7 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-VII 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.

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CONTENTS 1. Integers.............................................................................................. 5 2. Fractions and Decimals..................................................................... 21 3. Data Handling................................................................................... 43 4. Simple Equations.............................................................................. 71 5. Lines and Angles............................................................................... 87 6. Triangles............................................................................................ 111 7. Comparing Quantities....................................................................... 143 8. Rational Numbers............................................................................. 172 9. Perimeter and Area............................................................................ 185 10. Algebraic Expressions...................................................................... 212 11. Exponents and Powers...................................................................... 226 12. Practical Geometry, Symmetry and Visualising Solid Shapes.......... 236

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1

Integers EXERCISE

In the questions 1 to 25, there are four options, out of which only one is correct. Write the correct one. Q1. When the integers 10, 0, 5, – 5, – 7 are arranged in descending or ascending order, then find out which of the following integers always remains in the middle of the arrangement. (a) 0 (b) 5 (c) – 7 (d) – 5 Sol. (a): Descending order = 10, 5, 0, – 5, – 7 Ascending order = – 7, – 5, 0, 5, 10 Q2. By observing the number line shown below state which of the following statements is not true. B

–10

A

0

(a) B is greater than –10 (c) B is greater than A

10

(b) A is greater than 0 (d) B is smaller than 0

Sol. (c): On the given number line, B represents – 4 and A represents 7. Clearly, B is not greater than A. Q3. By observing the above number line (Q2) state which of the following statements is true. (a) B is 2 (b) A is – 4 (c) B is – 13 (d) B is – 4 Sol. (d): B is on the fourth point left of 0. So, B is – 4. Q4. Next three consecutive numbers in the pattern 11, 8, 5, 2, --, --, -- are (a) 0, – 3, – 6 (b) – 1, – 5, – 8 (c) – 2, – 5, – 8 (d) – 1, – 4, – 7 Sol. (d): In the given pattern, 3 is subtracted from each number to get the next number. Q5. The next number in the pattern – 62, – 37, – 12 _________ is (a) 25 (b) 13 (c) 0 (d) – 13 Sol. (b): In the given pattern, 25 is added to each term. 5

Q6. Which of the following statements is not true? (a) When two positive integers are added, we always get a positive integer. (b) When two negative integers are added, we always get a negative integer. (c) When a positive integer and a negative integer is added, we always get a negative integer. (d) Additive inverse of an integer 2 is (– 2) and additive inverse of ( – 2) is 2. Sol. (c): When a positive integer and a negative integer are added, the nature of the resultant number depends upon the sign preceding the greater number. Q7. On the following number line value ‘Zero’ is shown by the point X

Y

Z

–15

W 10

(a) X (b) Y (c) Z (d) W Sol. (c): On the given number line, points are plotted by following pattern: W = 10 – 5 = 5; Z = 5 – 5 = 0; Y = 0 – 5 = – 5; X = – 5 – 5 = – 10 So, Z represents 0 on the number line. Q8. If ⊗, , 3 and • represent some integers on number line, then descending order of these numbers is ✓

0

(a) •, ⊗, 3 , (b) ⊗, •, 3 , (c) , 3 , ⊗, • (d) , •, ⊗, 3 Sol. (c): Descending order of the integers is , 3 , ⊗, •. Q9. On the number line, the value of (–3) × 3 lies on right hand side of (a) – 10 (b) – 4 (c) 0 (d) 9 Sol. (a): (– 3) × 3 = – 9 which lies to the right of – 10. Q10. The value of 5 ÷ (–1) does not lie between (a) 0 and – 10 (b) 0 and 10 (c) – 4 and – 15 (d) – 6 and 6 Sol. (b): 5 ÷ (– 1) = – 5 does not lie between 0 and 10. Q11. Water level in a well was 20 m below ground level. During rainy season, rain water collected in different water tanks was drained into the well and the water level rises 5 m above the previous level. The wall of the well is 1 m 20 cm high and a pulley is fixed at a height of

6 n NCERT Exemplar Problems Mathematics–VII

80 cm. Raghu wants to draw water from the well. The minimum length of the rope that he can use is (a) 17 m (b) 18 m (c) 96 m (d) 97 m

