case study class 9 chapter 10 maths

• Number Systems Class 9 Assertion Reason Questions Maths Chapter 1

Last Updated on August 26, 2024 by XAM CONTENT

Hello students, we are providing case study questions for class 9 maths. Assertion Reason questions are the new question format that is introduced in CBSE board. The resources for assertion reason questions are very less. So, to help students we have created chapterwise assertion reason questions for class 9 maths. In this article, you will find assertion reason questions for CBSE Class 9 Maths Chapter 1 Number Systems. It is a part of Assertion Reason Questions for CBSE Class 9 Maths Series.

Table of Contents

Assertion Reason Questions on Number Systems

Questions :

Q. 1. Assertion (A): The rationalising factor of $8-\sqrt{7}$ is $8+\sqrt{7}$. Reason (R): If the product of two irrational numbers is rational, then each one is said to be the rationalising factor of the other.

Q. 2. Assertion (A): The sum of two irrational numbers $3-\sqrt{5}$ and $5+\sqrt{5}$ is rational number. Reason (R): The sum of two irrational numbers is always an irrational number.

Q. 3. Assertion (A): The simplified form of $7^4 \times 7^5$ is $7^{20}$. Reason (R): If $a>0$ be a real number and $p$ and $q$ be rational numbers. Then $a^p \times a^q=a^{p+q}$.

1. (a) Assertion (A): It is true that the rationalising factor of $8-\sqrt{7}$ is $8+\sqrt{7}$. Reason (R): It is true to say that each one is rationalising factor in the product of two irrational numbers. Hence, both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of Assertion (A).

2. (c) Assertion (A): Here, $3-\sqrt{5}+5+\sqrt{5}=8$, which is a rational number. So, Assertion (A) is true. Reason (R): It is not always true to say that sum of two irrational number is always an irrational number. Hence, Assertion (A) is true but Reason (R) is false.

3. (d) Assertion (A) is false but Reason (R) is true.

Polynomials Class 9 Assertion Reason Questions Maths Chapter 2

Topics from which assertion reason questions may be asked.

• Representation on number line
• Concept of rationalizing the denominator
• Rationalizing the denominator of expressions with square roots
• Applying the laws of exponents to simplify expressions
• Rationalizing surds

The sum or difference of a rational number and an irrational number is irrational. The product or quotient of a non-zero rational number with an irrational number is irrational.

Assertion reason questions from the above given topic may be asked.

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Frequently Asked Questions (FAQs) on Number Systems Assertion Reason Questions Class 9

Q1: what are assertion reason questions.

A1: Assertion-reason questions consist of two statements: an assertion (A) and a reason (R). The task is to determine the correctness of both statements and the relationship between them. The options usually include: (i) Both A and R are true, and R is the correct explanation of A. (ii) Both A and R are true, but R is not the correct explanation of A. (iii) A is true, but R is false. (iv) A is false, but R is true. or A is false, and R is also false.

Q2: Why are assertion reason questions important in Maths?

A2: Students need to evaluate the logical relationship between the assertion and the reason. This practice strengthens their logical reasoning skills, which are essential in mathematics and other areas of study.

Q3: How can practicing assertion reason questions help students?

A3: Practicing assertion-reason questions can help students in several ways: Improved Conceptual Understanding:  It helps students to better understand the concepts by linking assertions with their reasons. Enhanced Analytical Skills:  It enhances analytical skills as students need to critically analyze the statements and their relationships. Better Exam Preparation:  These questions are asked in exams and practicing them can improve your performance.

Q4: What strategies should students use to answer assertion reason questions effectively?

A4: Students can use the following strategies: Understand Each Statement Separately:  Determine if each statement is true or false independently. Analyze the Relationship:  If both statements are true, check if the reason correctly explains the assertion.

Q5: What are common mistakes to avoid when answering Assertion Reason questions?

A5: Common mistakes include: Not reading the statements carefully and missing key details. Assuming the Reason explains the Assertion without checking the logical connection. Confusing the order or relationship between the statements. Overthinking and adding information not provided in the question.

Q6: Are all integers also rational numbers?

A6: Yes, all integers are rational numbers because they can be expressed as a fraction where the denominator is 1. For example, 5 can be written as 5/1​, making it a rational number.

Q7: What are the key concepts covered in Chapter 1 of CBSE Class 9 Maths regarding number systems?

