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101.

Number of even numbers between 58 and 80 is(A) 10 (B) 11 (C) 12 (D) 13

Answer»

(A) 10

Even numbers between 58 and 80 are 60, 62, 64, 66, 68, 70, 72, 74, 76, 78.

102.

The factors of a number are 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80. What is the number?(a) 80(b) 100(c) 128(d) 160

Answer»

(a) 80

The number is 80.

103.

Fill in the blanks to make the statements true.2 is the only _____ number which is even

Answer»

2 is the only Prime number which is even.

104.

The numbers having more than two factors are called _____ numbers. 

Answer»

The numbers having more than two factors are called composite numbers. 

105.

Fill in the blanks to make the statements true.A number for which the sum of all its factors is equal to twice the number is called a _____ number.

Answer»

A number for which the sum of all its factors is equal to twice the number is called a perfect number.

106.

Fill in the blanks to make the statements true.The numbers having more than two factors are called _____ numbers.

Answer»

The numbers having more than two factors are called Composite numbers.

107.

Factors pairs of 60 by division method.

Answer»

We can find the factor in pairs, by multiplying two numbers in a combination to get the result as 60.

Let us see the factor pairs of 60.

1 × 60 = 60

2 × 30 = 60

3 × 20 = 60

4 × 15 = 60

5 × 12 = 60

6 × 10 = 60

Therefore, the factors pairs of 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12) and (6, 10).

108.

Between which two whole numbers the given numbers lie? (i) 0.2 (ii) 3.4 (iii) 3.9 (iv) 2.7 (v) 1.7 (vi) 1.3

Answer»

(i) 0.2 lies between 0 and 1. 

(ii) 3.4 lies between 3 and 4. 

(iii) 3.9 lies between 3 and 4. 

(iv) 2.7 lies between 2 and 3.

(v) 1.7 lies between 1 and 2. 

(vi) 1.3 lies between 1 and 2.

109.

Between which two whole numbers, the following decimal numbers lie? (i) 3.3 (ii) 2.5 (iii) 0.9

Answer»

(i) 3.3 – 3.3 lies between 3 and 4. 

(ii) 2.5 – 2.5 lies between 2 and 3. 

(iii) 0.9 – 0.9 lies between 0 and 1.

110.

Write the decimal numbers represented by the points P, Q, R and S on the given number line.

Answer»

The unit length between 1 and 2 is divided into 10 equal parts and the third part is taken as Q. 

∴ Q represents 1 + 0.3 = 1.3 

The unit lenth between 3 and 4 is divided into 10 equal parts and the 6th part is taken as P. 

∴ P represents 3 + 0.6 = 3.6 

The unit length between 4 and 5 is divied into 10 equal parts and the second part is taken as S. 

∴ S represents 4 + 0.2 = 4.2 

The unit length between 6 and 7 is divided into 10 equal parts and the 8th part is taken.

∴ R represents 6 + 0.8 = 6.8 

P(3.6), Q(1.3), R(6.8), S(4.2).

111.

Which is greater 28 km or 42.6 km.

Answer»

Comparing the whole number part 42 > 28. 

42.6 km is greater than 28 km.

112.

Which is greater? (i) 0.888 (or) 0.28 (ii) 23.914 (or) 23.915

Answer»

(i) 0.888 (or) 0.28 

The whole number parts is equal for both the numbers. 

Comparing the digits in the tenths place we get, 8 > 2. 

0.888 > 0.28 

∴ 0.888 is greater. 

(ii) 23.914 or 23.915 

The whole number part is equal in both the numbers.

Also the tenth place and hundredths place are also equal.

∴ Comparing the thousandths place, we get 5 > 4. 

23.915 > 23.914 

∴ 23.915 is greater.

113.

Find the difference between the largest number of seven digits and the smallest number of eight digits.

Answer»

The smallest number of eight digits = 10000000

The largest number of seven digits = 9999999

The required difference = 10000000 – 9999999 = 1

114.

In a 25 m swimming competition, the time taken by 5 swimmers A, B, C, D and E are 15.7 seconds, 15.68 seconds, 15.6 seconds, 15.74 seconds and 15.67 seconds respectively. Identify the winner.

