CBSE Class 10 Mathematics Real Numbers MCQs Set 12

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
Explanation: The prime factorisition of 196 is:
\( 196 = 2^2 \times 7^2 \)
So, sum of the exponents of prime factors 2 and 7 is \( 2 + 2 \) i.e., 4

Question. The total number of factors of a prime number is
(a) 1
(b) 0
(c) 2
(d) 3
Answer: (c) 2
Explanation: Factors of a prime number are 1 and the number itself.

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
Explanation:
Here, \( 12 = 2^2 \times 3 \)
\( 21 = 3 \times 7 \)
\( 15 = 3 \times 5 \)
So, HCF = 3; LCM = \( 2^2 \times 3 \times 7 \) i.e., 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) \( xy^2 \)
Explanation:
Given that \( a = x^3y^2 = x \times x \times x \times y \times y \)
and \( b = xy^3 = x \times y \times y \times y \)
\( \Rightarrow \) HCF of \( a \) and \( b = \text{HCF} (x^3y^2, xy^3) \)
\( = x \times y \times y = xy^2 \)
We know that HCF is the product of the smallest power of each common prime factor involved in the numbers.

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) \( a^3b^2 \)
Explanation: Given that
\( p = ab^2 = a \times b \times b \)
and \( q = a^3b = a \times a \times a \times b \)
We know that LCM is the product of the greatest power of each Prime factor of the numbers.
\( \Rightarrow \) LCM of \( p \) and \( q = \text{LCM} (ab^2, a^3b) \)
\( = a \times b \times b \times a \times a = a^3b^2 \)

Question. \( 7 \times 11 \times 13 \times 15 + 15 \) is a:
(a) Composite number
(b) Whole number
(c) Prime number
(d) (a) and (b) both
Answer: (d) (a) and (b) both
Explanation: \( 7 \times 11 \times 13 \times 15 + 15 \)
\( = 15 (7 \times 11 \times 13 + 1) = 15 \times 1002 \)
Also \( 15 \times 1002 \) is a whole number.
The number having factors more than two therefore, this is composite number and whole number.

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
Explanation: \( (2^3 \times 3 \times 5) \) and \( (2^4 \times 5 \times 7) \)
LCM \( = 2^4 \times 3 \times 5 \times 7 = 1680 \)

Question. 1.23451326... is
(a) an integer
(b) an irrational number
(c) a rational number
(d) none of these
Answer: (b) an irrational number
Explanation: Number neither terminating nor repeated, therefore this is 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
Explanation: We know that,
LCM \( (a, b) \times \text{HCF} (a, b) = a \times b \)
\( \Rightarrow 36 \times 2 = a \times 18 \)
\( \Rightarrow a = \frac{36 \times 2}{18} \)
\( \Rightarrow a = 4 \)

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
Explanation: Product of a non–zero rational and an irrational number is always irrational.
For example:
\( \frac{7}{9} \) is rational and \( \sqrt{2} \) is irrational numbers.
Their product is an irrational number.
\( \frac{7}{9} \times \sqrt{2} = \frac{7\sqrt{2}}{9} \), which is an irrational number.

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:
\( 1 = 1 \)
\( 2 = 1 \times 2 \)
\( 3 = 1 \times 3 \)
\( 4 = 1 \times 2 \times 2 \)
\( 5 = 1 \times 5 \)
\( 6 = 1 \times 2 \times 3 \)
\( 7 = 1 \times 7 \)
\( 8 = 1 \times 2 \times 2 \times 2 \)
\( 9 = 1 \times 3 \times 3 \)
\( 10 = 1 \times 2 \times 5 \)
LCM of number 1 to 10
\( = \text{LCM} (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) \)
\( = 1 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \)
\( = 2520 \)

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
Explanation:
Simplifying the given fraction:
\( \Rightarrow \frac{14587}{1250} = \frac{14587}{5^4 \times 2} \)
\( = \frac{14587}{5^4 \times 2} \times \frac{2^3}{2^3} \)
\( = \frac{116696}{5^4 \times 2^4} = \frac{116696}{10^4} \)
\( = 11.6696 \)
Hence, the given rational number will terminate after 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
Explanation: We know that,
\( \text{LCM} (a, b) = \frac{a \times b}{\text{HCF} (a, b)} \)
So, \( \text{LCM} (a, b) = \frac{30375}{45} = 675 \)

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
Explanation: The cube of any positive integer is of the form 9q or 9q + 1 or 9q + 8. So, 9q + 3 is incorrect.

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
Explanation: Since 3, 5, 15, 25 and 75 are the only common factors of 525 and 3000, 75 is the HCF.

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 above
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
\( 35 = 1 \times 5 \times 7, 39 = 1 \times 3 \times 13 \)
Here, common factor is 1.

Fill in the Blanks

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
\( = \text{HCF} \times \text{LCM} = 27 \times 162 \)
\( \Rightarrow 54x = 27 \times 162 \)
\( \Rightarrow x = 81 \)

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: \( a = (2^2 \times 3^3 \times 5^4) \)
\( b = (2^3 \times 3^2 \times 5) \)
\( \text{HCF}(a, b) = 2^2 \times 3^2 \times 5 \)
\( = 4 \times 9 \times 5 = 180 \)

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

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
\( = \text{HCF} \times \text{LCM} = 18144 \)
\( \Rightarrow 6 \times \text{LCM} = 18144 \)
\( \Rightarrow \text{LCM} = \frac{18144}{6} = 324 \)

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: \( \text{HCF}(a, b) \times \text{LCM}(a, b) = ab \)
\( \Rightarrow \frac{\text{HCF}(a, b) \times \text{LCM}(a, b)}{ab} = 1 \)

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: \( a^4b^3 \)