A7: Chapter 1 of CBSE Class 9 Maths covers concepts such as understanding rational numbers, irrational numbers and Laws of exponents. (i) Review of representation of natural numbers and Integers on number line (ii) Rational numbers on the number line. (iii) Rational numbers as recurring/ terminating decimals (iv) Operations on real numbers. (v) Definition of nth root of a real number (vi) Law of exponents with integral powers

Q8: Can a number be both rational and irrational?

A8: No, a number cannot be both rational and irrational. A rational number can be expressed as a fraction of two integers, while an irrational number cannot. They are mutually exclusive categories.

Q9: What are the important keywords for CBSE Class 9 Maths Number Systems?

A9: List of important keywords given below – Natural Numbers:  Positive Counting number starting from 1. Whole Number:  All natural numbers together with 0. Integers (Z):  Set of all whole numbers and negative of natural numbers Rational Number:  Numbers which can be expressed in p/q form, where q  ≠  0 and p and q are integers. Fraction:  Numbers which can be expressed in form of p/q but are only positive Equivalent Rational Numbers:  Two rational numbers are said to be equivalent, if numerator and denominators of both rational numbers are in proportion or they are reducible to be equal.

Q10: Are there any online resources or tools available for practicing linear equations in one variable assertion reason questions?

A10: A9: We provide assertion reason questions for CBSE Class 8 Maths on our  website . Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit  Physics Gurukul  website. they are having a large collection of case study questions for all classes.

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NCERT Solutions for Class 6 Hindi Chapter 5: Rahim Ke Dohe (Malhar)

NCERT Solutions for Class 6 Chapter 5 Hindi - FREE PDF Download

Vedantu provides NCERT Solutions for Class 6 Hindi Malhar Chapter 5, which explores the profound and timeless wisdom encapsulated in the couplets (dohe) of the renowned poet and saint, Rahim. Known for his insightful and reflective poetry, Rahim’s dohe offers valuable lessons on morality, life, and human nature. In this chapter, students will learn the meanings and interpretations of Rahim’s couplets, which convey deep philosophical thoughts and practical advice in a concise and poetic form.

Our solutions for Class 6 Hindi NCERT Solutions PDF breaks down the lesson into easy-to-understand explanations, making learning fun and interactive. Students will develop essential language skills with engaging activities and exercises. Check out the revised CBSE Class 6 Hindi Syllabus and start practising Hindi Class 6 Chapter 5.

Glance on Class 6 Hindi (Malhar) Chapter 5 - Rahim Ke Dohe

Chapter 5, "Rahim Ke Dohe," introduces students to the poetic couplets of Rahim, a prominent figure in Hindi literature known for his insightful and moralistic poetry. 

Rahim's dohe are short, yet profound verses that reflect his deep understanding of life, ethics, and human nature.

The chapter begins with an introduction to Rahim, explaining his significance as a poet and philosopher.

It highlights his contributions to Hindi literature and the moral values expressed in his poetry.

The core of the chapter revolves around the dohe (couplets) written by Rahim. Each doha is presented with its text and followed by an explanation of its meaning. 

These couplets offer practical wisdom and reflect Rahim's perspective on various aspects of life.

The chapter discusses the central themes of Rahim’s dohe, such as humility, wisdom, the value of time, and the importance of learning from experience. 

It emphasises how these teachings are relevant in everyday life and can guide individuals in their personal growth.

Access NCERT Solutions for Class 6 Hindi Chapter 5 Rahim Ke Dohe

मेरी समझ से.

(क) नीचे दिए गए प्रश्नों का सबसे सही (सटीक) उत्तर कौन-सा है ? उसके सामने तारा (★) बनाइए—

प्रश्न 1. “रहिमन जिह्वा बावरी, कहि गइ सरग पताल। आपु तो कहि भीतर रही, जूती खात कपाल।” दोहे का भाव है-

सोच-समझकर बोलना चाहिए।

मधुर वाणी में बोलना चाहिए।

धीरे – धीरे बोलना चाहिए।

सदा सच बोलना चाहिए।

उत्तर: सोच-समझकर बोलना चाहिए।

प्रश्न 2. “रहिमन देखि बड़ेन को, लघु न दीजिये डारि । जहाँ काम आवे सुई, कहा करे तलवारि।” इस दोहे का भाव क्या है?