Sol. (a): Required length of the rope = (20 – 5) m + 1. 2 m + 0.8 m = 17 m Q12. (– 11) × 7 is not equal to (a) 11 × (– 7) (b) – (11 × 7) (c) (– 11) × (– 7) (d) 7 × (– 11) Sol. (c): – 11 × 7 = – 77 ≠ (– 11) × (– 7) Q13. (– 10) × (– 5) + (– 7) is equal to (a) – 57 (b) 57 (c) – 43 (d) 43 Sol. (d): (– 10) × (– 5) + (– 7) = 43 Q14. Which of the following is not the additive inverse of a? (a) – (– a) (b) a × (– 1) (c) – a (d) a ÷ (– 1) Sol. (a): – (– a) = a is not the additive inverse of a. Q15. Which of the following is the multiplicative identity for an integer a? (a) a (b) 1 (c) 0 (d) – 1 Sol. (b): Multiplicative identity for any integer is 1. Q16. [(– 8) × ( – 3)] × (– 4) is not equal to (a) (– 8) × [(– 3) × (– 4)] (b) [(– 8) × (– 4)] × (– 3) (c) [(– 3) × (– 8)] × (– 4) (d) ( – 8) × (– 3) – (– 8) × (– 4) Sol. (d): [(– 8) × ( – 3)] × (– 4) ≠ ( – 8) × (– 3) – (– 8) × (– 4). Q17. (– 25) × [6 + 4] is not same as (a) (– 25) × 10 (b) (– 25) × 6 + (– 25) × 4 (c) (– 25) × 6 × 4 (d) – 250 Sol. (c): (– 25) × [6 + 4] is not same as (– 25) × 6 × 4.



Integers  n 7

Q18. – 35 × 107 is not same as (a) – 35 × (100 + 7) (b) (– 35) × 7 + ( – 35) × 100 (c) – 35 × 7 + 100 (d) (– 30 – 5) × 107 Sol. (c): – 35 × 107 is not same as – 35 × 7 + 100. Q19. (– 43) × (– 99) + 43 is equal to (a) 4300 (b) – 4300 (c) 4257 (d) – 4214 Sol. (a): (– 43) × (– 99) + 43 = 4257 + 43 = 4300. Q20. (– 16) ÷ 4 is not same as (a) (– 4) ÷ 16 (b) – ( 16 ÷ 4) (c) 16 ÷ (– 4) (d) – 4 Sol. (a): (– 16) ÷ 4 is not same as ( – 4) ÷ 16. Q21. Which of the following does not represent an integer? (a) 0 ÷ (– 7) (b) 20 ÷ (– 4) (c) (– 9) ÷ 3 (d) (– 12) ÷ 5 Sol. (d): (– 12) ÷ 5 = – 2.4 which is not an integer. Q22. Which of the following is different from the others? (a) 20 + (– 25) (b) (– 37) – (– 32) (c) (– 5) × (– 1) (d) (45) ÷ (– 9) Sol. (c): Except all, the resultant of (– 5) × (–1) is positive. Q23. Which of the following shows the maximum rise in temperature? (a) 23° to 32° (b) – 10° to + 1° (c) – 18° to – 11° (d) – 5° to 5° Sol. (b): 1° – (– 10°) = 11°, which shows the maximum rise in temperature. Q24. If a and b are two integers, then which of the following may not be an integer? (a) a + b (b) a – b (c) a × b (d) a÷b Sol. (d): a ÷ b may be a fraction. Q25. For a non-zero integer a which of the following is not defined? (a) a ÷ 0 (b) 0 ÷ a (c) a ÷ 1 (d) 1 ÷ a Sol. (a): a ÷ 0 is not defined. Encircle the odd one of the following (Questions 26 to 30). Q26. (a) (–3, 3) (b) (–5, 5) (c) (–6, 1) (d) (–8, 8) Sol. (c) Q27. (a) (–1, –2) (b) (–5, +2) (c) (–4, +1) (d) (–9, +7) Sol. (d): All the pairs except (– 9, + 7), give same answer on adding. Q28. (a) (– 9) × 5 × 6 × (– 3) (b) 9 × (–5) × 6 × (– 3) (c) (– 9) × (– 5) × (– 6) × 3 (d) 9 × (–5) × (– 6) × 3 Sol. (c)

