Q1. If groundwater continues to be overused in Punjab, which of the following will be the MOST immediate effect? (a) Decrease in coal reserves (b) Falling water table levels (c) Increase in rainfall (d) Rise in forest cover
Ans:(b) Falling water table levels The Punjab caselet states that overusing groundwater for crops lowers the water table (e.g., to ~30 meters deep in 80% of Punjab), making it the most immediate effect.
Q2. Which of the following BEST explains why coal is considered a non-renewable resource? (a) It is very expensive (b) It pollutes the air (c) It takes millions of years to form (d) It is found only in India
Ans: (c) It takes millions of years to form Non-renewable resources like coal form over millions of years and cannot be replenished quickly, unlike renewable resources.
Q3. When communities stopped fishing during spawning seasons, their main aim was: (a) To punish fishermen (b) To increase the fish trade (c) To allow fish populations to grow back (d) To avoid water pollution
Ans: (c) To allow fish populations to grow back The communities historically stopped fishing during spawning seasons to help fish populations (e.g., tuna) regenerate, preventing depletion.
Q4. Which of the following situations BEST shows fair access to resources? (a) A city giving 24-hour water to luxury apartments but limited supply to slums (b) Villagers and towns sharing the same river water through agreed-upon timings (c) A factory using all groundwater for industry (d) Farmers cutting down sacred groves for cultivation
Ans: (b) Villagers and towns sharing the same river water through agreed-upon timings The fair access to resources like water through stewardship, as equitable sharing reflects responsible management, unlike unequal or exploitative practices.
Q5. Which of the following is the MOST sustainable practice? (a) Mining coal rapidly for export profits (b) Using solar energy in place of diesel generators (c) Building factories in forests (d) Throwing untreated waste in rivers
Ans: (b) Using solar energy in place of diesel generators Solar energy (e.g., Bhadla Solar Park) is a renewable, sustainable alternative to polluting fossil fuels, unlike other environmentally harmful options.
Q6. Why does nature become a “resource” only when humans know how to use it? Ans: Nature becomes a resource when humans use its elements (e.g., trees, water) for purposes like living or making things, requiring knowledge of how to access and apply them. This involves three conditions: technological accessibility (e.g., machines to cut wood), economic feasibility (affordable extraction), and cultural acceptability (e.g., respecting sacred groves). The natural elements like trees or rivers are not resources until humans know how to use them (e.g., wood for furniture, water for farming).
Q7. “Sikkim’s organic farming journey is both an environmental and economic success.” Analyse. Ans:
Environmental Success: Sikkim’s 100% organic farming (since 2016) restored biodiversity (e.g., more insects and birds) by using natural fertilisers like compost and neem-based pest repellents, avoiding chemical pollution and supporting ecosystem services like soil health.
Economic Success: Organic farming increased farmers’ incomes by 20% on average and boosted tourism, as visitors were attracted to Sikkim’s sustainable model.
Analysis: By adopting traditional practices like multi-cropping, Sikkim balanced environmental health with economic growth, making it a global model for sustainability. The Sikkim caselet highlights how organic farming reversed low yields and debts, benefiting both nature and the economy.
Q8. If you were in charge of reducing cement pollution in India, what two measures would you take? Why? Ans: 1. Enforce Central Pollution Control Board Guidelines: Strictly implement guidelines to minimise cement factory dust, which harms lungs, plants, and soil. This reduces pollution and protects ecosystems. 2. Promote Sustainable Materials: Encourage using mud, sandstone, or recycled plastic for construction, as seen in Auroville and Jaisalmer Fort, to lower environmental impact. Why: These measures address health and ecological risks while promoting sustainable alternatives that focus on responsible resource use. The cement’s pollution suggests traditional materials to reduce environmental harm.
Q9. Compare renewable and non-renewable resources. Which is more threatened today? Ans: The careless use of renewables (e.g., water drying up) risks depletion, while non-renewables are inherently limited.
Renewable Resources: Can be replenished naturally in a short time (e.g., solar energy, forests) if nature’s cycles are maintained. Overuse, like deforestation, can deplete them.
Non-Renewable Resources: Formed over millions of years (e.g., coal, petroleum) and cannot be replenished quickly, with coal reserves potentially lasting only 50 years.
Comparison: Renewables are sustainable with proper management; non-renewables are finite and deplete with use.
More Threatened: Renewable resources are more threatened today because human actions (e.g., overusing groundwater in Punjab, deforestation) disrupt their regeneration cycles, making them behave like non-renewables.
Q10. How can the uneven distribution of natural resources lead to cooperation and conflict? Ans: The uneven distribution shapes trade and conflicts, emphasising the need for fair resource management.
Cooperation: Uneven resource distribution encourages trade and collaboration. For example, resource-rich areas (e.g., Jharkhand’s coal) trade with resource-scarce areas, fostering economic growth and job creation, as seen with India’s Wootz steel trade historically.
Conflict: Resources like water crossing borders (e.g., Kaveri River disputes among Karnataka, Tamil Nadu, Kerala, and Puducherry) cause tensions when access is unequal. Sacred sites or land lost to mining can also spark protests.
Solution: Fair management and negotiations can promote peace and equitable sharing.
Q11. “Overuse of renewable resources can make them behave like non-renewable resources.” Explain. Ans: Renewable resources (e.g., water, forests) can regenerate naturally if used sustainably, but overuse disrupts their cycles, making them behave like non-renewable resources, which deplete permanently. For example, overharvesting timber faster than forests regrow leads to forest loss. Similarly, overusing groundwater in Punjab lowers water tables, reducing availability, as seen in the caselet. Sustainable practices like rainwater harvesting are needed to maintain renewability.
Q12. How is the idea of lokasangraha useful in resource use? Ans: Lokasangraha, from the Bhagavad Gita, means working for the good of all, not just oneself. In resource use, it encourages responsible stewardship by promoting fair access to resources (e.g., clean water, air) and sustainable practices. For example, sharing river water equitably (like the Brahmaputra example) or adopting organic farming in Sikkim ensures resources benefit everyone and last for future generations. It discourages selfish exploitation and calls for collective responsibility.
Q13. Punjab groundwater case: What were the short-term benefits and long-term harms? Ans: The Punjab caselet highlights how groundwater overuse provided immediate food security but caused long-term environmental and health damage.
Short-Term Benefits: In the 1960s, Punjab’s farmers used groundwater for high-yielding crops, supported by free electricity, ensuring food security and increased agricultural output.
Long-Term Harms: Over-pumping lowered water tables to ~30 meters deep in 80% of Punjab, marked as “overexploited.” Chemical fertilisers polluted groundwater, posing health risks and reducing sustainability, as water became scarce for future use.
Q14. Design a school awareness campaign on sustainable resource use. Ans: Campaign Name: “Green Today, Thriving Tomorrow” Activities:
Workshops: Educate students on sustainable practices like rainwater harvesting and organic farming (e.g., Sikkim’s model), using posters and videos.
Tree-Planting Drive: Plant trees to promote ecosystem services like oxygen production, inspired by the chapter’s example of trees producing 275 litres of oxygen daily.
Recycle Club: Encourage reducing, reusing, and recycling to mimic nature’s no-waste cycles, as described in the chapter.
Goal: Raise awareness about responsible resource use, emphasising stewardship and sustainability.
