## RD Sharma Class 10 Solutions Chapter 1 Real Numbers VSAQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 1 Real Numbers VSAQS

Other Exercises

Answer each of the following questions either in one word or one sentence or as per requirement of the questions :

Question 1.

State Euclid’s division lemma.

Solution:

Euclid’s division lemma:

Let a and b be any two positive integers, then there exist unique integers q and r such that

a = bq + r, 0 ≤ r < b

If ba, then r = 0, otherwise x. satisfies the stronger inequality 0 < r < b.

Question 2.

State Fundamental Theorem of Arithmetic.

Solution:

Fundamental Theorem of Arithmetics :

Every composite number can be expressed (factorised) as a product of primes and this factorization is unique except for the order in which the prime factors occur.

Question 3.

Write 98 as product of its prime factors.

Solution:

Question 4.

Write the exponent of 2 in the prime factorization of 144.

Solution:

Question 5.

Write the sum of the exponents of prime factors in the prime factorization of 98

Solution:

98 = 2 x 7 x 7 = 2^{1} x 7^{2}

Sum of exponents = 1 + 2 = 3

Question 6.

If the prime factorization of a natural number n is 2^{3} x 3^{2} x 5^{2} x 7, write the number of consecutive zeros in n.

Solution:

n = 2^{3} x 3^{2} x 5^{2} x 7

Number of zeros will be 5^{2} x 2^{2} = 10^{2} two zeros

Question 7.

If the product of two numbers is 1080 and their H.C.F. is 30, find their L.C.M.

Solution:

Product of two numbers = 1080

H.C.F. = 30

Question 8.

Write the condition to be satisfied by q so that a rational number \\(\\frac { p }{ q }) has a terminating decimal expansion. [C.B.S.E. 2008]

Solution:

In the rational number \\(\\frac { p }{ q }) , the factorization of denominator q must be in form of 2^{m} x 5^{n} where m and n are non-negative integers.

Question 9.

Write the condition to be satisfied by q so that a rational number \\(\\frac { p }{ q }) has a non-terminating decimal expansion.

Solution:

In the rational number \\(\\frac { p }{ q }) , the factorization of denominator q, is not in the form of 2^{m} x 5^{n} where m and n are non-negative integers.

Question 10.

Complete the missing entries in the following factor tree.

Solution:

Question 11.

The decimal expression of the rational number \\(\\frac { 43 }{ { 2 }^{ 4 }times { 5 }^{ 3 } }) will terminate after how many places of decimals. [C.B.S.E. 2009]

Solution:

The denominator of \\(\\frac { 43 }{ { 2 }^{ 4 }times { 5 }^{ 3 } }) is 24 x 53 which is in the form of 2^{m} x 5^{n} where m and n are positive integers

\\(\\frac { 43 }{ { 2 }^{ 4 }times { 5 }^{ 3 } }) has terminating decimals

The decimal expansion of \\(\\frac { 43 }{ { 2 }^{ 4 }times { 5 }^{ 3 } }) terminates after 4 (the highest power is 4) decimal places

Question 12.

Has the rational number \\(\\frac { 441 }{ { 2 }^{ 5 }times { 5 }^{ 7 }times { 7 }^{ 2 } }) of a terminating or a non terminating decimal representation ? [CBSE 2010]

Solution:

Question 13.

Write whether \\(\\frac { 2surd 45+3surd 20 }{ 2surd 5 }) on simplification gives a rational or an irrational number. [CBSE 2010]

Solution:

Question 14.

What is an algorithm ?

Solution:

Algorithm : An algorithm is a series of well defined slips which gives a procedure for solving a type of problem.

Question 15.

What is a lemma ?

Solution:

A lemma is a proven statement used for proving another statement.

Question 16.

If p and q are two prime numbers, then what is their HCF ?

Solution:

If p and q are two primes, then their HCF will be 1 as they have no common factor except 1.

Question 17.

If p and q are two prime numbers, then what is their LCM ?

Solution:

If p and q are two primes, their LCM will be their product.

Question 18.

What is the total number of factors of a prime number ?

Solution:

Total number of factors of a prime number are 2, first 1 and second the number itself.

Question 19.

What is a composite number ?

Solution:

A composite number is a number which can be factorised into more than two factors.

Question 20.

What is the HCF of the smallest composite number and the smallest prime number ?

Solution:

We know that 2 is the smallest prime number and 4 is the smallest composite number

HCF of 2 and 4 = 2

Question 21.

HCF of two numbers is always a factor of their LCM (True / False).

Solution:

True.

Question 22.

π is an irrational number (True / False).

Solution:

True as value of π is neither terminating nor repeating.

Question 23.

The sum of two prime numbers is always a prime number (True / False).

Solution:

False. Sum of two prime numbers can be a composite number

e.g. 3 and 5 are prime numbers but their sum 3 + 5 = 8 is a composite number.

Question 24.

The product of any three consecutive natural numbers is divisible by 6 (True / False).

Solution:

True.

Question 25.

Every even integer is of the form 2m, where m is an integer (True / False).

Solution:

True, as 2m is divisible by 2.

Question 26.

Every odd integer is of the form 2m – 1, where m is an integer (True / False).

Solution:

True, as 2m is an even number but if we subtract 1 from it, it will be odd number.

Question 27.

The product of two irrational numbers is an irrational number (True / False).

Solution:

False, as it is not always possible that the product of two irrational number be also an irrational number, it may be a rational number

for example

√3 x √3 = 3, √7 x √7 = 7

Question 28.

The sum of two irrational numbers is an irrational number (True / False).

Solution:

False, as it is not always possible that the sum of two irrational is also an irrational number, it may be rational number also.

For example

(2 + √3) + (2 – √3) = 2 + √3 + 2 – √3 = 4

Question 29.

For what value of n, 2^{n} x 5^{n} ends in 5.

Solution:

In 2^{n} x 5^{n} ,

There is no such value of n, which satisifies the given condition.

Question 30.

If a and b are relatively prime numbers, then what is their HCF ?

Solution:

a and b are two prime numbers

Their HCF =1

Question 31.

If a and b are relatively prime numbers, then what is their LCM ?

Solution:

a and b are two prime numbers

Their LCM = a x b

Question 32.

Two numbers have 12 as their HCF and 350 as their LCM (True / False).

Solution:

HCF of two numbers = 12

and LCM is 350

False, as the HCF of two numbers is a factor of their LCM and 12 is not a factor of 350

Hope given RD Sharma Class 10 Solutions Chapter 1 Real Numbers VSAQS are helpful to complete your math homework.

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