 # Important Questions for Real Numbers Class 10 Maths

Given below are the Class 10 Maths Important Questions for 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. $\frac {343445}{140}$

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

Question 6True or False statement
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 $x^2 =3 \;, \; y^2=9 \; ,\; z^3=27$, 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

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

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

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

Question 16 Prove that $\sqrt {5}$is an irrational number.

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

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

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

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

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.

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.

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

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.

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

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.

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Reference Books for class 10

Given below are the links of some of the reference books for class 10 math.

You can use above books for extra knowledge and practicing different questions.

### Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20

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