Class 10 Maths Important Questions for Real Numbers

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Given below are the important questions for class 10 maths real numbers 
a. Concepts questions
b. Calculation problems
c. Multiple choice questions(MCQ) 
d. Long answer questions
e. Irrational number proof questions
f. True and false questions

Long answer questions

Question 1
Without actually performing division, state which of these number will terminating decimal expression or non terminating repeating decimal expression

1. $\frac{7}{25}$
2. $\frac{3}{7}$
3. $\frac{29}{343}$
4. $\frac{6}{15}$
5. $\frac{77}{210}$
6. $\frac{11}{67}$
7. $\frac{15}{27}$
8. $\frac{11}{6}$
9. $\frac{343445}{140}$

SOLUTION

Those rational number which can be expressed in form x/2^m X5ⁿ    are terminating expression and those can not be are non terminating decimal expression
Terminating decimal:  (a), (d)
Non terminating repeating decimal: (b), (c), (e), (f), (g).(h) ,(i)

Question 2
Using Euclid’s theorem to find the HCF between the following numbers
a. 867 and 225
b. 616 and 32

Question 3

Write 10 rational number between
a. 4 and 5
b. 1/2 and 1/3
Question 4
Represent in rational form.
a. 1.232323….
b. 1.25
c. 3.67777777
Question 5 
a. Prove that 2+√3 is a irrational number
b. Prove that 3√3 a irrational number

True or False statement

Question 6
Mark T/F as appropiate:
a. Number of the form $2n+1$ where n is any positive integer are always odd number
b. Product of two prime number is always equal to their LCM
c. $\sqrt{3}\times \sqrt{12}$ is a irrational number
d. Every integer is a rational number
e. The HCF of two prime number is always 1
f. There are infinite integers between two integers
g. There are finite rational number between 2 and 3
h. √3 Can be expressed in the form √3/1,so it is a rational number
i. The number 6ⁿ for n in natural number can end in digit zero
j. Any positive odd integer is of the form 6m+1 or 6m+3 or 6m +5 where q is some integer

Multiple choice Questions

Question 7
the HCF (a, b) =2 and LCM (a, b) =27. What is the value $a\times b$
a. 25
b. 9
c. 27
d. 54

Question 8
2+√2 is a
a. Non terminating repeating
b. Terminating
c. Non terminating non repeating
d. None of these

Question 9

if a and b are co primes which of these is true
a. LCM (a, b) =aXb
b. HCF (a, b)= aXb
c. a=br
d. None of these

Question 10
A rational number can be expressed as terminating decimal when the factors of the denominator are
a. 2 or 5 only
b. 2 or 3 only
c. 3 or 5 only
d. 3 or 7 only

Question 11
if x2=3,y2=9,z3=27, which of these is true
a. x is a irrational number
b. y is a rational number
c. z is rational number
d.All of the above

Short answer question

Question 12
Find the HCF and LCM of these by factorization technique
a.27,81
b. 120 ,144
c. 29029 ,580
Solution

Question 13
Find all the positive integral values of p for which  $p^2+16$ is a perfect square?
Solution

Question 14
Find the nature of the product $(\sqrt{2}-\sqrt{3})(\sqrt{3}+\sqrt{2})$ ?
Solution

Question 15
Prove that the sum of a rational number and an irrational number is always irrational.
Solution

Question 16
Prove that 5 is an irrational number.
Solution

Question 17
Show that $3+5\sqrt{2}$ is an irrational number. Is sum of two irrational numbers always an irrational number?
Solution

Question 18
Prove that 3 is an irrational number and hence show that $2\sqrt{3}$ is also an irrational number.
Solution

Question 19
Prove that $5-\sqrt{3}$ is an irrational number.
Solution

Question 20
Prove that $2\sqrt{5}$ is an irrational number.
Question 21
Show that $(\sqrt{3}+\sqrt{5}{)}^2$ is an irrational number.
Solution

Question 22
Prove that $4-\sqrt{5}$ is an irrational number.
Question 23
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Solution

Question 24
Prove that  √2 + 1/√2 is an irrational number
Question 25
Prove that for any positive integer n, n³ – n is divisible by 6.
Solution

Question 26 
If n is rational and √m   is irrational, then prove that (n + √m) is irrational.
Question 27
Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer
Solution

Question 28
Prove that √11 is irrational.
Question 29 
Show that 3√2 is irrational.
Question 30 Show that 4ⁿ can never end with the digit zero for any natural number n.
Solution

Question 31
The product of a non-zero rational and an irrational number is
(A) always irrational
(B) always rational
(C) rational or irrational
(D) one
Solution

Question 32
Prove that √p + √q is irrational, where p, q are primes.
Question 33
Prove that one of any three consecutive positive integers must be divisible by 3.
Solution

Cross-word Puzzle to check your Real number knowledge

Across
2. Number which are not divisible by any other number except 1
6. decimal expression can be expressed in the form 1/2^m5ⁿ
Down
1. Number which can be written as product of prime
3. In Euclid division lemma a=bq + r , it is the value r
4. HCF can be found using this division algorithm
5. In Euclid division lemma a=bq + r , it is the value b
7. Numbers of the forms p/q