• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer

CBSE Tuts

CBSE Maths notes, CBSE physics notes, CBSE chemistry notes

  • NCERT Solutions
    • NCERT Solutions for Class 12 English Flamingo and Vistas
    • NCERT Solutions for Class 11 English
    • NCERT Solutions for Class 11 Hindi
    • NCERT Solutions for Class 12 Hindi
    • NCERT Books Free Download
  • TS Grewal
    • TS Grewal Class 12 Accountancy Solutions
    • TS Grewal Class 11 Accountancy Solutions
  • CBSE Sample Papers
  • NCERT Exemplar Problems
  • English Grammar
    • Wordfeud Cheat
  • MCQ Questions

Real Numbers Class 10 Maths CBSE Important Questions with Solutions

Important Questions for Class 10 Maths Chapter 1 Real Numbers with solutions includes all the important topics with detailed explanation that aims to help students to score more marks in Board Exams 2020. Students who are preparing for their Class 10 exams must go through Important Questions for Class 10 Math Chapter 1 Real Numbers.

Important Questions for Class 10 Maths Chapter 1 Real Numbers

Expert teachers at CBSETuts.com collected and solved 2 Marks and 4 mark important questions for Class 10 Maths Chapter 1 Real Numbers. All the solutions given in this page are solved based on CBSE marking scheme and NCERT guidelines.

2016

Short Answer Type Questions I [2 Marks]

Question 1.
Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
Solution:
Maximum capacity of a container, which can measure the petrol in exact number of times.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 1

Question 2.
Find the value of: (-1) + (-1)2n +(-l) 2n+1 + (-l)4n+1 , where n is any positive odd integer.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 2

Question 3.
Find whether decimal expansion of 13/64 is a terminating or non-terminating decimal. If it terminates, find the number of decimal places its decimal expansion has.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 3

Short Answer Type Question II [3 Marks]

Question 4.
Explain whether the number 3 x 5 x 13 x 46 + 23 isa prime number or a composite number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 4

Long Answer Type Question [4 Marks]

Question 5.
Prove that the product of any three consecutive positive integers is divisible by 6. Solution: Let three consecutive numbers are n, n + 1, n + 2
Solution:
1st Case: If n is even
This means n + 2 is also even.
Hence n and n + 2 are divisible by 2
Also, product of n and (n + 2) is divisible by 2.
.’. n(n + 2) is divisible by 2.
This conclude n(n + 2) (n + 1) is divisible by 2 …(i)
As, n, n + 1, n + 2 are three consecutive numbers. n(n + 1) (n + 2) is a multiple of 3.
This shows n(n + 1) (n + 2) is divisible by 3. …(ii)
By equating (i) and (ii) we can say
n(n + 1) (n + 2) is divisible by 2 and 3 both.
Hence, n(n + 1) (n + 2) is divisible by 6.
2nd Case: When n is odd.
This show (n + 1) is even
Hence (n + 1) is divisible by 2. …(iii)
This conclude n(n + 1) (n + 2) is an even number and divisible by 2.
Also product of three consecutive number is a multiple of 3.
n(n + 1)(n + 2) is divisible by 3. …(iv)
Equating (iii) and (iv) we can say
n(n + 1) (n + 2) is divisible by both 2 and 3 Hence, n(n + 1)(n + 2) is divisible by 6.

2015

Short Answer Type Questions I [2 Marks]

Question 6.
Apply Euclid’s division algorithm to find HCF of numbers 4052 and 420.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 5

Question 7.

Show that (√3+√5)² is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 6

Short Answer Type Question [3 Marks]

For example, LCM of 15 and 20 is 60, and LCM of 5 and 7 is 35.

Question 8.
Three bells toll at intervals of 12 minutes, 15 minutes and 18 minutes respectively. If they start tolling together, after what time will they next toll together?
Solution:
LCM of 12, 15, 18 = 2²x 3² x 5
=4x9x5 = 180
So, next time the bells will ring together after 180 minutes.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 7

2014

Short Answer Type Questions I [2 Marks]

Question 9.
If HCF of 144 and 180 is expressed in the form 13m – 3, find the value of m.
Solution:
On applying Euclid’s division algorithm,
180 = 144 x 1 + 36
144 = 36 x 4 + 0
At the last stage, the divisor is 36.
∴ HCF of 144 and 180 is 36.
∵ 36 = 13 x 3 – 3
So, m = 3
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 8

Question 10.
Show that 9n can not end with digit 0 for any natural number n.
Solution:
Since prime factorisation of 9n is given by 9n= (3 x 3)n = 3271.
Prime factorisation of 9″ contains only prime number 3.
9 may end with the digit 0 for some natural number V if 5 must be in its prime factorisation, which is not present.
So, there is no natural number N for which 9n ends with the digit zero.

Question 11.
Determine the values otp and q so that the prime factorisation of2520 is expressible as 23 X y X q x 7.
Solution:
Prime factorisation of 2520 is given by
2520 = 23 x 32 x 5 x 7
Given that 2520 = 23 x 3p x q x 7
On comparing both factorisation we get p = 2 and q = 5.