8 n NCERT Exemplar Problems Mathematics–VII

Q29. (a) (–100) ÷ 5 (b) (–81) ÷ 9 (c) (–75) ÷ 5 (d) (–32) ÷ 9 Sol. (d): Gives a fraction. Q30. (a) (–1) × (–1) (b) (–1) × (–1) × (–1) (c) (–1) × (–1) × (–1) × (–1) (d) (–1) × (–1) × (–1) × (–1) × (–1) × (–1) (–1) × (–1) Sol. (b) In Questions 31 to 71, fill in the blanks to make the statements true. Q31. (– a) + b = b + Additive inverse of ____________. Sol. a Q32. ____________ ÷ (–10) = 0 Sol. 0 Q33. (–157) × (–19) + 157 = ____________ Sol. 3140 Q34. [(–8) + ________] + _________ = _________ + [(–3) + ________] = –3 Sol. [(–8) + –3] + 8 = –8 + [(–3) + 8] = –3 Q35. On the following number line, (– 4) × 3 is represented by the point. A –20

B

C

D

E

F

G

H

I

J 2

Sol. D: (– 4) × 3 = – 12 On the given number line J = 0, I = –2, H = –4, G = –6,..., D = –12 Q36. If x, y and z are integers, then (x +_____ ) + z = ______ + (y + ______) Sol. (x + y) + z = x + (y + z) Q37. (– 43) + _____ = – 43 Sol. (– 43) + 0 = – 43 Q38. (– 8) + (– 8) + (– 8) = _____ × (– 8) Sol. (– 8) + (– 8) + (– 8) = 3 × (– 8) Q39. 11 × (– 5) = – ( _____ × _____ ) = _____ Sol. 11 × (– 5) = – (11 × 5) = – 55 Q40. (– 9) × 20 = _____ Sol. (– 9) × 20 = – 180 Q41. (– 23) × (42) = (– 42) × _____ Sol. (– 23) × (42) = (– 42) × 23



Integers  n 9

Q42. While multiplying a positive integer and a negative integer, we multiply them as ________ numbers and put a ________ sign before the product. Sol. whole, negative : As x × (– y) = – (x × y) = – xy. Q43. If we multiply ________ number of negative integers, then the resulting integer is positive. Sol. even Q44. If we multiply six negative integers and six positive integers, then the resulting integer is _______. Sol. positive Q45. If we multiply five positive integers and one negative integer, then the resulting integer is _______. Sol. negative Q46. __________ is the multiplicative identity for integers. Sol. 1 Q47. We get additive inverse of an integer a when we multiply it by _________. Sol. – 1: Additive inverse of a = a × (– 1) = – a Q48. (– 25) × (– 2) = ____________ Sol. 50 Q49. (– 5) × (– 6) × (– 7) = ____________ Sol. – 210 Q50. 3 × (– 1 ) × (– 15) = ____________ Sol. 45 Q51. [12 × ( – 7)] × 5 = ____________ × [(– 7) × ____________] Sol. [12 × (– 7)] × 5 = 12 × [(– 7) × 5] Q52. 23 × (– 99) = ________ × (– 100 + _________) = 23 × _________ + 23 × _________ Sol. 23 × (– 99) = 23 × (– 100 + 1) = 23 × – 100 + 23 × 1 Q53. ____________ × (– 1) = – 35 Sol. 35 Q54. ____________ × (– 1) = 47 Sol. (– 47) × (– 1) = 47 Q55. 88 × _________ = – 88 Sol. – 1 Q56. _________ × (– 93) = 93 Sol. – 1 Q57. (– 40) × _________ = 80 Sol. 2 Q58. _________ × (– 23) = – 920 Sol. 40 Q59. When we divide a negative integer by a positive integer, we divide them as whole numbers and put a _________ sign before quotient. Sol. minus: (– x) ÷ (y) = – (x ÷ y)

