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Real Numbers

Mathematics · Explore the original guide and official source links in this read-only preview.

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This no-login demo shows the five main Class X subjects. The full app keeps the complete CBSE subject catalogue so each student can select the subjects on her school timetable.

✓Mathematics6 original guides + official linksMapped in demo
✓Science6 original guides + official linksMapped in demo
✓Social Science5 original guides + official linksMapped in demo
✓English Language & Literature2 original guides + official linksMapped in demo
✓Hindi Course A2 original guides + official linksMapped in demo
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MMathematics6 original chapter guidesOriginal + official
CBSECBSE Class X curriculum and syllabusUse the current session document for the official scope and scheme of studies.NCERTNCERT Class X textbook PDFOpen the official textbook or choose Class X and the subject.CBSECBSE sample papers and marking schemesCheck the session label before using a sample paper.CBSECBSE Class X question bankOfficial question-bank landing page for Class X resources.NCERTNCERT exemplar problemsAdditional official practice problems; use them after understanding the chapter.
SScience6 original chapter guidesOriginal + official
CBSECBSE Class X curriculum and syllabusUse the current session document for the official scope and scheme of studies.NCERTNCERT Class X textbook PDFOpen the official textbook or choose Class X and the subject.CBSECBSE sample papers and marking schemesCheck the session label before using a sample paper.CBSECBSE Class X question bankOfficial question-bank landing page for Class X resources.NCERTNCERT exemplar problemsAdditional official practice problems; use them after understanding the chapter.
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Real Numbers

Mathematics · Euclid's division algorithm and fundamental theorem of arithmetic.

StudyNest guide

Real Numbers

Every positive integer can be broken into prime factors in one unique way. Euclid’s algorithm turns repeated division into a reliable method for finding the HCF; the HCF–LCM relationship then connects factor structure with products.

Try these practice prompts
  1. Use Euclid’s division algorithm to find the HCF of 867 and 255.
  2. Explain, in two or three steps, why a rational number plus 2√5 cannot be rational when the rational number is non-zero.
Official source links

These buttons open the official NCERT or CBSE page. StudyNest does not copy or host the textbook.

CBSECBSE Class X curriculum and syllabusUse the current session document for the official scope and scheme of studies.NCERTNCERT Class X textbook PDFOpen the official textbook or choose Class X and the subject.CBSECBSE sample papers and marking schemesCheck the session label before using a sample paper.CBSECBSE Class X question bankOfficial question-bank landing page for Class X resources.NCERTNCERT exemplar problemsAdditional official practice problems; use them after understanding the chapter.
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StudyNest originals

Summaries and practice prompts

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These short explanations and questions are original StudyNest content. They are revision aids, not substitutes for the official textbook, school teaching or the current CBSE syllabus.

Mathematics

Real Numbers

Every positive integer can be broken into prime factors in one unique way. Euclid’s algorithm turns repeated division into a reliable method for finding the HCF; the HCF–LCM relationship then connects factor structure with products.

Try these
  1. Use Euclid’s division algorithm to find the HCF of 867 and 255.
  2. Explain, in two or three steps, why a rational number plus 2√5 cannot be rational when the rational number is non-zero.
Mathematics

Polynomials

A polynomial’s zeros are the input values that make it vanish. For a quadratic, the sum and product of zeros can be read from its coefficients, which lets you check or construct equations without solving every time.

Try these
  1. If the zeros of 2x² − 5x + k are reciprocals of each other, find k and justify your answer.
  2. Create a quadratic polynomial whose zeros have sum 7 and product 10, then verify both relationships.
Mathematics

Pair of Linear Equations

A pair of linear equations represents two lines. One intersection gives a unique solution, parallel lines give no solution, and coincident lines give infinitely many; algebraic methods should agree with that geometric picture.