Question 12.
Show that 2√2 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 9
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 10

Question 13.
Show that any positive odd integer is of the form 4m + 1 or 4m + 3, where m is some integer.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 11

Short Answer Type Questions II [3 Marks]

Question 14.
By using, Euclid’s algorithm, find the largest number which divides 650 and 1170.
Solution:
Given numbers are 650 and 1170.
On applying Euclid’s division algorithm,
we get 1170 = 650 x 1 + 520
650 = 520 x 1 + 130
520 = 130 x 4 + 0
∵ At the last stage, the divisor is 130.
∴ The HCF of 650 and 1170 is 130.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 12

Question 15.
Show that reciprocal of 3+2√2 is an irrational number
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 13
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 14

Long Answer Type Question [4 Marks]

Question 16.
Find HCF of 378,180 and 420 by prime factorisation method. Is HCF x LCM of three numbers equal to the product of the three numbers?
Solution:
378 = 2 x 33 x 7
180 = 22 x 32 x 5
420 = 22 x 3 x 5 x 7
∴ HCF (378, 180, 420) = 2 x 3 = 6.
No. HCF (p, q, r) x LCM (p, q, r) ≠ p x q x r. where p, q, r are positive integers.

2013

Short Answer Type Questions I [2 Marks]

Question 17.
Find the HCF of 255 and 867 by Euclid’s division algorithm
Solution:
Given numbers are 255 and 867.
On applying Euclid’s division algorithm, we have
867 = 255 x 3 + 102
255 = 102 x 2 + 51
102 = 51 x 2 + 0
∵ At the last stage, the divisor is 51
∴ The HCF of 255 and 867 is 51.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 15

Question 18.
Find the HCF (865, 255) using Euclid’s division lemma.
Solution:
Given numbers are 255 and 865.
On applying Euclid’s division algorithm, we have
865 = 255 x 3 + 100
255 = 100 x 2 + 55
100 = 55 x 1 + 45
55 = 45 x 1 + 10
45 = 10 x 4 + 5
10 = 5 x 2 + 0
∵At the last stage, the divisor is 5
∴ The HCF of 255 and 865 is 5.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 16

Short Answer Type Questions II [3 Marks]

Question 19.
Find HCF of 65 and 117 and find a pair of integral values of m and n such that HCF = 65m + 117n.
Solution:
Given numbers are 65 and 117.
On applying Euclid’s division algorithm, we get
117 = 65 x 1 + 52
65 = 52 x 1 + 13
52 = 13 x 4 + 0
∵At the last stage, the divisor is 13.
∴ The HCF of 65 and 117 is 13.
The required pair of integral values of m and n is
(2,-1) which satisfies the given relation HCF = 65m + 117n.
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 17

Question 20.
By using Euclid’s algorithm, find the largest number which divides 650 and 1170
Solution:

Since prime factorisation of 9n is given by 9n= (3 x 3)n = 3271.
Prime factorisation of 9″ contains only prime number 3.
9 may end with the digit 0 for some natural number V if 5 must be in its prime factorisation, which is not present.
So, there is no natural number N for which 9n ends with the digit zero.

 

2012

Short Answer Type Question I [2 Marks]

Question 21.
If Real Numbers Class 10 Maths CBSE Important Questions with Solutions 18 find the values of m and n where m and n are non-negative integers.Hence write its decimal expansion without actual division.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 19

Short Answer Type Questions II [3 Marks]

Question 22.
Express the number 0.3178 in the form of rational number a/b.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 20

Question 23.
Using Euclid’s division algorithm, find whether the pair of numbers 847,2160 are coprimes or not.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 21

Question 24.
The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then find the other number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 22

2011

Short Answer Type Questions I [2 Marks]

Question 25.
Prove that 15 + 17√3 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 23

Question 26.
Find the LCM and HCF of 120 and 144 by using Fundamental Theorem of Arithmetic.
Solution:
120 = 23 x 3 x 5
144 = 24 x 32
∴ HCF = 23 x 3 = 24
LCM = 24 x 5 x 32 = 720

Short Answer Type Questions II [3 Marks]

Question 27.
An army contingent of 1000 members is to march behind an army band of 56 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Solution:
1000 =2x2x2x5x5x5
56 = 2x2x2x7
HCF of 1000 and 56 = 8
Maximum number of columns = 8.

Question 28.
Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is a positive integer.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 24

Question 29.
Prove that 2√3/5 is irrational
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 25

2010

Very Short Answer Type Questions [1 Mark]

Question 30.
Has the rational number Real Numbers Class 10 Maths CBSE Important Questions with Solutions 26 a terminating or a non-terminating decimal repressentation
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 27

Question 31.
Write whether Real Numbers Class 10 Maths CBSE Important Questions with Solutions 28 on simplification gives a rational or an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 29

Question 32.
The HCF of 45 and 105 is 15. Write their LCM.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 30

Short Answer Type Questions II [3 Marks]

Question 33.
Prove that 2-3√5 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 31

Question 34.
Prove that 2√3 – 1 is an irrational number.
Solution:

Let 2√3 – 1 is a  irrational number.

