6 NCERT EXEMPLAR PROBLEMS-SOLUTIONS

Mathematics Dr. Subhendu Chakroborty Ph.D., AMRSC

Full Marks Pvt Ltd (Progressive Educational Publishers)

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Note from the Publisher National Council of Educational Research and Training (NCERT) developed Exemplar Problems in Science and Mathematics. The prime objective is to provide the students with number of quality problems to facilitate the concept of learning. Easy Marks NCERT Solutions to Exemplar Problems Mathematics-VI is mainly based on the idea to present the considerable requirements of the Exemplar Problems in a simple and detailed manner. Salient features of the book: ● Scientific and methodological solutions to the textual questions are provided. ● Multiple Choice Questions (MCQs) with explanation for understanding the concept better. ● The explanation of the answers are provided with diagram, wherever needed. ● Very Short, Short and Long Answer Type Questions are given to provide students with more practical problems. It has always been our endeavour to provide better quality material to the students. If there are any suggestions for the betterment of the book, we will certainly try to incorporate them.

CONTENTS 1. Number System................................................................................. 5 2. Geometry........................................................................................... 25 3. Integers.............................................................................................. 37 4. Fractions and Decimals..................................................................... 44 5. Data Handling................................................................................... 59 6. Mensuration...................................................................................... 77 7. Algebra.............................................................................................. 90 8. Ratio and Proportion......................................................................... 100 9. Symmetry and Practical Geometry................................................... 114

1

Number System EXERCISE

In questions 1 to 38, out of the four options, only one is correct. Write the correct answer: Q1. The product of the place values of two 2’s in 428721 is (a) 4 (b) 40000 (c) 400000 (d) 40000000 Sol. (c): Place values of two 2‘s in 428721 are 20,000 and 20. \ Product of 20,000 and 20 = 20,000 × 20 = 4,00,000 Q2. 3 × 10000 + 7 × 1000 + 9 × 100 + 0 ×10 + 4 is the same as: (a) 3794 (b) 37940 (c) 37904 (d) 379409 Sol. (c): 3 × 10000 + 7 × 1000 + 9 × 100 + 0 × 10 + 4 = 30000 + 7000 + 900 + 0 + 4 = 37904 Q3. If 1 is added to the greatest 7- digit number, it will be equal to: (a) 10 thousand (b) 1 lakh (c) 10 lakh (d) 1 crore Sol. (d): The greatest 7-digit number is 9999999. 9999999+1=10000000 Q4. The expanded form of the number 9578 is: (a) 9 × 10000 + 5 × 1000 + 7 × 10 + 8 × 1 (b) 9 × 1000 + 5 × 100 + 7 × 10 + 8 × 1 (c) 9 × 1000 + 57 × 10 + 8 × 1 (d) 9 × 100 + 5 × 100 + 7 × 10 + 8 × 1 Sol. (b): 9578 = 9 × 1000 + 5 × 100 + 7 × 10 + 8 × 1 Q5. When rounded off to nearest thousands, the number 85642 is (a) 85600 (b) 85700 (c) 85000 (d) 86000 Sol. (d): 85642 is closer to 86000. Q6. The largest 4-digit number, using any one digit twice, from digits 5,9, 2 and 6 is: (a) 9652 (b) 9562 (c) 9659 (d) 9965 Sol. (d): The largest 4-digit number, using any one digit twice, from digits 5, 9, 2 and 6 is 9965. Q7. In Indian System of Numeration, the number 58695376 is written as: (a) 58,69, 53, 76 (b) 58,695,376 (c) 5,86,95,376 (d) 586,95,376 Sol. (c): In Indian System of Numeration, the number 58695376 is written as 5, 86, 95,376. Q8. One million is equal to: (a) 1 lakh (b) 10 lakh (c) 1 crore (d) 10 crore Sol. (b): 1 million is equal to 10 lakh.