Q15. What if India had not invested in renewable energy like solar parks? Ans: If India had not invested in renewable energy like solar parks (e.g., Bhadla Solar Park), several consequences would arise:
Increased Non-Renewable Depletion: India would rely more on coal and petroleum, depleting reserves faster (e.g., coal may last only 50 years) and increasing import costs for oil.
Environmental Damage: Greater fossil fuel use would worsen pollution and climate change, accelerating Himalayan glacier melting, threatening water security for rivers like the Brahmaputra.
Economic Challenges: Without solar energy’s job creation and cheaper electricity (e.g., Bhadla meets ~15% of Rajasthan’s needs), India would face higher energy costs and slower economic growth.
Missed Global Leadership: India’s role in the International Solar Alliance (ISA) would weaken, reducing its influence in promoting clean energy globally.
Sustainability Risks: Overuse of non-renewables would harm ecosystems, reducing services like oxygen production and soil health, making sustainable development harder.
Conclusion: Investing in renewables has reduced environmental harm, supported economic growth, and ensured long-term resource availability emphasis on stewardship.
II. Work in pairs and discuss the following things. Share your answers with your classmates and teacher.
1. The reasons for the change in the means of communication over the years. Ans: Over time, technology has improved, making communication faster and easier. People wanted quicker ways to connect, so old methods like pigeons and postcards were replaced by phones and emails.
2. The means of communication that you prefer to use and the reason for your choice. Ans: I prefer mobile phones because they are fast, easy to use, and let me call or message anyone instantly.
3. Communication through social media: its benefits and precautions that one should take against its misuse. Ans: Social media helps share ideas and stay connected. Precautions include not sharing personal information, avoiding strangers online, and checking facts to avoid misinformation.
Let us discuss (Page 85)
A fact is something that can be proven true or false. A fact is not based on people’s beliefs. An opinion, on the other hand, is a belief, feeling or judgement, and can vary from one person to another. An opinion cannot be proven.
I. Identify which of the following statements are facts and which are opinions. Share your answers with your classmates and teacher. 1. But speaking on the telephone only gives us an immediate but fleeting joy. Ans: Opinion (expresses a personal view)
2. I started my working life soon after our country became independent. Ans: Fact (verifiable event)
3. To be quite honest, service to our nation’s farmers was not the career I had envisioned for myself. Ans: Opinion (personal thought)
4. I knew I could make a more meaningful contribution by working here in Anand, Gujarat. Ans: Opinion (personal belief)
5. … values that I saw in my mentor and supporter here in Anand—Tribhuvandas Patel. Ans: Opinion (subjective observation)
6. Life is a privilege, and to waste it would be wrong. Ans: Opinion (personal belief)
7. I hope that you, too, will discover, as I did, that failure is not about not succeeding. Ans: Opinion (personal perspective)
8. … the magnificent ceremony in which the President awarded me the Padma Vibhushan in 1999? Ans: Fact (verifiable event)
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Let Us think and reflect (Page 86)
I. Read the given extracts and answer the questions:
1. In today’s fast-paced world, we have become so addicted to instant communication that we prefer to use a telephone. But speaking in today’s fast-paced world, we have become so addicted to instant communication that we prefer to use a telephone. But speaking on the telephone only gives us immediate yet fleeting joy. Writing is different. Writing—even if it is a letter—not only conveys our present concerns and views of the events taking place around us, but it also becomes a possession that can be treasured and re-read over the years, providing great, abiding pleasure. The telephone only gives us an immediate but fleeting joy. Writing is different. Writing—even if it is a letter—not only conveys our present concerns and views of the events taking place around us, but it becomes a possession that can be treasured and re-read over the years, with great, abiding pleasure. (i) Fill in the blank given below by selecting the correct option from the brackets. When the writer refers to a ‘fast-paced world’, he is speaking of a world where _________________. (Communication happens quicklybut lacks depth/people have more time to write and reflect) Ans: communication happens quickly but lacks depth (ii) Select the option that applies correctly to both Assertion (A) and Reason (R). (A): Writing letters can be treasured and re-read over the years. (R): Speaking on the telephone provides long-lasting joy. A. Both (A) and (R) are true, and (R) is the correct explanationof (A). B. Both (A) and (R) are true, but (R) is not the correct explanationof (A). C. (A) is true, but (R) is false. D. Both (A) and (R) are false. Ans: C. (A) is true, but (R) is false. Reason: Writing can be treasured, but telephone joy is described as fleeting, not long-lasting. (iii) Identify whether the statement given below is true or false. The writer feels that letter-writing is a class apart from other kinds of writing. Ans: True Reason: the extract emphasises the unique, lasting value of letters. (iv) Which of the following is the most appropriate title for the given extract? A. The Rise of Instant Communication B. The Fleeting Delight of Phone Calls C. The Lasting Joy of Written Words D. Living in a Fast-paced World Ans: C. The Lasting Joy of Written Words.
2. Your grandmother, too, made an important choice. She knew, in those early days, life in Anand could not offer even the simple comforts that we take for granted today. However, she ardently supported my choice to live and work in Anand. That choice of your grandmother to stand by me has given me an everlasting strength, always ensuring that I shouldered my responsibilities with poise.
(i) How can we say that the grandmother shared her husband’s dedication to a greater cause? Ans: She supported his choice to work in Anand despite the lack of comforts, showing her commitment to his mission.
(ii) Complete the following with the correct option from the two given in brackets. The fact that the writer (grandfather) praises his wife tells us that he _____________________ (truly values/quietly accepts) her role in his life. Ans: The fact that the writer (grandfather) praises his wife tells us that he truly values her role in his life.
(iii) Complete the following sentence with the correct option from those given below. Based on the information in the extract, we can infer that the grandmother had all of the following traits EXCEPT ____________. A. encouraging B. understanding C. comforting D. thoughtful Ans: C. comforting
(iv) Select the phrase that conveys— ‘showed grace under pressure and carried out my duties with dignity and confidence’. Ans: The phrase that conveys “showed grace under pressure and carried out my duties with dignity and confidence” is “shouldered my responsibilities with poise”.
II. Answer the following questions. Q1. What event does Dr. Kurien refer to when he mentions receiving the Padma Vibhushan? Ans: Dr. Kurien refers to the occasion when he was honoured with the Padma Vibhushan, India’s second-highest civilian award, in recognition of his significant contributions to the White Revolution and to the development of India’s dairy industry. He recalls how his grandson proudly wore the medal and innocently asked if he could keep it.
Q2. How did Dr. Kurien’s decision to join the cooperative in Anand affect his life? Ans: Dr. Kurien’s decision to join the small cooperative of dairy farmers in Anand completely transformed his life. Though he had other career options like metallurgy, the army, or going abroad, he chose to work for the farmers. This decision led him to a life of service, brought him national recognition, and allowed him to contribute meaningfully to the development of rural India.
Q3. Dr. Kurien’s decision to work in Anand reveals his values and sense of duty. Explain. Ans: Dr. Verghese Kurien’s choice to remain in Anand, rather than pursue a more lucrative or comfortable career, reflects his deep sense of responsibility, social justice, and empathy for farmers. Though initially posted there as a government employee, he chose to dedicate himself to the rural community—an act of selflessness, patriotism, and faith in grassroots development. By joining the small cooperative of dairy farmers in Anand, Kurien not only transformed his own life but also the lives of millions. Despite having other career options in metallurgy, the army, or opportunities abroad, he deliberately chose service over personal gain. This decision earned him national recognition, a life of fulfillment, and the chance to contribute profoundly to the development of rural India.