Real Numbers Class 10 Maths CBSE Important Questions with Solutions 32

Question 35.
Prove that √2 is irrational.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 33

Question 36.
Prove that 7 – 2√3 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 34

Question 37.
Show that 5 + 3√2 is an irrational number.
Solution:

Let us assume 5 + 3√2 is an irrational number.
There exists coprime integers a and b (b≠0)

Real Numbers Class 10 Maths CBSE Important Questions with Solutions 35

2009

Very Short Answer Type Questions [1 Mark]

Question 38.
The decimal expansion of the rational number Real Numbers Class 10 Maths CBSE Important Questions with Solutions 36 will terminate after how many places of decimals.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 37

Question 39.
Find the [HCF X LCM] for the numbers 100 and 190.
Solution:
HCF x LCM = one number x another number
= 100 x 190 = 19000

Question 40.
Find the [HCF and LCM] for the numbers 105 and 120. [All India]
Solution:
105 = 5 x 7 x 3
120 = 2x2x2x3x5
HCF = 3 X 5 = 15
LCM = 5x7x3x2x2x2 = 840

Question 41.
Write whether the rational number Real Numbers Class 10 Maths CBSE Important Questions with Solutions 38 will have a terminating decimal expansion or a non terminating repeating decimal expansion.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 39
Question 42.
The HCF and LCM of two numbers are 9 and 360 respectively. If one number is 45, write the other number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 40

Short Answer Type Questions II [3 Marks]

Question 43.
Show that 5 – 2√3 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 41

Question 44.
Show that 3 + 5√2 is an irrational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 42
Question 45.
Show that the square of any positive odd integer is of the form 8m + 1, for some integer m.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 43

Question 46.
Prove that 7 + 3√2 is not a rational number.
Solution:
Real Numbers Class 10 Maths CBSE Important Questions with Solutions 44

Primary Sidebar

NCERT Exemplar problems With Solutions CBSE Previous Year Questions with Solutoins CBSE Sample Papers
  • The Summer Of The Beautiful White Horse Answers
  • Job Application Letter class 12 Samples
  • Science Lab Manual Class 9
  • Letter to The Editor Class 12 Samples
  • Unseen Passage For Class 6 Answers
  • NCERT Solutions for Class 12 Hindi Core
  • Invitation and Replies Class 12 Examples
  • Advertisement Writing Class 11 Examples
  • Lab Manual Class 10 Science

Recent Posts

  • Understanding Diversity Question Answer Class 6 Social Science Civics Chapter 1 NCERT Solutions
  • Our Changing Earth Question Answer Class 7 Social Science Geography Chapter 3 NCERT Solutions
  • Inside Our Earth Question Answer Class 7 Social Science Geography Chapter 2 NCERT Solutions
  • Rulers and Buildings Question Answer Class 7 Social Science History Chapter 5 NCERT Solutions
  • On Equality Question Answer Class 7 Social Science Civics Chapter 1 NCERT Solutions
  • Role of the Government in Health Question Answer Class 7 Social Science Civics Chapter 2 NCERT Solutions
  • Vital Villages, Thriving Towns Question Answer Class 6 Social Science History Chapter 9 NCERT Solutions
  • New Empires and Kingdoms Question Answer Class 6 Social Science History Chapter 11 NCERT Solutions
  • The Delhi Sultans Question Answer Class 7 Social Science History Chapter 3 NCERT Solutions
  • The Mughal Empire Question Answer Class 7 Social Science History Chapter 4 NCERT Solutions
  • India: Climate Vegetation and Wildlife Question Answer Class 6 Social Science Geography Chapter 8 NCERT Solutions
  • Traders, Kings and Pilgrims Question Answer Class 6 Social Science History Chapter 10 NCERT Solutions
  • Environment Question Answer Class 7 Social Science Geography Chapter 1 NCERT Solutions
  • Understanding Advertising Question Answer Class 7 Social Science Civics Chapter 7 NCERT Solutions
  • The Making of Regional Cultures Question Answer Class 7 Social Science History Chapter 9 NCERT Solutions

Footer

Maths NCERT Solutions

NCERT Solutions for Class 12 Maths
NCERT Solutions for Class 11 Maths
NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 9 Maths
NCERT Solutions for Class 8 Maths
NCERT Solutions for Class 7 Maths
NCERT Solutions for Class 6 Maths

SCIENCE NCERT SOLUTIONS

NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Chemistry
NCERT Solutions for Class 11 Physics
NCERT Solutions for Class 11 Chemistry
NCERT Solutions for Class 10 Science
NCERT Solutions for Class 9 Science
NCERT Solutions for Class 7 Science
MCQ Questions NCERT Solutions
CBSE Sample Papers
NCERT Exemplar Solutions LCM and GCF Calculator
TS Grewal Accountancy Class 12 Solutions
TS Grewal Accountancy Class 11 Solutions