Answer»

The winner is one who took less time for swimming 25 m. 

Comparing the time taken by A, B, C, D, E the whole number part is equal for all participants. 

Comparing digit in tenths place we get 6 < 7. 

∴ Comparing 15.68, 15.6, 15.67, that is comparing the digits in hundredths place we get

15.60 < 15.67 < 15.68 

One who took 15.6 seconds is the winner. 

∴ C is the winner.

115.

A mobile number consists of ten digits. First four digits are 9,9, 7 and 9. Make the smallest mobile number by using only one digit twice from 8, 3, 5, 6, 0.

Answer»

A mobile number consists of 10 digits.

If the first four digits are 9, 9, 7 and 9.

Thus, the smallest mobile number by using only one digit twice from 8, 3, 5, 6, 0 is 9979003568.

116.

A mobile number consists of ten digits. The first four digits of the number are 9, 9, 8 and 7. The last three digits are 3, 5 and 5. The remaining digits are distinct and make the mobile number, the greatest possible number. What are these digits?

Answer»

A mobile number consists of 10 digits.

If the first four digits of the number are 9, 9, 8 and 7 and the last three digits of the number are 3, 5 and 5.

Thus, for the greatest possible number, the remaining distinct digits are 6, 4 and 2.

117.

Fill in the blanks to make the statements true.9128 × _____ = 9128

Answer»

9128 x 1 = 9128

118.

Fill in the blanks to make the statements true.125 + (68+17) = (125 + _____ ) + 17

Answer»

125 + (68+17) = (125 + 68) + 17

119.

Find the product using suitable properties: 628×102

Answer»

628 × 102

= 628 × (100 + 2)

= 628 × 100 + 628 × 2

= 62800 + 1256

= 64056

120.

A baker uses 3.924 kg of sugar to bake 10 cakes of equal size. How much sugar is used in each cake?

Answer»

For 10 cakes sugar required = 3.924 kg 

For 1 cake sugar required = 3.924 ÷ 10 

\(\frac{3.924}{10}\)

= 0.3924 kg 

For 1 cake sugar required = 0.3924 kg.

121.

Which is greater ? (i) 537 or 98 (ii) 2428 or 529 (iii) 2, 59, 467 or 10, 35, 729

Answer»

(i) 537 or 98 

Since 537 is three digit number and 98 is two digit number. 

Hence 537 > 98 and 537 is greater 

(ii) 2428 or 529 

Since 2498 is four digit number and 529 is three digit number. 

2498 > 529 ; 2498 is greater 

(iii) 2, 59, 467 or 10, 35, 729 

Since 10, 35, 729 is seven digit number and 2, 59, 467 is six digit number 

10, 35, 729 > 2, 59, 467 ; 10, 35, 729 is greater.

122.

Number of whole numbers between 38 and 68 is(A) 31 (B) 30 (C) 29 (D) 28

Answer»

(c) 29

Number of whole numbers between 38 and 68 is 29.

123.

The product of a non-zero whole number and its successor is always(A) an even number (B) an odd number(C) a prime number (D) divisible by 3

Answer»

(A) an even number

The product of a non-zero whole number and its successor is always an even number.

For example: – 4 × 5 = 20, 7 × 8 = 56

124.

Fill in the blanks to make the statements true._____ is the successor of the largest 3 digit number

Answer»

1000

We know that Successor of any no. is 1 greater than the number. The largest 3 digit number is 999. Therefore, its successor is 999 + 1 = 1000.

125.

Give an example of each, of two irrational numbers whose: (i) Difference is a rational number.(ii) Difference is an irrational number.(iii) Sum is a rational number.(iv) Sum is an irrational number.(v) Product is a rational number.(vi) Product is an irrational number.(vii) Quotient is a rational number.(viii) Quotient is an irrational number.

Answer»

(i) \(\sqrt3\) is an irrational number.

Now,\((\sqrt3)-(\sqrt3)\) = 0

0 is the rational number. 

(ii) Let two irrational numbers are \(5\sqrt{2}\) and \(\sqrt2\)

Now, \((5\sqrt{2})-(\sqrt2)\) = \(4\sqrt{2}\)

\(4\sqrt{2}\) is an irrational number.