Q4. Why did Dr. Kurien write the letter to his grandson? Ans: Dr. Kurien wrote the letter to his grandson Siddharth to share his life experiences and values. He wanted to teach him about the importance of service, hard work, and honesty. Through the letter, he hoped to inspire the younger generation to make meaningful choices in life and to take responsibility for society.
Q5. According to Dr. Kurien, “life is a privilege and it is wrong to waste it”.What is your opinion on this? Ans: (Example opinion – you can personalise it as needed) I agree with Dr. Kurien. Life is a gift full of opportunities to grow, serve, and make a difference. If we waste it on selfish pursuits, we lose both personal growth and the chance to help others. True happiness and purpose come when we use our talents to benefit society, just as Dr. Kurien did.
Q6. What message does Dr. Kurien want to convey about selfless service and its long-term impact on society? Ans: Dr. Kurien emphasises that true success comes from selfless service and working for the good of others. By dedicating his life to rural India, he showed that helping people can strengthen communities, bring economic progress, and ensure social justice. His life proves that when we act with honesty and purpose, the benefits can inspire and uplift generations.
Also read: NCERT Textbook: Verghese Kurien — I Too Had A Dream
Let us learn (Page 88)
I. Replace the underlined words in each sentence with the correct synonym from the box. You may refer to a dictionary.
1. He received the award in amagnificentceremony. 2. I felt a fleetingjoy when I received the appreciation for my song. 3. With today’s instantcommunication, we can share information across the world in seconds. 4. Our school focuses on the corevalues—collaboration, innovation, and unity. 5. My teacher’s advice on time management is avaluablereminder of the importance of planning. 6. Reading books and listening to music can give abidingpleasure that will last forever.
Ans: 1. Magnificent → Splendid He received the award in a splendid ceremony.
2. Fleeting → Momentary I felt a momentary joy when I received the appreciation for my song.
3. Instant → Swift With today’s swift communication, we can share information across the world in seconds.
4. Core → Essential Our school focuses on the essential values—collaboration, innovation, and unity.
5. Valuable → Useful My teacher’s advice on time management is a useful reminder of the importance of planning.
6. Abiding → Lifelong Reading books and listening to music can give lifelong pleasure that will last forever.
II. Match the expressions from the text in Column 1 with their meanings in Column 2.
Ans:
III. Make as many words as you can out of the following words. One example has been done for you 1. community — city, toy, tune, unity, mutiny, cot 2. important 3. magnificent 4. satisfied 5. generation 6. achievement Ans: 1. community → city, toy, tune, unity, mutiny, cot 2. important → part, port, ant, rant, tip, trap 3. magnificent → gift, neat, fan, cane, cent, game 4. satisfied → sad, fade, side, date, site, idea 5. generation → gate, rate, near, tone, rent, giant 6. achievement → achieve, name, time, heat, team, event
IV. Let us play a word game—antakshari.
1. Write the synonym of the word according to the clues given below. 2. For example, the synonym of the first word, ‘IMPORTANT’, ends with the letter ‘T’. The synonym of the second word begins with ‘T’, ‘TROUBLE’.Then write the synonym of the third word starting with the letter ‘E’. 3. Now, continue with the word game.
(i) major (synonym) (ii) problem (synonym) (iii) uneven (antonym) (iv) related to post office (v) our moral duty (vi) related to time (vii) prosperous (synonym) (viii) related to being alive Ans:
V. Study the highlighted punctuation marks in the following sentences from the text. 1. … but that you should not be satisfied in merely keeping my awards—the challenge before you was to earn your rewards for the work that you did in your lifetime. 2. Writing—even if it is a letter—not only conveys our present concerns and views of the events taking place around us, but it becomes a possession. The highlighted punctuation is a long dash that can be used to link words or numbers or to create a pause in a sentence. There are two main types:the shorter en dash (–) and the longer em dash (—). Note: Do not add spaces around the dash and avoid overusing them. Read the sentences below and insert either an em dash (—) or an en dash (–) suitably. (i) The conference will be held from March 15 March 20. Ans: The conference will be held from March 15–March 20. (en dash for range) (ii) I have lived in Kanpur Raipur for five years now. Ans: I have lived in Kanpur–Raipur for five years now. (en dash is normally used for connections between places) (iii) The artist known for her vibrant paintings will be showcasing her work in Guwahati. Ans: The artist—known for her vibrant paintings—will be showcasing her work. (em dash for extra information) (iv) The scores for the final match were 2 3. Ans: The scores for the final match were 2–3. (en dash for scores) (v) My favourite snacks, samosas, pakoras, and jalebis are all from India. Ans: My favourite snacks—samosas, pakoras, and jalebis—are all from India. (em dash for extra information) (vi) The train journey from Belagavi Bengaluru takes less than ten hours. Ans: The train journey from Belagavi–Bengaluru takes less than ten hours. (en dash is normally used for connections between places) (vii) I just finished reading the book it was absolutely amazing! Ans: I just finished reading the book—it was absolutely amazing! (viii) The event will take place on September 1 September 5. Ans: The event will take place on September 1–September 5. (en dash for range)
VI. Study the highlighted words in the following sentences from the text. 1. We told you that, of course, this medal was yours as much as it was mine… 2. …who were trying to gain control over their lives. 3. I have always emphasised that it’s a recognition of the achievements of many people… In the given sentences, ‘yours’ and ‘theirs’ indicate possession—something belonging to you and them, respectively. But, ‘it’s’ does not indicate possession rather is the contracted form of ‘it is’.Fill in the blanks by choosing the correct word from the options given in the brackets.
(i) Please remember to bring ___________ (your/you’re) assignment book. (ii) The team celebrated ___________ (their/they’re) victory after the match. (iii) ___________ (its/it’s) important to stay hydrated during the hot summer days. (iv) I can’t believe ___________ (your/you’re) going to the mango festival today. (v) The students said ___________ (they’re/their) excited for the field trip tomorrow (vi) The tree lost ___________ (its/it’s) leaves in the winter.
Ans:
(i) Please remember to bring your assignment book.
(ii) The team celebrated their victory after the match.
(iii) It’s important to stay hydrated during the hot summer days.
(iv) The students said they’re excited for the field trip tomorrow.
(v) The tree lost its leaves in the winter.
VII. Read the following phrases from the text. 1. In today’s fast-paced world 2. On the telephone 3. With great, abiding pleasure 4. In the chapters that follow 5. At a certain time 6. With equal respect and love The given phrases are Prepositional Phrases.Now, complete the following sentences with suitable prepositional phrases.
(i) The children played happily in _______________________. Ans: The children played happily in the park. (ii) Nisha placed the vase on _______________________. Ans: Nisha placed the vase on the table. (iii) We discussed the project in _______________________. Ans: We discussed the project in the classroom. (iv) Preethi found the lost book under _______________________. Ans: Preethi found the lost book under the bed. (v) The cat slept on _______________________. Ans: The cat slept on the mat.
VIII. Rearrange the following jumbled words and phrases into meaningful sentences. 1. barked/at the strangers/the dog/loudly/the house/walking by Ans: The dog barked loudly at the strangers walking by the house.
2. wrote/my classmate/during the/a story/summer vacations Ans: My classmate wrote a story during the summer vacations.
3. enjoyed/under the trees/they/by the lake/a picnic Ans: They enjoyed a picnic under the trees by the lake.
