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(iii) 30.043

30 kg and 43 g = 30 kg + \(\frac{43}{1000}\) kg 

= 30 + 0.043 

= 30.043

(iv) 2.16

(i) 8.71 

Underlining the digit to be rounded 8.71. Since the digit next to the underlined digit, 7 which is greater than 5, adding 1 to the underlined digit. Hence the nearest whole number 8.71 rounds to is 9.

(ii) 26.01 

Underlining the digit to be rounded 26.01. Since the digit next to the underlined digit, 0 which is less than 5, the underlined digit 6 remains the same. 

∴ The nearest whole number 26.01 rounds to is 26.

(iii) 69.48 

Underlining the digit to be rounded 69.48. Since the digit next to the underlined digit, 4 which is less than 5, the underlined digit 9 remains the same. 

∴ The whole number is 69.48 rounds to is 69.

(iv) 103.72 

Underlining the digit to be rounded 103.72 since the digit next to the underlined digit, 7 which is greater than 5, we add 1 to the under lined digit.

Hence the nearest whole number 103.72 rounds to is 104.

(v) 49.84 

Underlining the digit to be rounded 49.84. Since the digit next to the underlined digit 8 which is greater than 5, we add 1 to the underlined digit. Hence the nearest whole number 49.84 rounds to 50.

(vi) 101.35 

Underlining the digit to be rounded 101.35. Since the digit next to the underlined digit 3 is less than 5, the underlined digit 1 remains the same. Hence the nearest whole number 101.35 rounds to is 101.

(vii) 39.814 

Underlining the digit to be rounded 39.814. Since the digit next to the underlined digit 8 is greater than 5, we add 1 to the underlined digit. 

Hence the nearest whole number 39.814 rounds to is 40.

(viii) 1.23 

Underlining the digit to be rounded 1.23. Since the digit next to the underlined digit 2, is less than 5, the underlined digit 1 remains the same. Hence the nearest whole number 1.23 rounds to is 1.

(i) -8 

(ii) -1 

(iii) 2 

(iv) -20 

(v) -8

(ii) 2.64 

264 cm = \(\frac{264}{100}\) m 

= 2.64 m

(i) -95 

(ii) 0 

(iii) Any integer; the same integer 

(iv) -140

Separating positive and the negative integers, we get -27, -4, -22, -9, -35 

Arranging the numbers in descending order -4 > -9 > -22 > -27 > -35 

The positive numbers are 19,12,47, 3 

Arranging in descending order, we get 47 > 19 > 12 > 3 

0 stands in the middle.

∴ Descending order: 47 > 19 > 12 > 3 > 0 > -4 > -9 > -22 > -27 > -35

The least number will be on the extreme left. 

∴ -17 will be on the extreme left.

We know that 

(i) product of two positive integers is positive

(ii) product of two negative integers is 

(iii) product of two negative integers is positive 

(iv) product of integers with opposite sign is negative.

∴ The table will be as follows:

One million = 10,00,000 = 10 lakhs 

Pravali’s father distributed 10 lakhs amount to his 3 children equally. 

So, the share of each children = 10 lakhs ÷ 3 = Rs. 3,33,333 

= Rs. 3,00,000 (approximately)

Yes. 

3 + √2 and 3 - √2 are two irrational numbers.

(3 + √2) + (3 - √2) = 6, a rational number. 

(3 + √2) x (3 - √2) = 7, a rational number. 

So, we have two irrational numbers whose sum and product both are rationals

True.

The composite number 4 has 3 factors namely 1, 2 and 4.

(b) 2

The difference between two successive odd number is 2.

(b) 92

92 is not prime number.

True.

The sum of any two consecutive odd numbers is divisible by 4

For example 11 + 13 = 24, divisible by 4

Also, all the consecutive odd numbers are of the form 4n + 1 or 4n + 3

Their sum = 4x + 4 which is divisible by 4.

15 is an odd number not prime.

(c) 2

The only even prime number is 2.

1002

The first 4 digit number divisible by 3 is 1002.

Even numbers greater then 2 upto 16 are 4, 6, 8, 10, 12, 14 and 16

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 (or) 5 + 5
12 = 5 + 7
14 = 7 + 7 (or) 3 + 11
16 = 5 + 11 (or) 3 + 13

(c) 71, 81

71, 81 is co-prime.

Date of birth 25.05.2007

Sum of digits = 2 + 5 + 0 + 5 + 2 + 0 + 0 + 7 = 21

Again 2 + 1 = 3, divisible by 3.

My date of birth is divisible by 3.

Three prime numbers 2, 37, 41

Sum 2 + 37+ 41 = 80

The difference between two of them 41 – 37 = 4

(d) all of these

In 684398

Sum of digits in odd places = 8 + 3 + 8 = 19

Sum of digits in even places = 6 + 4 + 9 = 19.

Difference = 19 – 19 = 0.

684398 is divisible by 11.

(i) 

1 + 3 = 4
5 + 11 = 16
21 + 47 = 68

An odd number + another odd number = An Even number

From this observation, we conclude that the sum of any two odd numbers is always an even number.

(ii) 

5 × 3 = 15
7 × 9 = 63
11 × 13 = 143

An odd number × Another odd number = An odd number

From this observation, we conclude that “the product of any two odd numbers is always an odd number.”

