

InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
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. |
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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. |
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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. |
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104. |
The numbers having more than two factors are called _____ numbers. |
Answer» The numbers having more than two factors are called composite numbers. |
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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. |
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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. |
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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). |
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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. |
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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. |
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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). |
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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. |
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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. |
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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 |
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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. |
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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. |
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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. |
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117. |
Fill in the blanks to make the statements true.9128 × _____ = 9128 |
Answer» 9128 x 1 = 9128 |
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118. |
Fill in the blanks to make the statements true.125 + (68+17) = (125 + _____ ) + 17 |
Answer» 125 + (68+17) = (125 + 68) + 17 |
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119. |
Find the product using suitable properties: 628×102 |
Answer» 628 × 102 = 628 × (100 + 2) = 628 × 100 + 628 × 2 = 62800 + 1256 = 64056 |
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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. |
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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. |
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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. |
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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 |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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 |
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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. |
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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. |
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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 |
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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…. \(90\times = 43\) x \(= \frac{43}{90}\) |
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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 |
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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. |
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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 |
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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… |
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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. |
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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. |
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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. |
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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 7q2 = 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. |
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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. |
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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. |
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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 |
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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. |
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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. |
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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. |
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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 |
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