4. in his/grandfather’s room/hung/he/the painting Ans: He hung the painting in his grandfather’s room.
5. studied/with great determination/for/students/the/exam Ans: The students studied with great determination for the exam.
Let us listen (Page 93)
I. You will listen to the passage about Evelyn Glennie’s journey in overcoming her hearing loss through music and turning a physical disability into a success story. After listening, match the words given in Column 1 with their meanings given in Column 2.
Ans:
II. Listen to the passage once again and state whether the statements given below are true or false. 1. Evelyn Glennie became deaf suddenly. Ans: False
Her hearing loss was gradual.
2. Her headmistress ignored the signs of her hearing loss. Ans: False
Teachers typically notice such signs.
3. Ron Forbes was a percussionist who helped Evelyn. Ans: True
4. Evelyn used her skin and bones to sense music. Ans: True
5. She gave up on music after her diagnosis. Ans: False
She continued her music career.
III. Answer the following questions in one or two words. 1. How did Evelyn turn her disability into a strength? Ans: By sensing vibrations
2. What role did Ron Forbes play in her life? Ans: Mentor/Percussionist
3. What lessons can we learn from Evelyn’s story about how to treat others who have different abilities or challenges? Ans: Empathy, support
I. Read the highlighted word in the following sentence from the text. That common good will present itself to you in many forms every day. The highlighted word is pronounced with the stress on the second syllable. PreSENT (verb) the stress is on the second syllable. PREsent (noun), the stress is on the first syllable. Now, read aloud the given words with the correct syllable stress according to the part of speech given in the brackets.
Ans: Words with “Zhuh” sound (-sure ending): measure, leisure, reassure, enclosure, exposure, composure
Words with “Shuh” sound (-sure but sharp ‘sh’): pressure, fissure, ensure, unsure
II. Read the highlighted words in the following sentence from the text. …but it becomes a possession that can be treasured and re-read over the years, with great, abiding pleasure. treasure [treh-Zhuh] pleasure [pleh-Zhuh] In the highlighted words, -sure has a ‘Zhuh’ sound. There are some words which end in -sure, but have a ‘shuh’ sound as in pressure [preh-shuh]. Now, read aloud the words given below with the ‘shuh’ or ‘zhuh’ sound.
Ans: This is a classroom activity. Read the words with correct pronunciation:
III. Pausing and chunking are techniques used when speaking to organise information for the listener. Speakers can split what they say into smaller segments—either individual words or groups of words—to convey ideas or emphasise key points clearly.
Q1. Read aloud the following lines from the poem.Now, read the lines of the poem once again by taking a shorter pause (/)or a long pause (//). Notice which version is easier for a listener to follow and understand. Ans: When things go wrong,/as they sometimes will,// when the road you’re trudging/seems all uphill,// when the funds are low/and the debts are high,// and you want to smile/but you have to sigh,// when care is pressing you down a bit/ —rest if you must,/ but don’t you quit.// Q2. Chunk the stanzas with (/) or (//) and read it aloud.
Ans: Life is queer with its twists and turns, / As everyone of us sometimes learns. // And many a fellow turns about / when he might have won / had he stuck it out. // Don’t give up / though the pace seems slow— / you may succeed / with another blow. // Often the goal is nearer than it seems / to a faint and faltering man. //.
Let us write (Page 95)
I. Write a letter to your grandparents thanking them for the valuable lessons they have taught you. Include how their wisdom has shaped your values and behaviour in life. 1. Thank them for teaching you important lessons, sharing stories, or helping others in the neighbourhood. 2. Share how their actions inspire you to contribute to the betterment of the society. Ans: Your Address [City, State, ZIP Code] 1 August 2025
Dear Grandparents,
I hope you are both doing well. I want to thank you for all the lessons you have taught me. Grandpa, your honesty and responsibility inspire me to always do what is right. Grandma, your kindness and patience show me how to care for others.
Because of you, I have learned the value of hard work, humility, and helping those in need. I try to follow your example by respecting my teachers, being kind to my friends, and helping in my community, like when we clean the park.
Thank you for guiding me and making me a better person.
With love, [Your Name]
Also read: NCERT Textbook: Verghese Kurien — I Too Had A Dream
Let us explore (Page 96)
I. The White Revolution led to an increase in milk production in India primarily due to the efforts of Dr. Verghese Kurien. Therefore, he is known as ‘the Milkman of India’.
The Green Revolution in India introduced high-yielding varieties of wheat and rice. It was initiated by Dr. M.S. Swaminathan. He was awarded the Padma Shri in 1967, Padma Bhushan in 1972, Padma Vibushan in 1989, and Bharat Ratna in 2024. The Blue Revolution focused on increasing fisheries production and productivity in India. This initiative was led by Dr. Hiralal Chaudhuri. In groups of four, research more information about these revolutions. Share your findings with your classmates and teacher. Ans: White Revolution (Operation Flood)
Leader: Dr. Verghese Kurien (Milkman of India).
Period: Started in 1970.
Key Idea: Increase milk production through a nationwide dairy development program.
Impact:
India became the largest producer of milk in the world.
Improved income for rural farmers, especially women.
Strengthened cooperative societies like Amul.
Helped reduce dependence on milk imports.
Green Revolution
Leader: Dr. M.S. Swaminathan (Father of Green Revolution in India).
Period: Mid-1960s.
Key Idea: Use of high-yielding variety (HYV) seeds, chemical fertilizers, pesticides, and modern irrigation.
Impact:
Boosted production of wheat and rice, reducing food shortages.
Helped India move from food-deficient to self-sufficient in grains.
Created surplus stocks in the Food Corporation of India (FCI).
Challenges: Regional imbalance (benefited Punjab, Haryana, and western UP more), overuse of fertilizers and water, environmental concerns.
Blue Revolution
Leader: Dr. Hiralal Chaudhuri (Father of Induced Breeding in Fish).
Period: Started in 1970s, expanded in 1980s.
Key Idea: Improve fish production and aquaculture.
Impact:
Increased inland and marine fish production.
Popularized aquaculture techniques like induced breeding.
Boosted rural employment and nutrition (fish as protein source).
Helped India become one of the world’s largest producers of fish.
II. Vocational training for people in rural regions is provided by the JanShikshan Sansthan (JSS), under the Ministry of Skill Development and Entrepreneurship, Government of India. The objective of JSS is to economically uplift the rural population by imparting essential skill-based training. This enables local trades to grow and creates new opportunities for the people in the region. Find out more about JSS with the help of the link given: https://jss.gov.in/staticPage Ans: JSS provides skill-based training to rural people to help them earn a living. It teaches skills like tailoring, farming, and crafts, creating job opportunities.
III Financial Literacy Read the passage and discuss the questions given below. Financial literacy is an essential lifelong skill that should be taught to children from an early age. Preparing children with financial literacy skills is an investment in their future well-being. By imparting an understanding of money, saving, budgeting, and wise spending, we empower them to navigate the financial world with confidence as they grow up. Whether it is managing a cooperative society, a start-up, or a household budget, financial literacy is an ongoing learning process that evolves as children grow and their understanding deepens. An introduction to concepts of saving, budgeting, investing, and responsible spending equips them with essential life skills. Children learn best through practical examples. Real-life situations help them understand the value of money and the importance of making wise financial choices. Thinking critically before spending helps cultivate responsible habits. The concept of saving and investing can be explained through relatable examples, such as planting seeds that grow into trees bearing fruit. As children grow older, they will learn to differentiate between needs and wants, compare prices before making a purchase and understand the importance of financial literacy. Discuss in groups. 1. The importance of learning about money and saving from a young age. Share one example of it. 2. Imagine you received some money as a gift. How would you decide whether to save it, spend it, or invest it? What factors would you consider? 3. If you were in charge of planning a small event at school with a limited budget, how would you manage the money? 4. Why is financial literacy important for everyone? Ans:
1. Learning about money early It builds good habits like saving and budgeting. For example, saving pocket money teaches patience and helps buy something useful later.