(iii) Take the odd number 5 and the even number 10

Their sum = 5 + 10 = 15, which is odd.

∴ Sum of an odd number and an even number is always an odd number.

(iv) Take the odd number 5 and the even number 10.

Their product = 5 × 10 = 50, which is even

Thus the product of an odd and an even number is always an even number.

(v) Consider 7 × 5 × 3

We know that the product of any two odd numbers is an odd number

7 × 5 = 35, odd number.

Also we have 35 × 3 = 105

∴ 7 × 5 × 3 = 105, an odd number.

So the product of three odd numbers is always an odd number.

Since, it can be represented as 0.27272727... which is a non - terminating repeating decimal.

A real number is a value that represents a quantity along a line.

(i) Every point on the number line corresponds to a real number which many be either rational or irrational. 

(ii) The decimal form of an irrational number is neither terminating nor repeating. 

(iii) The decimal representation of a rational number is either terminating or non-terminating recurring. 

(iv) Every real number is either rational number or an irrational number.

106160.

We know that Successor of any no. is 1 greater than the number. 

Therefore, Successor of 106159 is 106159 + 1 = 106160.

True.

As per the rule, Of the given two natural numbers, the one having more digits is greater.

True.

The successor of a whole number is the number obtained by adding 1 to it.

The largest 3-digit number = 999

Then, its successor = 999 + 1

= 1000

(i) True. 

(ii) True. 

(ii) False.

(2 + √2)² 

= 2² + (√2)² + 2 x 2 x √2  [using identity, (a+b)² = a² + b² + 2ab]

= 4 +  2 + 4√2

= 6 + 4√2 

Here, 6 is a rational number but 4√2 is an irrational number.  Since, the sum of a rational and an irrational number is an irrational number, therefore, (2+ √2)² is an irrational number.

False.

The successor of a whole number is the number obtained by adding 1 to it.

Example: – consider 3-digit number 999 it is a one digit, then its successor = 999 + 1 = 1000.

(i) \(\sqrt4 = 2 = \frac{2}{1}\)

\(\sqrt4\) can be written in the form of \(\frac{p}q\), so it is a rational number. Its decimal expansion is 2.0

(ii) \(3\sqrt{18}\) \(= 3\sqrt{{2 \times 3\times 3}}\)

\(=3 \times 3 \sqrt2\)

\(= 9\sqrt2\)

Since, the product of a rational and an irrational is an irrational number. 

Therefore,\(9\sqrt2\) is an irrational;

\(3\sqrt{18}\)  is an irrational number

(iii) We have,

\(\sqrt{1.44}\) \(= \frac{12}{10}\)

= 1.2

Every terminating decimal is a rational number, so 1.2 is a rational number. 

(iv) we have,

\(\sqrt\frac{9}{27}\) \(= \frac{3}{\sqrt{27}}\) \(\frac{3}{\sqrt{3\times 3\times 3}}\)

\(= \frac{1}{3}\)

Quotient of a rational and an irrational number is irrational number. Therefore, it is an irrational number.

(v) -\(\sqrt{64}\) \(= - \sqrt{8\times8}\)

= - 8 = -8/1

AS it can be expressed in the form of \(\frac{p}q,\) so it is a rational number

(vi) \(\sqrt{100}\) = 10 = \(\frac{10}{1}\)

Thus it can be expressed in the form of \(\frac{p}q,\) so it is a rational number.

false.

The number which comes immediately before a particular number is called its predecessor.

Example: – consider 2-digit number 10, then its predecessor = 10 – 1 = 9.

A number which can neither be expressed as a terminating decimal nor as a repeating decimal is called an irrational number. For example, 

0.33033003300033… 

On the other hand, every rational number is expressible either as a terminating decimal or as a repeating decimal. For example, 3.24̅and 6.2876 are rational numbers.

400 is the predecessor of 401.

78787 rounded off to the nearest hundreds = 78800

95833 rounded off to the nearest hundreds = 95800

The estimated increase in population = 95800 – 78800 = 17000

758 can be rounded off to 800 and 6784 can be rounded off to 7000

∴ Estimated product = 800 × 7000 = 5600000

(B) 998000

The number which comes immediately before a particular number is called its predecessor.

The successor of a whole number is the number obtained by adding 1 to it.

So, Successor of 999 = 999 + 1 = 1000

Predecessor = 999 – 1 = 998

Then, product of successor and predecessor of 999 is = 1000 × 998

= 998000

Number of shirts produced by the factory = 216315

Number of trousers produced by the factory =182736

Number of jackets produced by the factory = 58704

Total production of the factory = 216315 + 182736 + 58704 = 457755

Since, successor of 32 is 33, predecessor of 49 is 48, predecessor of the predecessor of 56 is 54 and successor of the successor of 67 is 69.

∴ The required sum = 33 + 48 + 54 + 69 = 204

Quantity of fruit juice in a vessel = 13 L 200 mL = (13 × 1000 + 200) mL = 13200 mL

Capacity of one glass = 60 mL

∴ The required number of glasses = 13200 ÷ 60 = 220

Therefore, 220 glasses can be filled by fruit juice.