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Q9. The greatest number which on rounding off to nearest thousands gives 5000, is: (a) 5001 (b) 5559 (c) 5999 (d) 5499 Sol. (d): Numbers 1 to 499 are nearer to 0 than to 1000, so these numbers are rounded off as 0. Hence, the greatest number that would be rounded off to 5000 is 5499. Q10. Keeping the place of 6 in the number 6350947 same, the smallest number obtained by rearranging other digits is: (a) 6975430 (b) 6043579 (c) 6034579 (d) 6034759 Sol. (c): Keeping the place of 6 in the number 6350947 same, the smallest number obtained by rearranging other digits is 6034579. Q11. Which of the following numbers in Roman numerals is incorrect? (a) LXXX (b) LXX (c) LX (d) LLX Sol. (d): LLX is not a correct Roman number. Q12. The largest 5-digit number having three different digits is: (a) 98978 (b) 99897 (c) 99987 (d) 98799 Sol. (c): The largest 5-digit number having three different digits is 99987. Q13. The smallest 4-digit number having three different digits is: (a) 1102 (b) 1012 (c) 1020 (d) 1002 Sol. (d): The smallest 4-digit number having three different digits is 1002. Q14. Number of whole numbers between 38 and 68 is: (a) 31 (b) 30 (c) 29 (d) 28 Sol. (c): Number of whole numbers between 38 and 68 is 29. Q15. The product of successor and predecessor of 999 is: (a) 999000 (b) 998000 (c) 989000 (d) 1998 Sol. (b): Successor of 999 is 1000. Predecessor of 999 is 998. Their product = 998 × 1000 = 998000. Q16. 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 Sol. (a): The product of a non-zero whole number and its successor is always an even number. For example product of 1 and 2 is 2 which is even. Product of 7 and 8 is 56 which is even. Q17. A whole number is added to 25 and the same number is subtracted from 25. The sum of the resulting numbers is: (a) 0 (b) 25 (c) 50 (d) 75 Sol. (c): The sum of the resulting numbers is 50.

6 n NCERT Exemplar Problems Mathematics–VI

Q18. Which of the following is not true? (a) (7 + 8) + 9 = 7 + (8 + 9) (b) (7 × 8) × 9 = 7 × (8 × 9) (c) 7 + 8 × 9 = (7 + 8) × (7 + 9) (d) 7 × (8 + 9) = (7 × 8) + (7 × 9) Sol. (c): (a) and (b) are associative law and (d) is distributive law. But (c) is incorrect. Q19. By using dot (.) patterns, which of the following numbers can be arranged in all the three ways namely a line, a triangle and a rectangle? (a) 9 (b) 10 (c) 11 (d) 12 Sol. (b): 10 can be arranged in all the three ways namely a line, a triangle and a rectangle. Q20. Which of the following statements is not true? (a) Both addition and multiplication are associative for whole numbers. (b) Zero is the identity for multiplication of whole numbers. (c) Addition and multiplication both are commutative for whole numbers. (d) Multiplication is distributive over addition for whole numbers. Sol. (b): 1 is the identity for multiplication of whole numbers. Q21. Which of the following statements is not true? (a) 0 + 0 = 0 (b) 0 – 0 = 0 (c) 0 × 0 = 0 (d) 0 ÷ 0 = 0 Sol. (d) Q22. The predecessor of 1 lakh is: (a) 99000 (b) 99999 (c) 999999 (d) 100001 Sol. (b): Predecessor of 1 lakh is 99999. Q23. The successor of 1 million is: (a) 2 millions (b) 1000001 (c) 100001 (d) 10001 Sol. (b): The successor of 1 million is 1000001. Q24. Number of even numbers between 58 and 80 is: (a) 10 (b) 11 (c) 12 (d) 13 Sol. (a): Number of even numbers between 58 and 80 is 10. Q25. Sum of the number of primes between 16 to 80 and 90 to 100 is: (a) 20 (b) 18 (c) 17 (d) 16 Sol. (c): Prime numbers between 16 to 80 are 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 which are sixteen in numbers and prime number between 90 to 100 is only one, which is 97. Therefore sum of the number of primes between 16 to 80 and 90 to 100 is (16 + 1) = 17. Q26. Which of the following statements is not true? (a) The HCF of two distinct prime numbers is 1 (b) The HCF of two co-prime numbers is 1 (c) The HCF of two consecutive even numbers is 2 (d) The HCF of an even and an odd number is even. Sol. (d): The HCF of an even and an odd number is even is not true. Number System  n 7