2. Gift money – save, spend, or invest? I would save some for future needs, spend a little on something I enjoy, and invest the rest. My choice would depend on my needs, goals, and how much I’ve already saved.
3. Planning a school event on a budget List essentials, spend only on priorities, look for cost-saving options (borrow, recycle, donations), and track expenses. Keep a small amount aside for emergencies.
4. Why financial literacy matters It helps everyone manage money wisely, avoid debt, save for the future, and make smart financial choices for a secure life.
Q1: A quadrilateral has three acute angles, each measure 80°. What is the measure of the fourth angle?
Solution:
Let x be the measure of the fourth angle of a quadrilateral.
Sum of the four angles of a quadrilateral = 360°
80° + 80° + 80° + x = 360°
x = 360° – (80° + 80° + 80°)
x = 360° – 240°
x = 120°
Hence, the fourth angle is 120°.
Q2: In a quadrilateral ABCD, the measure of the three angles A, B and C of the quadrilateral is 110°, 70° and 80°, respectively. Find the measure of the fourth angle.
Solution: Let,
∠A = 110°
∠B = 70°
∠C = 80°
∠D = x
We know that the sum of all internal angles of quadrilateral ABCD is 360°.
∠A + ∠B+ ∠C+∠D = 360°
110° + 70° + 80° + x = 360°
260° + x = 360°
x = 360° – 260°
x = 100°
Therefore, the fourth angle is 100°.
Q3: The opposite angles of a parallelogram are (3x + 5)° and (61 – x)°. Find the measure of four angles.
Solution: Given, (3x + 5)° and (61 – x)° are the opposite angles of a parallelogram. We know that the opposite angles of a parallelogram are equal. Therefore, (3x + 5)° = (61 – x)° 3x + x = 61° – 5° 4x = 56° x = 56°/4 x = 14° ⇒ 3x + 5 = 3(14) + 5 = 42 + 5 = 47 61 – x = 61 – 14 = 47 The measure of angles adjacent to the given angles = 180° – 47° = 133° Hence, the measure of four angles of the parallelogram are 47°, 133°, 47°, and 133°.
Q4: A diagonal and a side of a rhombus are of equal length. Find the measure of the angles of the rhombus.
Solution: Let ABCD be the rhombus. Thus, AB = BC = CD = DA
Given that a side and diagonal are equal. AB = BD (say) Therefore, AB = BC = CD = DA = BD Now, all the sides of a triangle ABD are equal. Therefore, ΔABD is an equilateral triangle. Similarly, ΔBCD is also an equilateral triangle. Thus, ∠A = ∠ABD = ∠ADB = ∠DBC = ∠C = ∠CDB = 60° ∠B = ∠ABD + ∠DBC = 60° + 60° = 120° And ∠D = ∠ADB + ∠CDB = 60° + 60° = 120° Hence, the angles of the rhombus are 60°, 120°, 60° and 120°.
Q5: In a trapezium PQRS, PQ || RS, ∠P : ∠S = 3 : 2 and ∠Q : ∠R = 4 : 5. Find the angles of the trapezium.
Solution:
Given,
∠P : ∠S = 3 : 2
∠Q : ∠R = 4 : 5
Let 3x and 2x be the measures of angles P and S.
And let 4x and 5x be the measures of angle Q and R.
As we know, the sum of adjacent angles at the corners of non-parallel sides of a trapezium = 360°
∠P + ∠S = 180°
3x + 2x = 360°
5x = 180°
x = 180°/5 = 36°
So, 3x = 3(36°) = 108°
2x = 2(36°) = 72°
Similarly,
4x + 5x = 180°
9x = 180°
x = 180°/9 = 20°
So, 4x = 4(20°) = 80°
5x = 5(20°) = 100°
Therefore, the angles of the trapezium are ∠P = 108°, ∠Q = 80, ∠R = 100 and ∠S = 72.
Q6: The diagonals of a rhombus are 12 cm and 7.5 cm. Find the area of a rhombus.
Solution:
Given: Length of diagonal 1 = 12 cm
Length of diagonal 2 = 7.5 cm
We know that,
Area of a rhombus = (1/2) × Diagonal 1× Diagonal 2 square units
A = (½)×12×7.5
A = 6×7.5
A = 45 cm2
Hence, the area of a rhombus is 45 cm2.
Q7: ABCD is a quadrilateral, whose angles are ∠A = 5(a+2)°, ∠B = 2(2a+7)°, ∠C = 64°, ∠D = ∠C-8°. Determine the value of ∠A.
Build numbers by repeating symbols. For example 324 which equals 100 + 100 + 100 + 10 + 10 + 4 is written as
Limit: Needs infinite symbols for very large numbers.
Abacus
Decimal-based calculating tool.
Each line = power of 10.
Counters above line = value of 5× that landmark.
Mesopotamian System
Location: Ancient civilisation in present-day Iraq and nearby regions.
Time Period: Around 4000 years ago.
Base: Base-60 (sexagesimal system).
Symbols:
Two main wedge-shaped symbols (cuneiform writing) for numbers.
Numbers formed by repeating and combining these symbols.
Special Use:
Still used today in measuring time (60 seconds in a minute, 60 minutes in an hour) and angles (360° circle).
Mayan Number System Basics
Base: Modified base-20.
1st place: 1’s (units)
2nd place: 20’s
3rd place: 360’s (not 400, due to calendar reasons)
4th place: 7200’s, etc.
Symbols:
Dot (•) = 1
Bar (—) = 5
Shell = 0 (placeholder)
Numbers are written vertically, lowest value at bottom.
Chinese Rod Numeral System – Key Points1. Purpose
Two systems existed:
Written system – for recording quantities.
Rod numeral system – for performing calculations efficiently.
2. Rod Numerals
Base: Decimal (base-10), like our modern system.
Digits 1–9: Represented using vertical or horizontal rods (small sticks or lines).
Place value:
Vertical rods → used for units and hundreds places.
Horizontal rods → used for tens and thousands places. (This alternation prevented confusion between adjacent digits.)
3. Zero Representation
Like the Mesopotamians: used a blank space to indicate an empty place value.
Advantage: Due to uniform rod sizes, the blank space was easier to identify.
Note: If they had an actual symbol for zero, it would have been a fully developed place value system like the Hindu–Arabic numerals.
Spread of Hindu–Arabic Numerals
Base: Base-10 (decimal system).
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Place Value System: Value of a digit depends on its position. Example: In 375, 3 → Hundreds place = 3 × 100 = 300 7 → Tens place = 7 × 10 = 70 5 → Ones place = 5 × 1 = 5
Use of Zero: A major contribution by Indian mathematicians (Aryabhata, Brahmagupta).
Spread: Carried to Europe by Arab traders → became the Hindu–Arabic numerals we use today.
Q1: A new sapling starts with a height of 2 cm. How tall might the plant be after 4 years if its height doubles each year?