10 n NCERT Exemplar Problems Mathematics–VII

Q60. When –16 is divided by _________ the quotient is 4. Sol. – 4 Q61. Division is the inverse operation of ____________ Sol. Multiplication Q62. 65 ÷ (– 13) = ____________ Sol. 5 Q63. (– 100) ÷ (– 10) = ____________ Sol. 10 Q64. (– 225) ÷ 5 = ____________ Sol. – 45 Q65. ____________÷ (– 1) = – 83 Sol. 83 Sol. – 75 Q66. ____________ ÷ (– 1) = 75 Q67. 51 ÷ ____________ = – 51 Sol. – 1 Sol. – 113 Q68. 113 ÷ ____________ – 1 Q69. (– 95) ÷ ____________ = 95 Sol. – 1 Q70. (– 69) ÷ (69) = ____________ Sol. – 1 Q71. (– 28) ÷ (– 28) = ____________ Sol. 1 In Questions 72 to 108, state whether the statements are True or False. Q72. 5 – (– 8) is same as 5 + 8. Sol. True Q73. (– 9) + (– 11) is greater than (– 9) – (– 11). Sol. False: (– 9) + (– 11) = – 9 – 11 = – 20 and (– 9) – (– 11) = – 9 + 11 = 2 Q74. Sum of two negative integers always gives a number smaller than both the integers. Sol. True Q75. Difference of two negative integers cannot be a positive integer. Sol. False Q76. We can write a pair of integers whose sum is not an integer. Sol. False Q77. Integers are closed under subtraction. Sol. True Q78. (– 23) + 47 is same as 47 + (– 23). Sol. True Q79. When we change the order of integers, their sum remains the same. Sol. True: Addition is commutative for integers. Q80. When we change the order of integers their difference remains the same. Sol. False: Subtraction is not commutative for integers.



Integers  n 11

Q81. Going 500 m towards east first and then 200 m back is same as going 200 m towards west first and then going 500 m back. Sol. True: 500 m – 200 m = – 200 m + 500 m. Sol. True Q82. (– 5) × (33) = 5 × (– 33) Sol. True Q83. (– 19) × (– 11) = 19 × 11 Sol. False Q84. (– 20) × (5 – 3) = (– 20) × ( – 2) Sol. False Q85. 4 × (– 5) = (– 10) × (– 2) Sol. False Q86. (– 1) × (– 2) × (– 3) = 1 × 2 × 3 Sol. True Q87. – 3 × 3 = – 12 – (– 3) Q88. Product of two negative integers is a negative integer. Sol. False Q89. Product of three negative integers is a negative integer. Sol. True Q90. Product of a negative integer and a positive integer is a positive integer. Sol. False: (– x) × (y) = – xy Q91. When we multiply two integers their product is always greater than both the integers. Sol. False Q92. Integers are closed under multiplication. Sol. True Q93. (– 237) × 0 is same as 0 × (–39) Sol. True Q94. Multiplication is not commutative for integers. Sol. False Q95. (–1) is not a multiplicative identity of integers. Sol. True Q96. 99 × 101 can be written as (100 – 1) × (100 + 1) Sol. True Q97. If a, b, c are integers and b ≠ 0, then a × (b – c) = a × b – a × c Sol. True Sol. True Q98. (a + b) × c = a × c + a × b Sol. True Q99. a × b = b × a Sol. False Q100. a ÷ b = b ÷ a Sol. False Q101. a – b = b – a Sol. True Q102. a ÷ (– b) = – (a ÷ b) Sol. True Q103. a ÷ (– 1) = – a

12 n NCERT Exemplar Problems Mathematics–VII

Q104. Multiplication fact (– 8) × (– 10) = 80 is same as division fact 80 ÷ (– 8) = (– 10) Sol. True Q105. Integers are closed under division. Sol. False Q106. [(–32) ÷ 8] ÷ 2 = –32 ÷ [8 ÷ 2] Sol. False Q107. The sum of an integer and its additive inverse is zero (0). Sol. True Q108. The successor of 0 × (– 25) is 1 × (– 25) Sol. False Q109. Observe the following patterns and fill in the blanks to make the statements true: (a) – 5 × 4 = – 20; – 5 × 3 = – 15 = – 20 – (– 5); – 5 × 2 = _________ = – 15 – (– 5); – 5 × 1 = _________ = _________; – 5 × 0 = 0 = _________; – 5 × – 1 = 5 = _________; – 5 × – 2 = _________ = _________ (b) 7 × 4 = 28 7 × 3 = _________ = 28 – 7; 7 × 2 = _________ = _________ – 7; 7 × 1 = 7 = _________ – 7; 7 × 0 = _________ = _________ – _________; 7 × – 1 = – 7 = _________ – _________; 7 × – 2 = _________ = _________ – _________; 7 × – 3 = _________ = _________ – _________ Sol. (a) – 5 × 2 = – 10 = – 15 – (– 5); – 5 × 1 = – 5 = – 10 – (– 5); – 5 × 0 = 0 = – 5 – (– 5); – 5 × – 1 = 5 = 0 – (– 5); – 5 × – 2 = 10 = 5 – (– 5) (b) 7 × 3 = 21 = 28 – 7; 7 × 2 = 14 = 21 – 7; 7 × 1 = 7 = 14 – 7; 7 × 0 = 0 = 7 – 7; 7 × – 1 = – 7 = 0 – 7; 7 × – 2 = – 14 = – 7 – 7; 7 × – 3 = – 21 = – 14 – 7 Q110. Science Application: An atom consists of charged particles called electrons and protons. Each proton has a charge of +1 and each electron has a charge of –1. Remember number of electrons is equal to number of protons, while answering these questions: (a) What is the charge on an atom? (b) What will be the charge on an atom if it loses an electron? (c) What will be the charge on an atom if it gains an electron? Sol. (a) An atom has same number of electrons and protons. So, there is 0 charge on an atom. (b) An atom loses an electron, means the atom has one less number of electron than that of proton. So, the charge on the atom = + 1. (c) An atom gains an electron means it has one more electron than proton. So, the charge on the atom = – 1.