Q27. The number of distinct prime factors of the largest 4-digit number is: (a) 2 (b) 3 (c) 5 (d) 11 Sol. (b): The largest 4-digit number is 9999. The prime factors of 9999 are 3 × 3 × 11 × 101. Hence, the distinct prime factors of 9999 are 3, 11 and 101. Q28. The number of distinct prime factors of the smallest 5-digit number is: (a) 2 (b) 4 (c) 6 (d) 8 Sol. (a): Smallest five digit number is 10,000. The prime factors of 10,000 are 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5. Hence, the distinct prime factors of 10,000 are 2 and 5. Q29. If the number 7254*98 is divisible by 22, the digit at * is: (a) 1 (b) 2 (c) 6 (d) 0 Sol. (c): The divisibility rule for 22 is that the number should be divisible by 2 and by 11. A number is divisible by 2 if the number ends in 0, 2, 4, 6 or 8 whereas a number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of all the digits in even positions is 0 or divisible by 11. The condition is fulfilled if the blank space is occupied by 6. (7 + 5 + 6 + 8) – (2 + 4 + 9) = 26 – 15 = 11 which is divisible by 11. Q30. The largest number which always divides the sum of any pair of consecutive odd numbers is: (a) 2 (b) 4 (c) 6 (d) 8 Sol. (b): Let us take two consecutive odd numbers to be 11 and 13. Their sum is 24 divisible by 4. Again 1 + 3 = 4 divisible by 4. Therefore, the largest number which always divides the sum of any pair of consecutive odd numbers is 4. Q31. A number is divisible by 5 and 6. It may not be divisible by: (a) 10 (b) 15 (c) 30 (d) 60 Sol. (d): 30 is a number divisible by 5, 6, 10, 15 and 30 but not by 60. Therefore a number may be divisible by 5 and 6 but may not be divisible by 60. Q32. The sum of the prime factors of 1729 is: (a) 13 (b) 19 (c) 32 (d) 39 Sol. (d): The prime factors of 1729 are 7, 13 and 19. Their sum is 39. Q33. The greatest number which always divides the product of the predecessor and successor of an odd natural number other than 1,is: (a) 6 (b) 4 (c) 16 (d) 8 Sol. (d): Let odd number = 3, so product of predecessor and successor = 2 × 4 = 8.

8 n NCERT Exemplar Problems Mathematics–VI

And another odd number = 5. Therefore the product of predecessor and successor = 4 × 6 = 24 To find the greatest number we find HCF of 8 and 24 8 = 2 × 2 × 2 24 = 2 × 2 × 2 × 3 HCF = 2 × 2 × 2 = 8 Q34. The number of common prime factors of 75, 60, 105 is: (a) 2 (b) 3 (c) 4 (d) 5 Sol. (a): Prime factors of 75 = 5 × 5 × 3. Prime factors of 60 = 5 × 3 × 2 × 2. Prime factors of 105 = 5 × 3 × 7. Common factors between these three numbers are 3 and 5. Q35. Which of the following pairs is not coprime? (a) 8, 10 (b) 11, 12 (c) 1, 3 (d) 31, 33 Sol. (a): Two numbers are coprime if they have no factors in common. In case of 8 and 10, factors of 8 are 1, 2, 4 and 8 whereas the factors of 10 are 1, 2, 5 and 10 i.e. in addition to 1 they have 2 in common. Q36. Which of the following numbers is divisible by 11? (a) 1011011 (b) 1111111 (c) 22222222 (d) 3333333 Sol. (c): A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of all the digits in even positions is 0 or divisible by 11. In 22222222 the rule is followed and has 8 – 8 = 0 therefore divisible by 11. Q37. LCM of 10, 15 and 20 is: (a) 30 (b) 60 (c) 90 (d) 180 Sol. (b): LCM is least common multiple of two or more numbers. 5 10, 15, 20 2 2, 3, 4 2 1, 3, 2 3 1, 3, 1 1, 1, 1 i.e. 5 × 2 × 2 × 3 = 60 Q38. LCM of two numbers is 180. Then which of the following is not the HCF of the numbers? (a) 45 (b) 60 (c) 75 (d) 90 Sol. (c): In (a), LCM/HCF = 180/45 = 4 , 4 ∈ N. In (b), LCM/HCF = 180/60 = 3 , 3 ∈ N. In (c), LCM/HCF = 180/75 = 2.4 , 2.4 ∈ N. In (d), LCM/HCF = 180/90 = 2 , 2 ∈ N.