Solution: Initial height = 2 cm
The height doubles each year, so after 4 years: = 21 × 24 = (2)1+4 = 32 cm
The plant will be 32 cm tall after 4 years.
Q2: The number of books in a library increases by 5 times every 2 years. A library starts with 100 books. How many books will there be in the library after:
(a) 6 years
(b) 10 years
Solution: Initial number of books = 100
The number of books increases by a factor of 5 every 2 years.
(a) After 6 years (3 periods of 2 years):
Number of books after 6 years = 100 × 5³ = 100 × 125 = 12500 books
After 6 years: 12500 books
(b) After 10 years (5 periods of 2 years):
Number of books after 10 years = 100 × 5⁵ = 100 × 3125 = 312500 books
After 10 years: 312500 books
Q3: The number of bacteria in a culture increases 3 times every hour. A culture starts with 1 bacterium. How many bacteria will be in the culture after 5 hours?
Solution:
Initial bacteria = 1
The bacteria triple every hour, so after 5 hours: Number of bacteria after 5 hours = 1 × 3⁵ = 1 × 243 = 243 bacteria
The number of bacteria will be 243 after 5 hours.
Q4: The planet Uranus is approximately 2,896,819,200,000 metres awayfrom the Sun. What is this distance in standard form?
Solution:
The distance of Uranus from the Sun is given as 2,896,819,200,000 meters.
To express this in standard form:
2,896,819,200,000 = 2.8968192 × 10¹²
So, the distance of Uranus from the Sun in standard form is:
2.8968192 × 10¹² meters
Q5: An inch is approximately equal to 0.02543 metres. Write this distance in standard form.
Solution:
We are given that 1 inch ≈ 0.02543 meters.
To express this in standard form:
0.02543 = 2.543 × 10⁻²
So, the distance in standard form is:
2.543 × 10⁻² meters
Q6: A particular star is at a distance of about 8.1 × 10¹³ km from the Earth. Assuring that light travels at 3 × 10⁸ m per second, find how long does light takes from that star to reach the Earth.
Solution:
Given, a particular star is at a distance of about 8.1 × 10¹³ km from the Earth.
Assuring that light travels at 3 × 10⁸ m per second.
We have to find the time the light takes from that star to reach the Earth.
We know, speed = distance / time
Given, speed = 3 × 10⁸ m/s
Distance = 8.1 × 10¹³ km
We know, 1 km = 1000 m
= 8.1 × 10¹³ × 10³
Using the law of exponents,
am × an = am + n
= 8.1 × 1013 + 3
= 8.1 × 1016 m
Time = distance / speed
= 8.1 × 1016 / 3 × 108
= (8.1 / 3) × (1016/108)
= 2.7 × (1016/108)
Using the law of exponents,
am ÷ an = am – n
= 2.7 × 1016 – 8
= 2.7 × 10⁸ seconds
Therefore, the required time is 2.7 × 10⁸ seconds.
Q7: In a stack, there are 4 books, each of thickness 15mm, and 6 paper sheets, each of thickness 0.010mm. What is the total thickness of the stack?
Solution:
Thickness of each book = 15mm
Number of books in the stack = 4
Thickness of 4 books = 4 × 15 = 60mm
Thickness of each paper sheet = 0.010mm
Thickness of 6 paper sheets = 6 × 0.010 = 0.060mm
Total thickness of the stack = 60mm + 0.060mm = 60.060mm
Q8: A number when divides ( –15) –1 results ( –5) –1. Find the number.
Solution:
Let x be the number such that
( –15) –1 ÷ x = ( –5) –1
⇒ –1/15 ÷ x = –⅕
⇒ –1/15 × 1/x = –⅕
⇒ –1/15x = –⅕
⇒ 15x = 5
⇒ x = ⅓ or 3 –1
Q9: A savings account balance quadruples every 3 years. The initial balance in a savings account is 1500 rupees. How much will the balance be after 9 years?
Solution:
Initial balance = 1500 rupees
The balance quadruples every 3 years, so after 9 years:
Balance after 9 years = 1500 × 4³ = 1500 × 64 = 96000 rupees
The balance will be 96000 rupees after 9 years
Q10: The volume of the Earth is approximately 7.67 × 10–7 times thevolume of the Sun. Express this figure in usual form.
Solution:
The volume of Earth is approximately 7.67 × 10–7 times the volume of the Sun. We are asked to express this in usual form.
To convert from scientific notation to usual form, we move the decimal point to the left by 7 places (since the exponent is -7):
7.67 × 10–7 = 0.000000767
So, the volume of Earth as a fraction of the volume of the Sun is:
Q1: Give a reason to show that the number given below is a perfect square: 5963 Sol: The unit digit of the square numbers will be 0, 1, 4, 5, 6, or 9 if we examine the squares of numbers from 1 to 10. Thus, the unit digit for all perfect squares will be 0, 1, 4, 5, 6, or 9, and none of the square numbers will end in 2, 3, 7, or 8. Given 5963 We have the property of a perfect square, i.e. a number ending in 3 is never a perfect square. Therefore the given number 5963 is not a perfect square.
Q2: 2025 plants are to be planted in a garden in a way that each of the rows contains as many plants as the number of rows. Find the number of rows and the number of plants in each row. Sol: Let the number of rows be x. Thus, the number of plants in each row = x. Total many contributed by all the students = x × x = x² Given, x² = Rs.2025 x2 = 3 × 3 × 3 × 3 × 5 × 5 ⇒ x2 = (3 × 3) × (3 × 3) × (5 × 5) ⇒ x2 = (3 × 3 × 5) × (3 × 3 × 5) ⇒ x2 = 45 × 45 ⇒ x = √(45 × 45) ⇒ x = 45 Therefore, Number of rows = 45 Number of plants in each row = 45
Q3: Find out the cube root of 13824 by the prime factorisation method. Sol: First, let us prime factorise 13824: 13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2 ³ × 2 ³ × 2 ³ × 3 ³ 3√13824 = 2 × 2 × 2 × 3 = 24
Q4: (13/10) ³ Sol: The cube of a rational number is the result of multiplying a number by itself three times. To evaluate the cube of (13/10) ³ Firstly we need to convert into proper fractions, i.e.(13/10) ³ We need to multiply the given number three times, i.e. (13/10) × (13/10) × (13/10) = (2197/1000) ∴ the cube of (1 3/10) is (2197/1000)
Q5: By what least number should the number be divided to obtain a number with a perfect square? In this, in each case, find the number whose square is the new number 4851. Sol: The number is a perfect square if and only if the prime factorization creates pairs; it is not exactly a perfect square if it is not paired up. Given 4851, Resolving 4851 into prime factors, we obtain 4851 = 3 X 3 X 7 X 7 X 11 = (32 X 72 X 11) To obtain a perfect square, we need to divide the above equation by 11 we obtain, 9075 = 3 X 3 X 7 X 7 The new number = (9 X 49) = (3² X 7² ) Taking squares on both sides of the above equation, we obtain ∴ The new number = (3 X 7)² = (21)² Therefore, the new number is a square of 21
Q6: Find the cube root of 10648 by the prime factorisation method. Sol:10648 = 2 × 2 × 2 × 11 × 11 × 11 Grouping the factors in triplets of number equal factors, 10648 = (2 × 2 × 2) × (11 × 11 × 11) Here, 10648 can be grouped into triplets of number equal factors, ∴ 10648 = 2 × 11 = 22 Therefore, the cube root of 10648 is 22.
