Without actually performing the long division, state whether 543 225 has a terminating decimal expansion or non terminating recurring decimal expansion.

Given rational number is 543 225 . Prime factorising the denominator 225 , 225 = 3 × 3 × 5 × 5 = 3 2 × 5 2 Since the prime factorised form of denominator is not of the form 2 m and 5 n , Therefore, the given rational number will have a non-terminating recurring decimal expansion.

Prove that 3 + 7 is an irrational number.

Let us assume that 3 + 7 is a rational number. Now, let 3 + 7 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 7 = a b - 3 Since a , b are integers, then a b - 3 is also a rational number. ⇒ 7 is a rational number But, we know that, 7 is an irrational number. This shows our initial assumption was incorrect. Therefore, 3 + 7 is an irrational number.

Prove that 2 3 - 4 is an irrational number.

Let us assume that 2 3 - 4 is a rational number. Now, let 2 3 - 4 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 3 = a b + 4 ⇒ 3 = 1 2 a b + 4 Since a , b are integers, then a b + 4 is also a rational number. ⇒ 1 2 a b + 4 is a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 2 3 - 4 is an irrational number.

Find the HCF and LCM of 306 and 54 . Verify that HCF × LCM = Product of the two numbers

Given numbers are: 306 , 54 . Applying the prime factorisation method to find the prime factors. 306 = 2 × 3 × 3 × 17 = 2 × 3 2 × 17 , and 54 = 2 × 3 × 3 × 3 = 2 × 3 3 . ∴ H . C . F . of 306 , 54 = 2 × 3 2 = 18 , and L . C . M . of 306 , 54 = 2 × 3 3 × 17 = 918 . The statement to verify is: HCF × LCM = Product of the two numbers L . H . S . = HCF × LCM = 18 × 918 = 16524 R . H . S = 306 × 54 = 16524 L . H . S . = R . H . S . Therefore, the statement is verified.

E M B I B E

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 7

EASY

10th CBSE

IMPORTANT

Earn 100

Find the LCM and HCF of $120 and 144$ by fundamental theorem of Arithmetic.

Important Questions on Number System

EASY

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 8

In the adjoining factor tree, find the numbers $m, n :$

` Question Image `

EASY

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 9

Without actually performing the long division, state whether

\frac{543}{225}

has a terminating decimal expansion or non terminating recurring decimal expansion.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 10

Use Euclid's Division Lemma to show that the square of any positive integer if of the form

4 m

or

4 m + 1

for some integer

m

.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 1

Prove that

3 + \sqrt{7}

is an irrational number.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 2

Prove that

2 \sqrt{3} - 4

is an irrational number.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 3

A merchant has

120

litres of oil of one kind,

180

litres of another kind and

240

litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE>Q 4

Show that any positive even integer is of the form