Number System  n 9

The HCF of the numbers divides the LCM completely in each case except in (c). So the answer is 75. In questions 39 to 98 state whether the given statements are true (T) or false (F). Q39. In Roman numeration, a symbol is not repeated more than three times. Sol. (T) Q40. In Roman numeration, if a symbol is repeated, its value is multiplied as many times as it occurs. Sol. (F): If a symbol is repeated, its value is added as many times as it occurs. Q41. 5555 = 5 × 1000 + 5 × 100 + 5 × 10 + 5 × 1 Sol. (T) Q42. 39746 = 3 × 10000 + 9 × 1000 + 7 × 100 + 4 × 10 + 6 Sol. (T) Q43. 82546 = 8 × 1000 + 2 × 1000 + 5 × 100 + 4 × 10 + 6 Sol. (F): 8 × 10000 + 2 × 1000 + 5 × 100 + 4 × 10 + 6 Q44. 532235 = 5 × 100000 + 3 × 10000 + 2 × 1000 + 2 × 100 + 3 × 10 + 5 Sol. (T) Q45. XXIX = 31 Sol. (F): XXIX is equal to 29. Q46. LXXIV = 74 Sol. (T): LXXIV = 50 + 10 + 4 = 74 Q47. The number LIV is greater than LVI. Sol. (F): The number LIV (54) is less than LVI (56). Q48. The numbers 4578, 4587, 5478, 5487 are in descending order. Sol. (F): The numbers 4578, 4587, 5478, and 5487 are in ascending order. Q49. The number 85764 rounded off to nearest hundreds is written as 85700. Sol. (F): The number 85764 rounded off to nearest hundreds is written as 85800. Q50. Estimated sum of 7826 and 12469 rounded off to hundreds is 20,000. Sol. (F): Sum of 7826 and 12469 = 7826 + 12469 = 20295 = 20300 rounded off to hundreds. Q51. The largest six digit telephone number that can be formed by using digits 5, 3, 4, 7, 0, 8 only once is 875403. Sol. (F): The largest six digit telephone number that can be formed by using digits 5, 3, 4, 7, 0, 8 only once is 875430. Q52. The number 81652318 will be read as eighty one crore six lakh

10 n NCERT Exemplar Problems Mathematics–VI

fifty two thousand three hundred eighteen. Sol. (F): The number 81652318 will be read as 8 crore sixteen lakh fifty two thousand three hundred eighteen. Q53. The largest 4-digit number formed by the digits 6, 7, 0, 9 using each digit only once is 9760. Sol. (T) Q54. Among kilo, milli and centi, the smallest is centi. Sol. (F): The smallest is milli. Q55. Successor of a one digit number is always a one digit number. Sol. (F): Successor of single digit number like 9 may be a two digit number like 10. Q56. Successor of a 3-digit number is always a 3-digit number. Sol. (F): Successor of three digit number like 999 may be a four digit number like 1000. Q57. Predecessor of a 2-digit number is always a 2-digit number. Sol. (F): Predecessor of a 2-digit number like 10 may be a single digit number like 9. Q58. Every whole number has its successor. Sol. (T) Q59. Every whole number has its predecessor. Sol. (F): Zero is a whole number having no predecessor. Q60. Between any two natural numbers, there is one natural number. Sol. (F): Between two natural numbers there exists many natural numbers. Q61. The smallest 4-digit number is the successor of the largest 3-digit number. Sol. (T): Smallest 4-digit number = 1000 which is 999 + 1. Q62. Of the given two natural numbers, the one having more digits is greater. Sol. (T) Q63. Natural numbers are closed under addition. Sol. (T) Q64. Natural numbers are not closed under multiplication. Sol. (F): The natural numbers are closed under multiplication. For example, 5 and 3 are natural numbers and 5 × 3 =15 and 5 + 3 = 8 are also natural numbers Q65. Natural numbers are closed under subtraction. Sol. (F): The natural numbers are not closed under subtraction. For 3 example 5 and 3 are natural numbers but 3 – 5 = – 2 and 5 are not natural numbers.