तलवार सुई से बड़ी होती है।

सुई का काम तलवार नहीं कर सकती।

तलवार का महत्व सुई से ज्यादा है।

हर छोटी-बड़ी चीज़ का अपना महत्व होता है।

उत्तर: हर छोटी-बड़ी चीज़ का अपना महत्व होता है।

(ख) अब अपने मित्रों के साथ चर्चा कीजिए और कारण बताइए कि आपने यही उत्तर क्यों चुने? उत्तर:

सोच-समझकर बोलना चाहिए ताकि बाद में पछतावा न पड़े।

हर छोटी-बड़ी चीज़ का अपना महत्व होता है अर्थात किसी को उसके रूप, आकार या आर्थिक स्थिति से नहीं आंकना चाहिए क्योंकि प्रत्येक का अपनी-अपनी जगह महत्व होता है।

मिलकर करें मिलान

पाठ में से कुछ दोहे स्तंभ 1 में दिए गए हैं और उनके भाव स्तंभ 2 में दिए गए हैं। अपने समूह में इन पर चर्चा कीजिए और रेखा खींचकर सही भाव से मिलान कीजिए।

उत्तर: 1. → 3 2. → 2 3. → 1

पंक्तियों पर चर्चा

नीच दिए गए दोहों पर समूह में चर्चा कीजिए और उनके अर्थ या भावार्थ अपनी लेखन पुस्तिका में लिखिए –

(क) “रहिमन बिपदाहू भली, जो थोरे दिन होय । हित अनहित या जगत में, जानि परत सब कोय। ” उत्तर: रहीमदास का मानना है कि थोड़े दिन की विपदा भी भली होती है जो हमें यह बता देती है कि संसार में कौन हमारा हितैषी है और कौन अहितैषी अर्थात कौन हमारा मुश्किल में साथ देने वाला है और कौन नहीं।

(ख) “रहिमन जिह्वा बावरी, कहि गइ सरग पताल। आपु तो कहि भीतर रही, जूती खात कपाल।” उत्तर: रहीमदास का कहना है कि हमारी जीभ बिलकुल बावरी अर्थात पागल जैसी होती है । यह कई बार ऐसा कुछ बोल देती है कि दिमाग को जूते खाने पड़ते हैं अर्थात मनुष्य को पछताना पड़ता है।

सोच-विचार के लिए

दोहों को एक बार फिर से पढ़िए और निम्नलिखित के बारे में पता लगाकर अपनी लेखन पुस्तिका में लिखिए-

प्रश्न 1. “रहिमन धागा प्रेम का, मत तोड़ो छिटकाय। टूटे से फिर ना मिले, मिले गाँठ परि जाय।”

(क) इस दोहे में ‘मिले’ के स्थान पर ‘जुड़े’ और ‘छिटकाय’ के स्थान पर ‘चटकाय’ शब्द का प्रयोग भी लोक में प्रचलित है। जैसे— “रहिमन धागा प्रेम का, मत तोड़ो चटकाय। टूटे से फिर ना जुड़े, जुड़े गाँठ पड़ जाय ।” इसी प्रकार पहले दोहे में ‘डारि’ के स्थान पर ‘डार’, ‘तलवार’ के स्थान पर ‘तरवार’ और चौथे दोहे में ‘’मानुष’ के स्थान पर ‘मानस’ का उपयोग भी प्रचलित हैं। ऐसा क्यों होता है? उत्तर: डारि के स्थान पर डार तलवारि के स्थान पर तलवार मानुष के स्थान पर मानस आदि शब्दों का प्रयोग थोडी-सी दूरी पर बोली बदल जाने के कारण होता है।

(ख) इस दोहे में प्रेम के उदाहरण में धागे का प्रयोग ही क्यों किया गया है? क्या आप धागे के स्थान पर कोई अन्य उदाहरण सुझा सकते हैं? अपने सुझाव का कारण भी बताइए। उत्तर: कवि ने प्रेम के टूटने को धागे द्वारा दर्शाया है कि जिस प्रकार धागा एक बार टूट जाए तो उसे जोड़ने के लिए गाँठ लगानी पड़ती है। ऐसे ही प्रेम संबंधों में दरार आ जाए तो भले ही उन्हें फिर से जोड़ लिया जाए परंतु मन-मुटाव रह ही जाता है। इसे हम अन्य उदाहरणों द्वारा भी समझ सकते है जैसे-

नदी के जल से एक लोटा पानी ले लिया जाए तो उन्हें दोबारा नदी में मिलाया तो जा सकता है परंतु उसे उसकी सहोदर (मित्र) बूँदों से नहीं मिलाया जा सकता। ऐसे ही किसी से संबंध अगर टूट जाए तो दोबारा वैसे नहीं बन पाते।