Try these
  1. Solve 2x + 3y = 13 and 3x − 2y = 4, then check the ordered pair in both equations.
  2. Choose a value of k that makes 3x + ky = 6 and 6x + 2ky = 12 have infinitely many solutions. Explain the condition used.
Mathematics

Quadratic Equations

A quadratic equation models a relationship with an x² term. Factorisation, the quadratic formula and the discriminant are connected tools; the discriminant predicts whether the roots are real and distinct, equal, or non-real.

Try these
  1. Solve x² − 7x + 12 = 0 by factorisation and state how many real roots it has.
  2. For 3x² − 4x + p = 0 to have equal roots, find p using the discriminant.
Mathematics

Triangles

Similarity preserves shape while scaling lengths. Once a pair of triangles is shown similar, corresponding sides and angles can be used to build proportionality results and solve unfamiliar diagrams.

Try these
  1. Two triangles have one equal angle and the including sides in the ratio 3:5. State the additional information needed for SAS similarity.
  2. In similar triangles, a corresponding side changes from 6 cm to 9 cm. What is the scale factor, and how does a 4 cm matching side change?
Mathematics

Statistics & Probability

Grouped data is summarised by representative values rather than listing every observation. Mean, median and mode answer different questions, while probability measures how often an event is expected relative to all equally likely outcomes.

Try these
  1. A fair die is rolled once. Find the probability of getting a prime number and explain the favourable outcomes.
  2. When would the median be a better summary than the mean for a data set containing one unusually large value?
Science

Chemical Reactions

Chemical reactions rearrange atoms into new substances, so a balanced equation must conserve each element. Clues such as colour change, gas, temperature or precipitate help identify a reaction, while oxidation and reduction occur as paired changes.

Try these
  1. Balance: Al + HCl → AlCl₃ + H₂, and name the visible or measurable evidence you would look for in a safe demonstration.
  2. In a displacement reaction, what must be true about the reactivity of the element doing the displacing?
Science

Acids, Bases & Salts

Acids and bases show characteristic behaviour in water, measured with indicators and the pH scale. Neutralisation produces salt and water, and the useful properties of salts depend on the ions and preparation process involved.

Try these
  1. A solution changes blue litmus to red and has pH 3. Classify it and predict what happens when a small amount of base is added.
  2. Write a word equation for a neutralisation reaction and identify the two broad products.
Science

Life Processes

Life processes keep an organism supplied with energy and materials. Nutrition provides raw material, respiration releases usable energy, transport moves substances, and excretion removes metabolic wastes; diagrams are useful only when labels explain the function.

Try these
  1. Why does the small intestine need both a large surface area and a good blood supply for absorption?
  2. Compare aerobic and anaerobic respiration by oxygen requirement and energy release.
Science

Light: Reflection & Refraction

Reflection changes the direction of light at a surface, while refraction changes its speed and usually its path between media. Ray diagrams, sign conventions and the mirror/lens relations connect the visual model to numerical problems.

Try these
  1. Draw a labelled ray diagram for an object beyond 2F of a convex lens and describe the image.
  2. Why does a pencil partly immersed in water appear bent when viewed from the side? Use refraction in your answer.
Science

Electricity

Current describes charge flow, potential difference supplies energy per unit charge, and resistance opposes the flow. Ohm’s law and series/parallel rules let you reason about circuits before substituting values.

Try these
  1. A 6 Ω resistor is connected to a 12 V supply. Find the current and power, showing the formulas used.
  2. Why does adding a parallel branch usually reduce the equivalent resistance of a circuit? Explain using available paths for current.
Science

Our Environment

An ecosystem links organisms with one another and with non-living conditions. Energy moves through food chains and decreases at higher trophic levels, while persistent waste and ozone-damaging substances can create effects beyond the original source.

Try these
  1. Why is a food chain rarely very long? Give an energy-based explanation.
  2. Suggest one household action that reduces non-biodegradable waste and explain the environmental benefit.
Social Science

Nationalism in Europe

Nationalism in Europe grew through political change, shared cultural ideas and struggles over territory. A useful answer separates the roles of state power, popular movements and symbols rather than treating nationalism as a single event.