(iii) Let two irrational numbers be \(\sqrt{11}\) and \(-\sqrt{11}\)

Now,\((\sqrt11)+(\sqrt11)\) = 0

0 is a rational number 

(iv) Let two irrational numbers are \(4\sqrt6\) and \(\sqrt6\)

Now,\((4\sqrt6)+(\sqrt6)\) = \(5\sqrt6\)

\(5\sqrt6\) is an irrational number. 

(v) Let two irrational numbers are \(2\sqrt3\) and \(\sqrt3\)

Now, \(2\sqrt3\times\sqrt3\) = \(2\times3\)

= 6 

6 is a rational number. 

(vi) Let two irrational numbers are \(\sqrt2\) and \(\sqrt5\)

Now, \(\sqrt2\times\sqrt5\) = \(\sqrt10\)

\(\sqrt10\) is a irrational number.

(vii) Let two irrational numbers are \(3\sqrt6\) and \(\sqrt6\)

Now, \(\frac{3\sqrt6}{\sqrt6}\) = 3

3 is a rational number.

(viii) Let two irrational numbers are \(\sqrt6\) and \(\sqrt2\)

Now, \(\frac{\sqrt6}{\sqrt2}\) = \(\frac{\sqrt3\times\sqrt2}{\sqrt{2}}\)

\(= \sqrt3\)

\(\sqrt3\) is an irrational number.

126.

Examine, whether the following numbers are rational or irrational:(i) √7(ii) √4(iii) 2 + √3(iv) √3 + √2(v) √3 + √5(vi) (√2-2)2(vii) (2-√2)(2+√2)(viii) (√3 + √2)2(ix) √5 - 2(x) √23(xi) √225(xii) 0.3796(xiii) 7.478478.....(xiv) 1.101001000100001.....

Answer»

(i) √7 

Not a perfect square root, so it is an irrational number.

(ii) √4

A perfect square root of 2.

We can express 2 in the form of 2/1, so it is a rational number. 

(iii) 2 + √3

Here, 2 is a rational number but √3 is an irrational number 

Therefore, the sum of a rational and irrational number is an irrational number. 

(iv) √3 + √2 

√3 is not a perfect square thus an irrational number. 

√2 is not a perfect square, thus an irrational number. 

Therefore, sum of √2 and √3 gives an irrational number. 

(v) √3 + √5

√3 is not a perfect square and hence, it is an irrational number

Similarly, √5 is not a perfect square and also an irrational number.

Since, sum of two irrational number, is an irrational number, therefore √3 + √5 is an irrational number.

(vi) (√2 – 2)2 

(√2 – 2)2 

= 2 + 4 – 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.

(vii) (2 – √2)(2 + √2)

We can write the given expression as; 

(2 – √2)(2 + √2) = ((2)2 − (√2)2

[Since, (a + b)(a – b) = a2 – b2

= 4 – 2

= 2 or 2/1

Since, 2 is a rational number, therefore, (2 – √2)(2 + √2) is a rational number.

(viii) (√3 + √2)2 

We can write the given expression as; 

(√3 + √2)2 

= (√3)2 + (√2)2 + 2√3 x √2 

= 3 + 2 + 2√6 

= 5 + 2√6 

[using identity, (a+b)2 = a2 + 2ab + b2

Since, the sum of a rational number and an irrational number is an irrational number, therefore, (√3 + √2)2 is an irrational number. 

(ix) √5 – 2 

√5 is an irrational number whereas 2 is a rational number. 

The difference of an irrational number and a rational number is an irrational number. 

Therefore, √5 – 2 is an irrational number. 

(x) √23 

Since, √23 = 4.795831352331… 

As decimal expansion of this number is non-terminating and non-recurring therefore, it is an irrational number. 

(xi) √225 

√225 = 15 or 15/1 

√225 is rational number as it can be represented in the form of p/q and q not equal to zero.

(xii) 0.3796

As the decimal expansion of the given number is terminating, therefore, it is a rational number. 

(xiii) 7.478478…… 

As the decimal expansion of this number is non-terminating recurring decimal, therefore, it is a rational number. 