Number System  n 11

Q66. Addition is commutative for natural numbers. Sol. (T) Q67. 1 is the identity for addition of whole numbers. Sol. (F): 0 is the identity for addition of whole numbers Q68. 1 is the identity for multiplication of whole numbers. Sol. (T) Q69. There is a whole number which when added to a whole number, gives the number itself. Sol. (T) Q70. There is a natural number which when added to a natural number, gives the number itself. Sol. (F): There is a whole number which when added to a natural number, gives the number itself. The whole number is 0. Q71. If a whole number is divided by another whole number, which is greater than the first one, the quotient is not equal to zero. Sol. (T) Q72. Any non-zero whole number divided by itself gives the quotient 1. Sol. (T) Q73. The product of two whole numbers need not be a whole number. Sol. (F): The product of two whole numbers needs to be a whole number. For example, 0 × 1 = 0 Q74. A whole number divided by another whole number greater than 1never gives the quotient equal to the former. Sol. (T) Q75. Every multiple of a number is greater than or equal to the number. Sol. (T) Q76. The number of multiples of a given number is finite. Sol. (F): The number of multiples of a given number is infinite. It is one of the property of the factor multiples. Q77. Every number is a multiple of itself. Sol. (T) Q78. Sum of two consecutive odd numbers is always divisible by 4. Sol. (T) Q79. If a number divides three numbers exactly, it must divide their sum exactly. Sol. (T) Q80. If a number exactly divides the sum of three numbers, it must exactly divide the numbers separately. Sol. (F): Let (3 + 5 + 4) / 4 = 12 / 4 = 3 However:

12 n NCERT Exemplar Problems Mathematics–VI

3 / 4 = 0.75 5 / 4 = 1.25 4 / 4 = 1 So, just because a number divides the sum of three other numbers evenly does not mean it will divide evenly the three other numbers separately. Q81. If a number is divisible both by 2 and 3, then it is divisible by 12. Sol.(F): For example, 6 is a number divisible by 2 and 3 but it is not divisible by 12. Q82. A number with three or more digits is divisible by 6, if the number formed by its last two digits (i.e., ones and tens) is divisible by 6. Sol. (F): A number with three or more digits is divisible by 6 if the entire number is divisible by both 2 and 3. Q83. A number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8. Sol. (T) Q84. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 9. Sol. (F): A number is divisible by 9 if the sum of its digits is divisible by 9. Q85. All numbers which are divisible by 4 may not be divisible by 8. Sol. (T) Q86. The Highest Common Factor of two or more numbers is greater than their Lowest Common Multiple. Sol. (F): The Highest Common Factor of two or more numbers is greater, lesser than or may be equal to their Lowest Common Multiple. Q87. LCM of two or more numbers is divisible by their HCF. Sol. (T) Q88. LCM of two numbers is 28 and their HCF is 8. Sol. (F) Q89. LCM of two or more numbers may be one of the numbers. Sol. (T) Q90. HCF of two or more numbers may be one of the numbers. Sol. (T) Q91. Every whole number is the successor of another whole number. Sol. (F): Zero is a whole number and is the successor of – 1 which is not a whole number. Q92. Sum of two whole numbers is always less than their product. Sol. (F): Sum of two whole numbers is not always less than their product. For example, 1 + 2 = 3 greater than 1 × 2 = 2. Q93. If the sum of two distinct whole numbers is odd, then their difference also must be odd. Number System  n 13