Q7: Without adding, find the sum of the following: (1+3+5+7+9+11+13+15+17+19+21+23) Sol: (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23) As per the given property of perfect square, for any natural number n, we have some of the first n odd natural numbers = n² But here n = 12 By applying the above the law, we get thus, (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23) = 12² = 144
Q8: By what least number should the given number be divided to get a perfect square number? In each of the following cases, find the number whose square is the new number 1575. Sol: A method for determining the prime factors of a given number, such as a composite number, is known as prime factorisation. Given 1575, Resolve 1575 into prime factors, we get 1575 = 3 X 3 X 5 X 5 X 7 = (3² X 5² X 7) To obtain a perfect square, we have to divide the above equation by 7 Then we get, 3380 = 3 X 3 X 5 X 5 New number = (9 X 25) = (3² X 5² ) Taking squares on both sides of the above equation, we get ∴ New number = (3 X 5)² = (15)²
Q9:If m is the required square of a natural number given by n, then n is (a) the square of m (b) greater than m (c) equal to m (d) √m Ans: (d) Sol: n² = m Then, = n = √m Q10: The cube of 100 will have _________ zeroes. Sol: The cube of 100 will have six zeroes. = 1003 = 100 × 100 × 100 = 1000000
Q11: Use the following identity and find the square of 189. (a – b)² = a² – 2ab + b² Sol: 189 = (200 – 11) 2 = 40000 – 2 x 200 x 11 + 112 = 40000 – 4400 + 121 = 35721
Q12: What would be the square root of the number 625 using the identity (a +b)² = a² + b² + 2ab? Sol: (625)² = (600 + 25)² = 600² + 2 x 600 x 25 +25² = 360000 + 30000 + 625 = 390625
Q13:Show that the sum of two consecutive natural numbers is 13². Sol: Let 2n + 1 = 13 So, n = 6 So, ( 2n + 1)² = 4n² + 4n + 1 = (2n² + 2n) + (2n² + 2n + 1) Substitute n = 6, (13)² = ( 2 x 6² + 2 x 6) + (2 x 6² + 2 x 6 + 1) = (72 + 12) + (72 + 12 + 1) = 84 + 85
Q15: Find the cube root of 91125 by the prime factorisation method. Sol: 91125 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 By grouping the factors in triplets of equal factors, 91125 = (3 × 3 × 3) × (3 × 3 × 3) × (5 × 5 × 5) Here, 91125 can be grouped into triplets of equal factors, ∴ 91125 = (3 × 3 × 5) = 45 Thus , 45 is the cube root of 91125.
Q16: A cuboid of plasticine made by Parikshit with sides 5 cm, 2 cm, and 5 cm. How many such cuboids will be needed to form a cube? Sol: The given side of the cube is 5 cm, 2 cm and 5 cm. Therefore, volume of cube = 5 × 2 × 5 = 50 The prime factorisation of 50 = 2 × 5 × 5 Here, 2, 5 and 5 cannot be grouped into triples of equal factors. Therefore, we will multiply 50 by 2 × 2 × 5 = 20 to get the perfect square. Hence, 20 cuboids are needed to form a cube.
Q17: State true or false. (i) The cube of any odd number is even (ii) A perfect cube never ends with two zeros. (iii) If the square of a number ends with 5, then its cube ends with 25. (iv) There is no perfect cube which ends with 8. (v) The cube of a two-digit number may be a three-digit number. (vi) The cube of a two-digit number may have seven or more digits. (vii) The cube of a single-digit number may be a single-digit number. Sol: (i) This statement is false. Taking a cube of any required odd numbers 3³= 3 x 3 x 3 = 27 7³=7 x 7 x 7= 343 5³=5 x 5 x 5=125 All the required cubes of any given odd number will always be odd. (ii) This statement is true. 10³= 10 x 10 x 10= 1000 20³ = 20 x 20 x 20 = 2000 150³ =150 x150 x150 = 3375000 Hence a perfect cube will never end with two zeros. (iii) This statement is false. 15²= 15 x15= 225 15³= 15 x 15 x 15= 3375 Thus, the square of any given number ends with 5; then the cube ends with the number 25 is an incorrect statement. (iv) This statement is false. 2³= 2x2x2= 8 12³ = 12 x 12 x 12= 1728 Accordingly, There are perfect cubes ending with the number 8 (v) This statement is false. The minimum two digits number is 10 And 10³=1000→4 Digit number. The maximum two digits number is 99 And 99³=970299→6 Digit number Accordingly, the cube of two-digit numbers can never be a three-digit number. (vi) This statement is false 10³=1000→4 Digit number. The maximum two digits number is 99 And 99³=970299→6 Digit number Accordingly, the cube of two-digit numbers can never have seven or more digits. (vii) This statement is true 1³ = 1 x 1 x 1= 1 2³ = 2 x 2 x 2= 8 According to the cube, a single-digit can be a single-digit number.
Q18: Find the cube of 3.5. Sol: 3.53 = 3.5 x 3.5 x 3.5 = 12.25 x 3.5 = 42.875
Q19: Find the smallest whole number from which 1008 should be multiplied to obtain a perfect square number. Also, find out the square root of the square number so obtained. Sol: Let us factorise the number 1008.1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7 = ( 2 × 2 ) × ( 2 × 2 ) × ( 3 × 3 ) × 7 Here, 7 cannot be paired. Therefore, we will multiply 1008 by 7 to get a perfect square. New number so obtained = 1008 ×7 = 7056 Now, let us find the square root of 70567056 = 2 × 2 × 2 × 2 × 3 × 3× 7 × 7 7056 = (2 × 2 ) × ( 2 × 2 ) × ( 3 × 3 ) ×( 7 × 7 ) 7056 = 2² × 2² × 3² × 7² 7056 = (2 × 2 × 3 × 7)² Therefore; √7056 = 2 × 2 × 3 × 7 = 84
Q21: There are _________ perfect cubes between 1 and 1000. Sol: There are 8 perfect cubes between 1 and 1000. 2 × 2 × 2 = 8 3 × 3 × 3 = 27 4 × 4 × 4 = 64 5 × 5 × 5 = 125 6 × 6 × 6 = 216 7 × 7 × 7 = 343 8 × 8 × 8 = 512 9 × 9 × 9 = 729
Q22: Is 392 a perfect cube? If not, find the smallest natural number by which 392 should be multiplied so that the product is a perfect cube. Sol: The prime factorisation of 392 gives: 392 = 2 x 2 x 2 x 7 x 7 As we can see, number 7 cannot be paired in a group of three. Therefore, 392 is not a perfect cube. We must multiply the 7 by the original number to make it a perfect cube. Thus, 2 x 2 x 2 x 7 x 7 x 7 = 2744, which is a perfect cube, such as 23 x 73 or 143. Hence, the smallest natural number, which should be multiplied by 392 to make a perfect cube, is 7.
Q23: Which of the following numbers are in perfect cubes? In the case of a perfect cube, find the number whose cube is the given number 256 Sol: A perfect cube can be expressed as a product of three numbers of equal factors Resolving the given number into prime factors, we obtain 256 = 2 × 2 × 2 × 2 × 2× 2 × 2 × 2 Since the number 256 has more than three factors ∴ 256 is not a perfect cube.