(xiv) 1.101001000100001……

As the decimal expansion of given number is non-terminating and non-recurring, therefore, it is an irrational number.

127.

In the following equations, find which variables x, y, z etc. represent rational or irrational numbers:(i) x2 = 5 (ii) y2 = 9 (iii) z2 = 0.04 (iv) u2 = \(\frac{17}{4}\)(v) v2 = 3 (vi) w2 = 27(vii) t2 = 0.4

Answer»

(i) We have,

x2 = 5 

Taking square root on both sides,

\(= \sqrt{x}^2 = \sqrt5\)

= x = \(\sqrt5\)

\(\sqrt5\)

is not a perfect square root, so it is an irrational number.

(ii) We have,

y2 = 9 

y = \(\sqrt9\)

= 3 = \(\frac{3}{1}\)

\(\sqrt9\) can be expressed in the form of \(\frac{p}q\) , so it is a rational number.

(iii) We have, 

z2 = 0.04 

Taking square root on both the sides, we get,

\(\sqrt{z}^2\) = \(\sqrt{0.04}\)

\(z = \sqrt{0.04}\)

\(= 0.2 = \frac{2}{10}\)

\(= \frac{1}{5}\)

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

(iv) We have,

\(u^2 = \frac{17}{4}\)

Taking square root on both the sides, we get

\(\sqrt{u}^2\) \(= \frac{\sqrt{17}}{\sqrt4}\)

u = \(\frac{\sqrt{17}}{\sqrt2}\)

Quotient of an rational number is irrational, so u is an irrational number.

(v) We have,

v 2 = 3 

Taking square roots on both the sides, we get,

\(\sqrt{v}^2\) = \(\sqrt3\)

v = \(\sqrt3\)

\(\sqrt3\) is not a perfect square root, so v is an irrational number.

(vi) We have,

w2 = 27 

Taking square roots on both the sides, we get,

\(\sqrt{w}^2\) = \(\sqrt{27}\)

w = \(\sqrt3\times\sqrt3\times\sqrt3\) = \(\sqrt[3]{3}\)

Product of a rational number and an irrational number is irrational number. So, it is an irrational number.

(vii) We have,

t2 = 0.4 

Taking square roots on both the sides, we get,

\(\sqrt{t}^2\) = \(\sqrt{0.4}\) = \(\frac{\sqrt4}{\sqrt{10}}\)

\(= \frac{2}{\sqrt{10}}\)

Since, quotient of a rational number and an irrational number is irrational number, so t is an irrational number.

128.

Define an irrational number.

Answer»

A number which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. It is non-terminating or non-repeating decimal.

129.

State whether the given statement are true (T) or false (F).Between any two natural numbers, there is one natural number.

Answer»

False.

Consider the two natural numbers 4 and 8.

Then, natural numbers between 4 and 8 are 5, 6, 7.

130.

State whether the given statement are true (T) or false (F).Every whole number has its predecessor.

Answer»

False.

Consider the whole number 0,

Then, its predecessor = 0 – 1 = -1

– 1 is an integer.

131.

Estimate each of the following products by rounding off each number to nearest tens: (a) 87 × 32 (b) 311×113 (c) 3239 × 28 (d) 1385 × 789

Answer»

(a) 2700 

(b) 34100 

(c) 97200 

(d) 1098100

132.

State whether the given statement are true (T) or false (F).The HCF of two numbers is smaller than the smaller of the numbers.

Answer»

False.

The HCF of two numbers is either greater than or equal to the smaller of the numbers.

133.

Between two rational numbers(A) there is no rational number(B) there is exactly one rational number(C) there are infinitely many rational numbers(D) there are only rational numbers and no irrational numbers

Answer»

(C) there are infinitely many rational numbers

Explanation:

Between two rational numbers there are infinitely many rational number.

Hence, (C) is the correct option.

134.

Examine, whether the following numbers are rational or irrational:(i) \(\sqrt{7}\)(ii) \(\sqrt{4}\)(iii) \(2+\sqrt{3}\)(iv) \(\sqrt{3}+\sqrt{2}\)(v) \(\sqrt{3}+\sqrt{5}\)(vi) \((\sqrt{2}-2)^2\)(vii) \((2-\sqrt{2})(2+\sqrt{2})\)(viii) \((\sqrt{2}+\sqrt{3})^2\)(ix) \(\sqrt{5}-2\)(x) \(\sqrt{23}\)(xi) \(\sqrt{225}\)(xii) 0.3796 (xiii) 7.478478 (xiv) 1.101001000100001…..