एक टूटे हुए लकड़ी के डंडे को प्रयत्न करके सिल भी लिया जाए तो हम पहले की भाँति उसका प्रयोग नहीं कर सकते। हर बार ध्यान से प्रयोग करना पड़ता है।

एक कीमती कपड़े के फट जाने पर उसे कितना भी सिल लिया जाए लेकिन मन में उसका फटा होना खटकता ही रहता है।

प्रश्न 2. “तरुवर फल नहिं खात हैं, सरवर पियहिँ न पान । कहि रहीम पर काज हित, संपति सँचहि सुजान।” इस दोहे में प्रकृति के माध्यम से मनुष्य के किस मानवीय गुण की बात की गई है? प्रकृति से हम और क्या-क्या सीख सकते हैं? उत्तर: प्रकृति के माध्यम से इस दोहे में मनुष्य के इस मानवीय गुण की बात की गई है कि जैसे पेड़ अपने फल नहीं खाते, सरोवर अपना जल ग्रहण नहीं करते। ऐसे ही सज्जन धन का संचय स्वयं के लिए न करके दूसरों की भलाई के लिए करते हैं। प्रकृति से हम और गुण भी सीख सकते हैं। जैसे-

नदियों के जल की भाँति निरंतर आगे बढ़ना चाहिए।

जिस प्रकार तपती गरमी से बचाने के लिए वृक्ष छाया देते हैं, वैसे ही दूसरों के कठिन समय में हमें उनकी मदद करनी चाहिए।

फूलों की भाँति अपने अच्छे कार्यों की सुगंध चारों ओर बिखेरनी चाहिए।

सूरज की भाँति अच्छे कार्य करने पर अपना नाम जग में चमकाना चाहिए।

चाँद की चाँदनी की ठंड कला की भाँति अपने विचारों से सबको प्रभावित करना चाहिए।

पर्वतों की भाँति अपने विचारों को दृढ़ रखना चाहिए।

सागर की भाँति अपने हृदय को विशाल बनाना चाहिए। जीवन में अच्छी-बुरी जिन भी घटनाओं का सामना हो उन्हें गहराई से अपने अंदर समेटना चाहिए।

शब्दों की बात

हमने शब्दों के नए-नए रूप जाने और समझे। अब कुछ करके देखें- 

कविता में आए कुछ शब्द नीचे दिए गए हैं। इन शब्दों को आपकी मातृभाषा में क्या कहते हैं? लिखिए।

शब्द एक अर्थ अनेक

“रहिमन पानी राखिये, बिनु पानी सब सून। पानी गए न ऊबरै, मोती, मानुष, चून।”

इस दोहे में ‘पानी’ शब्द के तीन अर्थ हैं— सम्मान, जल, चमक।

इसी प्रकार कुछ शब्द नीचे दिए गए हैं। आप भी इन शब्दों के तीन-तीन अर्थ लिखिए। आप इस कार्य में शब्दकोश, इंटरनेट, शिक्षक या अभिभावकों की सहायता भी ले सकते हैं।

कल – ________, _________, _________ उत्तर: कल – आने वाला कल , चैन या शांति , पुर्जा/मशीन

पत्र – ________, _________, _________ उत्तर: पत्र – पत्ता , चिट्ठी , दल

कर – ________, _________, _________ उत्तर: कर – हाथ , टैक्स , किरण

फल – ________, _________, _________ उत्तर: फल – परिणाम , एक खाने का फल (आम) , हल का अग्र भाग

“रहिमन देखि बड़ेन को, लघु न दीजिये डारि । जहाँ काम आवे सुई, कहा करे तलवारि ॥”

इस दोहे का भाव है— न कोई बड़ा है और न ही कोई छोटा है। सबके अपने-अपने काम हैं, सबकी अपनी-अपनी उपयोगिता और महत्ता है। चाहे हाथी हो या चींटी, तलवार हो या सुई, सबके अपने-अपने आकार-प्रकार हैं और सबकी अपनी-अपनी उपयोगिता और महत्व है। सिलाई का काम सुई से ही किया जा सकता है, तलवार से नहीं। सुई जोड़ने का काम करती है जबकि तलवार काटने का। कोई वस्तु हो या व्यक्ति, छोटा हो या बड़ा, सबका सम्मान करना चाहिए।