Q24: Find the smallest number by which 128 must be divided to get a perfect cube. Sol: The prime factorisation of 128 is given by: 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 By grouping the factors in triplets of equal factors, 128 = (2 × 2 × 2) × (2 × 2 × 2) × 2 Here, 2 cannot be grouped into triples of equal factors. Therefore, to obtain a perfect cube, we will divide 128 by 2.
Q25: There are _________ perfect squares between 1 and 100. Sol: There are 8 perfect squares between 1 and 100. 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25 6 × 6 = 36 7 × 7 = 49 8 × 8 = 64 9 × 9 = 81
Q26: Show that each of the numbers is a perfect square. In each case, find the number whose square is the given number: 7056 Sol: 7056, A perfect square is always expressed as a product of pairs of prime factors. Resolving 7056 into prime factors, we obtain 7056 = 11 X 539 = 12 X 588 = 12 X 7 X 84 = 84 X 84 = (84)² Thus, 84 is the number whose square is 5929 Therefore,7056 is a perfect square.
Example: 16, 25, 49 are squares; but 23, 47 are not.
(b) Digits Pattern
Numbers ending in 1 or 9 → square ends in 1.
Numbers ending in 4 or 6 → square ends in 6.
Example: 19² = 361 (ends in 1).
(c) Zeros Rule
If number ends in n zeros, square ends in 2n zeros.
Example: 100² = 10,000 (2 zeros → 4 zeros).
(d) Parity (Even/Odd)
Square of even number = even.
Square of odd number = odd.
Example: 12² = 144 (even), 25² = 625 (odd).
(e) Odd Number Differences
Difference of consecutive squares = odd number.
2² – 1² = 3, 3² – 2² = 5, 4² – 3² = 7.
Sum of first n odd numbers = n².
(f) Perfect Square Test (Subtraction Rule)
Subtract consecutive odd numbers from n.
If result becomes 0 → number is a perfect square.
Example: 25 – 1 – 3 – 5 – 7 – 9 = 0 → 25 is a square.
(g) Finding Next Square
(n+1)² = n² + (2n+1).
Example: 35² = 1225 → 36² = 1225 + 71 = 1296.
(h) Numbers Between Squares
Between n² and (n+1)² → always 2n numbers.
Example: Between 4² = 16 and 5² = 25 → 8 numbers.
(i) Triangular Numbers Relation
Triangular numbers: 1, 3, 6, 10, 15, …
Sum of two consecutive triangular numbers = perfect square.
Square Roots
Definition
If y = x², then x = √y.
Every square root has ± values, but usually positive root is used.
Example: √49 = ±7.
Methods to Check/Find Square Roots
Listing Squares → compare with nearby perfect squares. List squares of natural numbers: 1² = 1, 2² = 4, 3² = 9, 4² = 16, … Compare the given number with the list of squares. If it matches a square → the square root is the corresponding number. Quick Tip: If the number is not a perfect square, it lies between the squares of two numbers → √n is between those two numbers.
Successive Subtraction of Odd Numbers → subtract until 0. – Start with the given number. – Subtract 1, 3, 5, 7, 9… successively (odd numbers in order). – Count how many subtractions you can do until the result becomes 0. – The number of subtractions = the square root.Quick Tip:
This method works only for perfect squares.
The sequence of odd numbers always starts from 1.
Prime Factorisation → group factors in pairs.
Example: 256 = 2⁸ → √256 = 2⁴ = 16.
Estimation → use nearby perfect squares. – Identify perfect squares closest to the given number – one smaller, one larger. – Conclude that the square root lies between the roots of these perfect squares. – Refine by checking multiples to find the exact root (if it is a perfect square). Quick Tip:
For numbers not perfect squares, this method gives a good approximate value.
Works well with decimal approximations too.
For Non-Perfect Squares
When a number is not a perfect square, its square root can be estimated by comparing it with nearby perfect squares.
Steps:
Identify the perfect squares just below and above the number.
Conclude that the square root lies between the roots of these perfect squares.
Refine the estimate using linear approximation or trial and error.
Quick Tip:
This method gives a quick approximation.
For more precision, use long division or a calculator.
Cubic Numbers
Definition and Notation
Cube = n³ = n × n × n.
Represents volume of cube of side n.
Examples: 2³ = 8, 3³ = 27, 4³ = 64.
Properties
Cubes grow faster than squares.
Possible last digits of cubes → any digit (0–9).
Relation with Odd Numbers
n³ = sum of n consecutive odd numbers.
Example: 4³ = 13+15+17+19 = 64.
Taxicab Numbers
First discovered by Ramanujan (famous Hardy–Ramanujan number).
1729 = 1³+12³ = 9³+10³.
Smallest number expressible as sum of two cubes in two ways.
Note: Do not add spaces around the dash and avoid overusing them. 
Ans: Words with “Zhuh” sound (-sure ending):
Now, read the lines of the poem once again by taking a shorter pause (/)or a long pause (//). Notice which version is easier for a listener to follow and understand. 
Example: 5 cows → 5 sticks.
Example: “a, b, c…”
This is an early form of written numbers.
Hands, fingers, body parts used to count (e.g., Papua New Guinea).
Marks on bones/walls (e.g., Ishango bone, 20,000–35,000 years ago; Lebombo bone, 44,000 years ago).
1 = urapon, 2 = ukasar, 3 = ukasar-urapon, 4 = ukasar-ukasar, …
Similar systems in South America & South Africa.
Symbols: I (1), V (5), X (10), L (50), C (100), D (500), M (1000).
Landmark numbers: 1, 10, 100, … (powers of 10, base-10 system).
Example: Base-5 system → 1, 5, 25, 125, …
Counters on lines representing powers of 10.
Developed base-60 (sexagesimal) system.
Symbols: 1 and 10.
Numbers grouped by powers of 60 (e.g., 7530 = 2×3600 + 5×60 + 30).
Dots = 1, Bars = 5; numbers written vertically.
Vertical position of the symbol indicates powers of 10.
Uses place value to represent numbers.
Sticks/Pebbles/Seeds (Method 1)
Sounds/Names (Method 2) → Limited by available sounds/letters.
Written symbols (Method 3) → e.g., Roman numerals.
Lebombo Bone (44k yrs old) – lunar calendar.
Gumulgal (Australia): Count in 2’s → Numbers = combinations of 2’s & 1’s.
Common groups in history: 2, 5, 10, 20.
Base-n system: Landmark numbers = powers of n
Advantages: Consistent grouping, easier addition & multiplication.
Base-10, symbols for 1, 10, 100, 1000, 10,000…
Build numbers by repeating symbols. For example 324 which equals 100 + 100 + 100 + 10 + 10 + 4 is written as
Decimal-based calculating tool.
Two main wedge-shaped symbols (cuneiform writing) for numbers.
Numbers formed by repeating and combining these symbols.
Numbers are written vertically, lowest value at bottom.
Example: 8² ÷ 2² = (8 ÷ 2)² = 4²
Squares of natural numbers = perfect squares.
Sum of two consecutive triangular numbers = perfect square.
Listing Squares → compare with nearby perfect squares.
Successive Subtraction of Odd Numbers → subtract until 0.
Example: 256 = 2⁸ 
Estimation → use nearby perfect squares.
Cubes grow faster than squares.
Other examples: 4104, 13832.
Example: 3375 = (3×3×3)(5×5×5) → ³√3375 = 15.
Example: ³√8000 = 20.8000 = (2×2×2) (2×2×2) (5×5×5)= 2×2×2 = 20.