Answer»

(i) \(\sqrt7\) is not a perfect square root, so it is an irrational number.

(ii) We have,

\(\sqrt4 = 2 = \frac{2}{1}\)

\(\sqrt4\) can be expressed in the form of \(\frac{p}q,\) so it is a rational number.

The decimal expression of \(\sqrt4\) is 2.0

(iii) 2 ia a rational number, whereas \(\sqrt3\) is an irrational number. Because, sum of a rational number and an irrational number is an irrational number, so 2 + is an irrational number

(iv) \(\sqrt2\) is an irrational number. Also \(\sqrt3\) is an irrational number. The sum of two irrational numbers is irrational.

Therefore, \(\sqrt{3}+\sqrt{2}\) is an irrational number

(v) \(\sqrt5\) is an irrational number. Also, \(\sqrt3\) is an irrational number. The sum of two irrational numbers is irrational.

Therefore, \(\sqrt3+5\) is an irrational number.

(vi) We have,

\((\sqrt{2}-\sqrt{2})^2\) \(= (\sqrt{2})^2-2\times\sqrt2{\times2}+({2})^2\)

\(= 2-4\sqrt2{+4}\)

\(= 6 - 4\sqrt2\)

Now 6 is a rational number, whereas \(4\sqrt2\) is an irrational number

The difference of a rational number and an irrational number is an irrational number. So, it is an irrational number.

(vii) We have,

\((2-\sqrt2)(2+\sqrt2)\)

\(({2})^2-(\sqrt2)^2\) [Therefore, (a – b) (a + b) = a2 – b2]

\(= 4 - 2 = 2 =\frac{2}{1}\)

Since 2 is a rational number

Therefore, \((2-\sqrt2)(2+\sqrt2)\) is a rational number

(viii) We have,

\((\sqrt2+\sqrt3)^2\) \(=( \sqrt{2})^2+2 \times\sqrt2\times\sqrt3+(\sqrt3)^2\)

\(= 2 +2\sqrt6+3\)

\(= 5+ 2\sqrt6\)

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

(ix) The difference of a rational number and an irrational number is an irrational number.

Therefore, \(5 - \sqrt2\) is an irrational number

(x) \(\sqrt23\) = 4.79583152331….

Therefore, it is an irrational number

(xi) \(\sqrt225\) = 15 = \(\frac{15}{1}\)

Therefore, it is a rational number as it is represented in the form of \(\frac{p}q\), where q ≠ 0

(xii) 0.3796, as a decimal expansion of this number is terminating, so it is an irrational number.

(xiii) 7.478478….\(= 7.4\overline78\)

As, decimal expansion of this number is non – terminating recurring so it is a rational number

 (xiv) 1.101001000100001….. 

It is an irrational number

135.

Express each of the following decimals in the form \(\frac{p}q\);(i) 0.\(\overline4\)(ii) \(0.\overline{37}\)(iii) \(0.\overline{54}\)(iv) \(0.\overline{621}\)(v) \(125.\overline3\)(vi) \(4.\overline7\)(vii) \(0.4\overline7\)

Answer»

(i) Let x =\(0.\overline4\)

Now, x = \(0.\overline4\) = 0.444… (1)

Multiplying both sides of equation (1) by 10, we get,

10x = 4.444… (2)

Subtracting equation (1) by (2)

10x – x = 4.444… - 0.444…

\(9\times = 4\)

x = \(\frac{4}{9}\)

Hence, \(0.\overline4 = \frac{4}{9}\)

(ii) Let x = \(0.\overline{37}\)

Now, x = 0.3737…. (1) 

Multiplying equation (1) by 10 

10x = 3.737…. (2) 

Multiplying equation (2) by 10 

100x = 37.3737…. (3) 

100x – x = 37 

99x = 37

\(x = \frac{37}{99}\)

Hence, \(0.\overline{37}\) \(= \frac{37}{99}\)

(iii) Now x \(= 0.\overline{54}\)

= 0.5454… (i) 

Multiplying both sides of equation (i) by 100, we get 

100x = 54.5454…. (ii) 

Subtracting (i) by (ii), we get 

100x – x = 54.5454…. – 0.5454…. 

