCM = 27 × 162
\( \implies \) \( 54x = 27 \times 162 \)
\( \implies \) \( x = 81 \)

Question. If \( a = (2^2 \times 3^3 \times 5^4) \) and \( b = (2^3 \times 3^2 \times 5) \) then HCF (\( a, b \)) = .................... .
Answer: 180
Explanation: \( a = (2^2 \times 3^3 \times 5^4) \)
\( b = (2^3 \times 3^2 \times 5) \)
HCF(\( a, b \)) = \( 2^2 \times 3^2 \times 5 \)
= \( 4 \times 9 \times 5 = 180 \)

Question. A decimal number \( 0.\overline{8} \) can be expressed in its simplest form as .................... .
Answer: \( \frac{8}{9} \)
Explanation: Let \( x = 8.88888... \)
\( 10x = 8.88888... \)
\( \implies \) \( 10x - x = 8 \)
\( \implies \) \( 9x = 8 \)
\( \implies \) \( x = \frac{8}{9} \)

Question. Product of two numbers is 18144 and their HCF is 6, then their LCM is .................... .
Answer: 324
Explanation: Product of two numbers = 18144
HCF of two numbers is 6
Product of two numbers
= HCF × LCM = 18144
\( \implies \) \( 6 \times \text{LCM} = 18144 \)
\( \implies \) \( \text{LCM} = \frac{18144}{6} = 324 \)

Question. The decimal expression of the rational number \( \frac{23}{2^2 \times 5} \) will terminate after ............. decimal place(s).
Answer: 2
Explanation: Here the power of 2 is 2 and the power of 5 is 1.
\( 2 > 1 \)
Hence, \( \frac{23}{2^2 \times 5} \) has terminating decimal expansion which terminates after 2 places of decimals.

Question. The HCF of smallest composite number and the smallest prime number is .................... .
Answer: 2
Explanation: Smallest prime number = 2
Smallest composite number = 4
HCF (2, 4) = 2

Question. If \( a \) and \( b \) are positive integers, then \( \frac{\text{HCF} (a, b) \times \text{LCM} (a, b)}{ab} = \) ....................
Answer: 1
Explanation: HCF (\( a, b \)) × LCM (\( a, b \)) = \( ab \)
\( \implies \) \( \frac{\text{HCF} (a, b) \times \text{LCM} (a, b)}{ab} = 1 \)

Question. .................... is the H.C.F. of two consecutive even numbers.
Answer: 2
Explanation: All even numbers are divisible by 2. Therefore, HCF of two consecutive numbers is 2.

Question. If two positive integers \( p \) and \( q \) can be expressed as \( p = a^2b^3 \) and \( q = a^4b \); \( a, b \) being prime numbers, then LCM (\( p, q \)) is.....................
Answer: \( a^4b^3 \)