Sol. (T) Q94. Any two consecutive numbers are coprime. Sol. (T) Q95. If the HCF of two numbers is one of the numbers, then their LCM is the other number. Sol. (T) Q96. The HCF of two numbers is smaller than the smaller of the numbers. Sol. (F): The HCF of two or more numbers is smaller than or equal to the smallest of those numbers. Q97. The LCM of two numbers is greater than the larger of the numbers. Sol. (F): The LCM of two or more numbers is greater than or equal to the largest of those numbers. Q98. The LCM of two coprime numbers is equal to the product of the numbers. Sol. (T) In questions 99 to 150, fill in the blanks to make the statements true. Q99. (a) 10 million = _____ crore. (b) 10 lakh = _____ million. Sol. (a) 10 million = 1 crore. (b) 10 lakh = 1 million. Q100. (a) 1 metre = _____ millimetres. (b) 1 centimetre = _____ millimetres. (c) 1 kilometre = _____ millimetres. Sol. (a) 1 metre = 1000 millimetres. (b) 1 centimetre = 10 millimetres. (c) 1 kilometre = 10,00,000 millimetres Q101. (a) 1 gram = _____ milligrams. (b) 1 litre = _____ millilitres. (c) 1 kilogram = _____ miligrams. Sol. (a) 1 gram = 1000 milligrams. (b) 1 litre = 1000 millilitres. (c) 1 kilogram = 1000,000 miligrams Q102. 100 thousands = _____ lakh. Sol. 100 thousands = 1 lakh. Q103. Height of a person is 1m 65 cm. His height in millimetres is_______. Sol. Height of a person is 1m 65cm. His height in millimetres is 1650. Q104. Length of river ‘Narmada’ is about 1290 km. Its length in metresis_______.

14 n NCERT Exemplar Problems Mathematics–VI

Sol. Length of river ‘Narmada’ is about 1290 km. Its length in metres is 1290000. Q105. The distance between Srinagar and Leh is 422km. The same distance in metres is_____. Sol. The distance between Srinagar and Leh is 422km. The same distance in metres is 422000. Q106. Writing of numbers from the greatest to the smallest is called an arrangement in _____ order. Sol. Writing of numbers from the greatest to the smallest is called an arrangement in descending order. Q107. By reversing the order of digits of the greatest number made by five different non-zero digits, the new number is the _____ number of five digits. Sol. By reversing the order of digits of the greatest number made by five different non-zero digits, the new number is the smallest number of five digits. Q108. By adding 1 to the greatest_____ digit number, we get ten lakh. Sol. By adding 1 to the greatest 6-digit number, we get ten lakh. Q109. The number five crore twenty three lakh seventy eight thousand four hundred one can be written, using commas, in the Indian System of Numeration as _____. Sol. The number five crore twenty three lakh seventy eight thousand four hundred one can be written, using commas, in the Indian System of Numeration as 5,23,78,401. Q110. In Roman Numeration, the symbol X can be subtracted from_____,M and C only. Sol. In Roman Numeration, the symbol X can be subtracted from L, M and C only. Q111. The number 66 in Roman numerals is_____. Sol. The number 66 in Roman numerals is LXVI. Q112. The population of Pune was 2,538,473 in 2001. Rounded off to nearest thousands, the population was __________. Sol. The population of Pune was 2,538,473 in 2001. Rounded off to nearest thousands, the population was 2,538,000. Q113. The smallest whole number is_____. Sol. The smallest whole number is 0. Q114. Successor of 106159 is _____. Sol. Successor of 106159 is 106160. Q115. Predecessor of 100000 is_____. Sol. Predecessor of 100000 is 99999. Q116. 400 is the predecessor of _____.



Number System  n 15

Full Marks Pvt Ltd has prepared NCERT Exemplar Problems-Solutions series with an objective to provide the students with foolproof solutions to questions for the book NCERT Exemplar Problems. NCERT has developed Exemplar Problems in Science and Mathematics, whose prime objective is to provide the students with a number of quality problems to facilitate the concept of learning. It has always been our endeavour to provide better quality material to the students.

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