99x = 54

\(x = \frac{54}{99}\)

(iv) Now x \(= 0\overline{621}\)

= 0.621621…. (i) 

Multiplying both sides by 1000, we get 

1000x = 621.621621…. (ii) 

Subtracting (i) by (ii), we get 

1000x – x = 621.621621…. – 0.621621…. 

999x = 621

\(x = \frac{621}{999}\) \(= \frac{23}{37}\)

(v) Now x = \(125.\overline3\)

 = 125.3333…. (i)

Multiplying both sides of equation (i) by 10, we get 

10x = 1253.3333…. (ii) 

Subtracting (i) by (ii), we get 

10x – x = 1253.3333…. – 125.3333…. 

9x = 1128 

x = 1128 / 9 = 376/3 

(vi) Now x \(= 4.\overline7\)

= 4.7777…. (i) 

Multiplying both sides of equation (i) by 10, we get 

10x = 47.7777…. (ii) 

10x – x = 47.7777…. – 4.7777….

\(9\times = 43\)

\(x = \frac{43}{9}\)

(vii) Now, x = \(0.4\overline7\)

 = 0.47777…. 

Multiplying both sides by 10, we get 

10x = 4.7777…. (i) 

Multiplying both sides of equation (i) by 10, we get 

100x = 47.7777…. (ii) 

Subtracting (i) from (ii), we get 

100x – 10x = 47.7777…. – 4.7777….

\(90\times = 43\)

\(= \frac{43}{90}\)

136.

Estimate each of the following by rounding off each number to nearest tens: (a) 11963 – 9369 (b) 76877 – 7783 (c) 10732 – 4354 (d) 78203 – 16407

Answer»

(a) 2590 

(b) 69100 

(c) 6380 

(d) 61790

137.

Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:(i) √4(ii) 3√18(iii) √1.44(iv) √9/27(v) -√64(vi) √100

Answer»

(i) √4 

√4 = 2, which can be written in the form of a/b. Therefore, it is a rational number. Its decimal representation is 2.0.

(ii) 3√18 

3√18 = 9√2 

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

Therefore, 3√18 is an irrational. 

Or 3 x √18 is an irrational number. 

(iii) √1.44 

√1.44 = 1.2 

Since, every terminating decimal is a rational number, Therefore, √1.44 is a rational number. 

And, its decimal representation is 1.2. 

(iv) √9/27

√9/27 = 1/√3 

Since, we know, quotient of a rational and an irrational number is irrational numbers, therefore, √9/27 is an irrational number. 

(v) – √64 

– √64 = – 8 or – 8/1 

Therefore, – √64 is a rational number. 

Its decimal representation is –8.0. 

(vi) √100 

√100 = 10 

Since, 10 can be expressed in the form of a/b, such as 10/1, 

Therefore, √100 is a rational number.

And it’s decimal representation is 10.0.

138.

Decimal representation of a rational number cannot be(A) terminating(B) non-terminating(C) non-terminating repeating(D) non-terminating non-repeating

Answer»

(D) non-terminating non-repeating

Explanation:

The decimal representation of a rational number cannot be non-terminating and non- repeating.

Hence, (D) is the correct option

139.

Define an irrational number

Answer»

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

For example, 

1.01001000100001…

140.

State whether the given statement are true (T) or false (F).Every whole number has its successor.

Answer»

True.

Every whole number has its successor.

141.

State whether the given statement are true (T) or false (F).HCF of two or more numbers may be one of the numbers.

Answer»

True

HCF of two or more numbers may be one of the numbers.

142.

Which of the following statement is true? A. Product of two irrational numbers is always irrational. B. Product of a rational and an irrational number is always irrational. C. Sum of two irrational numbers can never be irrational. D. Sum of an integer and a rational number can never be an integer.