Try these
  1. How can a common language or cultural symbol help create a sense of national identity? Give two mechanisms.
  2. Distinguish between a political revolution and a diplomatic reorganisation when explaining nation-building.
Social Science

Nationalism in India

Indian nationalism developed through different movements, local experiences and forms of participation. Strong answers connect a movement’s cause, method, social groups involved and outcome instead of listing leaders alone.

Try these
  1. Compare one reason people joined a mass movement with one reason some groups may have remained cautious.
  2. Make a four-step timeline showing how a national movement can move from grievance to organisation to mass action to negotiation.
Social Science

Resources & Development

Resources become useful through technology, institutions and human decisions, not only because they exist in nature. Planning, conservation and fair access are necessary when land, soil and water are limited.

Try these
  1. Why is resource planning needed even in a region that appears resource-rich? Give two reasons.
  2. Propose two measures for reducing soil erosion on cultivated land and explain how each works.
Social Science

Power Sharing & Federalism

Power sharing reduces the risk of conflict by giving different groups a meaningful role. Federalism divides authority across levels of government, but its success also depends on institutions, language policy and democratic practice.

Try these
  1. Give one prudential and one moral reason for sharing power in a democracy.
  2. How is a federation different from a unitary system in the way responsibilities are organised?
Social Science

Development & Sectors

Development is broader than income: health, education, security and freedom also shape people’s lives. Economic sectors describe kinds of work, while money and credit influence how households and enterprises manage exchange and risk.

Try these
  1. Why might two people with the same income report different levels of development? Give two non-income factors.
  2. Classify a school teacher, a farmer and a factory worker by the broad sector of economic activity, and explain each choice.
English Language & Literature

Reading & Inference

Inference means combining what a passage says with what its details imply. A high-quality response points to evidence, avoids adding facts that are not supported, and explains the link between evidence and conclusion.

Try these
  1. Write one inference and one piece of supporting evidence after reading any short newspaper paragraph of your choice.
  2. How is an inference different from a personal opinion in a comprehension answer?
English Language & Literature

Writing & Grammar

Effective formal writing is purposeful, organised and audience-aware. A clear subject line, direct opening, logical paragraphs, appropriate tone and careful editing matter as much as vocabulary.

Try these
  1. Draft a formal email to a school coordinator requesting a one-week revision timetable, using a clear subject line and two specific reasons.
  2. Edit this sentence for clarity: ‘The students who had revised regularly they felt less anxious before the test.’
Hindi Course A

लेखन और व्याकरण

अच्छा लेखन विषय, उद्देश्य और पाठक को ध्यान में रखकर व्यवस्थित किया जाता है। प्रारम्भ, मुख्य विचार, उदाहरण और निष्कर्ष के साथ सही प्रारूप तथा भाषा की शुद्धता अंक बढ़ाती है।

Try these
  1. विद्यालय में पुस्तक-पठन को बढ़ावा देने के लिए लगभग 120 शब्दों का एक औपचारिक सुझाव लिखिए।
  2. किसी अनुच्छेद को संपादित करते समय वर्तनी, विराम-चिह्न और वाक्य-क्रम की जाँच क्यों आवश्यक है?
Hindi Course A

साहित्य पठन

साहित्यिक उत्तर केवल कथानक दोहराने के बजाय भाव, पात्र, भाषा और लेखक के उद्देश्य को जोड़ता है। पाठ से संकेत लेकर अपनी व्याख्या लिखना उत्तर को व्यक्तिगत होते हुए भी प्रमाण-आधारित बनाता है।

Try these
  1. किसी पढ़ी हुई रचना के पात्र में दिखाई देने वाले एक परिवर्तन को कारण और परिणाम सहित समझाइए।
  2. किसी कविता के भाव-सौंदर्य पर उत्तर लिखते समय विषय, बिंब और भाषा—इनमें से कम-से-कम दो आधार कैसे जोड़े जा सकते हैं?
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