Integers  n 13

Q111. An atom changes to a charged particle called ion if it loses or gains electrons. The charge on an ion is the charge on electrons plus charge on protons. Now, write the missing information in the table given below: Name of Ion

Proton Charge

Electron Charge

Ion Charge

Hydroxide ion

+9

—

–1

Sodium ion

+11

—

+1

Aluminium ion

+13

– 10

—

Oxide ion

+8

–10

—

Sol. + 9 + (– 10) = – 1; + 11 + (– 10) = + 1; +13 + (– 10) = + 13; + 8 + (– 10) = – 2. Q112. Social Studies Application: Remembering that 1AD came immediately after 1BC, while solving these problems take 1BC as –1 and 1AD as +1. (a) The Greeco-Roman era, when Greece and Rome ruled Egypt started in the year 330 BC and ended in the year 395 AD. How long did this era last? (b) Bhaskaracharya was born in the year 1114 AD and died in the year 1185 AD. What was his age when he died? (c) Turks ruled Egypt in the year 1517 AD and Queen Nefertis ruled Egypt about 2900 years before the Turks ruled. In what year did she rule? (d) Greek mathematician Archimedes lived between 287 BC and 212 BC and Aristotle lived between 380 BC and 322 BC. Who lived during an earlier period? Sol. (a) 395 AD = + 395 and 330 BC = – 330 So, the period of the Greeco-Roman era = 395 – (– 330) = 725 years (b) Age of Bhaskaracharya = 1185 AD – 1114 AD = 1185 – 1114 = 71 years (c) Rulling period of Queen Nefertis = 1517 AD – 2900 = 1517 – 2900 = – 1383 = 1383 BC (d) Aristotle Q113. The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature.

14 n NCERT Exemplar Problems Mathematics–VII

The Lowest Recorded Temperatures Continent

Temperature (in Fahrenheit)

Africa

–11°

Antarctica

–129°

Asia

–90°

Australia

– 9°

Europe

– 67°

North America

– 81°

South America

– 27°

Sol.  – 129°F < – 90°F < – 81°F < – 67°F < – 27°F < 11°F < – 9°F  Antarctica, Asia, North America, Europe, South America, Africa, Australia. Q114. Write a pair of integers whose product is – 12 and there lies seven integers between t hem (excluding the given integers). Sol. Such integers are – 2 and 6. Q115. From given integers in Column I match an integer of Column II so that their product lies between – 19 and – 6: Column I Column II – 5 1 6 –1 – 7 3 8 –2 Sol. –5 × 3, 6 × – 2, – 7 × 1, 8 × –1 (Some more matching are also possible) Q116. Write a pair of integers whose product is – 36 and whose difference is 15. Sol. Such a pair is (– 3, 12) Q117. Match the following: Column I Column II (a) a × 1 (i) Additive inverse of a (b) 1 (ii) Additive identity (c) (– a) ÷ (– b) (iii) Multiplicative identity (d) a × (– 1) (iv) a ÷ (– b) (e) a × 0 (v) a ÷ b (f) (– a) ÷ b (vi) a



Integers  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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