Answer»

Example: Take any rational number other than Zero let’s take 2 as rational number and√2 as irrational number than product as:

\(2\times\sqrt2\) = \(2\sqrt2\)

\(2\sqrt2\) is irrational number so if we can say that if rational number other than zero product with any irrational number the result is also irrational.

143.

Which of the following is irrational?A. \(\sqrt\frac{4}{9}\)B. \(\frac{4}5\)C. \(\sqrt7\)D. \(\sqrt{81}\)

Answer»

Proof: let us assume that √7 be rational. 

then it must in the form of p / q [q ≠ 0] [p and q are co-prime] 

√7 = p / q 

√7 x q = p 

squaring on both sides 

7q= p2 (i) 

p 2 is divisible by 7 

p is divisible by 7 

p = 7c [c is a positive integer] [squaring on both sides ] 

p 2 = 49 c2 (ii) 

Substitute p2 in eq (i), we get, 

7q2 = 49 c2 

q 2 = 7c2 

q is divisible by 7 

Thus q and p have a common factor 7. 

There is a contradiction 

As our assumption p & q are co - prime but it has a common factor. 

So that √7 is an irrational.

144.

State whether the given statement are true (T) or false (F).Every whole number is the successor of another whole number.

Answer»

False.

We know that, 0 is the whole number.

0 is no the successor of another whole number.

145.

The product of a non-zero whole number and its successor is always divisible by (A) 2 (B) 3 (C) 4 (D) 5

Answer»

The correct option is (A) 2.

In these two numbers obviously one no. will be odd and another will be even and (Even number) × (odd number) = Even no. ...

Hence will be divisible by 2.

146.

3 × 10000 + 0 × 1000 + 8 × 100 + 0 × 10 + 7 × 1 is same as (A) 30087 (B) 30807 (C) 3807 (D) 3087

Answer»

The correct option is (B) 30807. 

3 × 10000 + 0 × 1000 + 8 × 100 + 0 × 10 + 7 × 1

= 30000 + 0 + 800 + 0 + 7

= 30807

147.

By adding 1 to the greatest ________ digit number we get 1 lakh.

Answer»

By adding 1 to 99999 we get 1 lakh.

99999 is the greatest 5 digit no. Hence,

By adding 1 to the greatest 5 digit number, we get 1 lakh.

148.

State whether the given statement are true (T) or false (F).Natural numbers are not closed under multiplication.

Answer»

False.

We know that, multiplication of two natural numbers is always natural number.

Therefore, natural numbers are closed under multiplication.

149.

In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:(i) x2 = 5(ii) y2 = 9(iii) z2 = 0.04(iv) u2 = 17/14(v) v2 = 3(vi) w2 = 27(vii) t2 = 0.4

Answer»

(i) x2 = 5 

Taking square root both the sides, 

x = √5 

√5 is not a perfect square root, so it is an irrational number. 

(ii) y2 = 9 

y2 = 9 or y = 3 

3 can be expressed in the form of p/q, such as 3/1, so it a rational number. 

(iii) z2 = 0.04 

z2 = 0.04 

Taking square root both the sides, we get 

z = 0.2 

0.2 can be expressed in the form of p/q such as 2/10, so it is a rational number. 

(iv) u2 = 17/4 

Taking square root both the sides, we get 

u = √17/2

We know that, quotient of an irrational and a rational number is irrational, therefore, u is an Irrational number. 

(v) v2 = 3 

Taking square root both the sides, we get 

v = √3 

Since, √3 is not a perfect square root, so v is irrational number. 

(vi) w2 = 27 

Taking square root both the sides, we get 

w = 3√3 

We know that, the product of a rational and irrational is an irrational number. Therefore, w is an irrational number. 

(vii) t2 = 0.4 

Taking square root both the sides, we get 

t = √(4/10) 

t = 2/√10 

We know that, quotient of a rational and an irrational number is irrational number. Therefore, t is an irrational number.

150.

State whether the given statement are true (T) or false (F).Addition is commutative for natural numbers.

Answer»

True.

Let us assume ‘a’ and ‘b’ are the two natural numbers.

Then commutative for natural numbers is a + b = b + c.

Consider the two natural numbers 2 and 4.

Where, a = 2, b = 4

a + b = b + c

2 + 4 = 4 + 2

6 = 6