अपने मनपसंद दोहे को इस तरह की शैली में अपने शब्दों में लिखिए | दोहा पाठ से या पाठ से बाहर का हो सकता है। उत्तर: बड़े बड़ाई न करै; बड़ो न बोले बोल । रहिमन हीरा कब कहैं, लाख मेरो टकै का मोल रहीमदास जी कहते हैं कि जिनमें बड़प्पन होता है वे अपनी बड़ाई स्वयं कभी नहीं करते। उनके कार्य ही उनके कौशल को दर्शा देते हैं। जैसे हीरा कितना भी बहुमूल्य क्यों न हो लेकिन कभी अपने मुँह से अपने बारे में नहीं कहता। हमें भी अपने गुणों को दर्शाना नहीं चाहिए। वे स्वतः ही हमारे कार्यों के माध्यम से सबके समक्ष आ जाते हैं। जैसे- कुशल खिलाड़ी अपने खेल से, बावर्ची अपने स्वादिष्ट पकवानों से अच्छा नर्तक अपने नृत्य से श्रेष्ठ गायक अपने गायन से प्रतिभाशाली विद्यार्थी अपने परिणाम से ही जाना जाता है।

आज की पहेली

1. दो अक्षर का मेरा नाम, आता हूँ खाने के काम उल्टा होकर नाच दिखाऊँ, मैं क्यों अपना नाम बताऊँ।

2. एक किले के दो ही द्वार, उनमें सैनिक लकड़ीदार टकराएँ जब दीवारों से, जल उठे सारा संसार।

उत्तर: माचिस

खोजबीन के लिए

रहीम के कुछ अन्य दोहे पुस्तकालय या इंटरनेट की सहायता से पढ़ें, देखें व समझें।

उत्तर: रहीम के दोहे उनकी गहन सोच और जीवन के मूल्यवान सबक को दर्शाते हैं। यदि आप रहीम के कुछ अन्य दोहे पढ़ना और समझना चाहते हैं, तो निम्नलिखित संसाधनों की सहायता ले सकते हैं:

अपने स्थानीय पुस्तकालय में "रहीम के दोहे" या "रहीम की कविताएँ" से संबंधित किताबें खोजें। वहां आप रहीम के अन्य दोहे और उनकी व्याख्याएँ प्राप्त कर सकते हैं।

इंटरनेट पर "रहीम के दोहे" खोजें। कई वेबसाइट्स और साहित्यिक पोर्टल्स पर रहीम के दोहे और उनकी व्याख्या उपलब्ध हैं।

आप Google पर "Rahim ke dohe" या "Rahim's couplets" सर्च करके भी उनकी अन्य रचनाओं तक पहुंच सकते हैं।

Benefits of NCERT Solutions for Class 6 Hindi Chapter 5 Rahim Ke Dohe

Class 6 Hindi Lesson 5 Question Answers provide detailed explanations of the lesson, helping students learn the lesson easily.

The solutions Break down the text and vocabulary, making it easier for students to understand the context and characters' actions.

Offers clear answers to textbook questions, enabling students to complete their assignments accurately and efficiently.

Class 6 Hindi Chapter 5 Question Answers encourage critical thinking and discussion about empathy, cooperation, and real-life problem-solving.

NCERT Solutions include engaging exercises and activities related to the chapter, making learning fun and interactive for students.

It helps students feel more prepared and confident in understanding the chapter's themes and content.

The Class 6 Hindi Chapter 5 Question Answer PDF is available for FREE download so that students can easily access it when needed.

Important Study Material Links for Hindi Chapter 5 Class 6 - Rahim Ke Dohe

NCERT Solutions for Hindi Class 6 Chapter 5 is a comprehensive resource for understanding the lesson’s concepts. With step-by-step explanations and examples, students can learn the concepts effectively. Accessible in PDF format, students can review the material conveniently. These solutions for Hindi Malhar Chapter 5 Class 6 enhance understanding, and exam performance, making it easier for students studying in Class 6. You can easily access and download the FREE Class 6 Hindi Chapter 5 PDF from Vedantu updated for the 2024-25 syllabus. Students can refer to these solutions to perform better in their examinations.

Chapter-wise NCERT Solutions Class 6 Hindi - Malhar

After familiarising yourself with the Class 6 Hindi Chapter 5 Question Answers, you can access comprehensive NCERT Solutions from all Hindi Class 6 Malhar textbook chapters.

Related Important Links for Hindi (Malhar) Class 6

Along with this, students can also download additional study materials provided by Vedantu for Hindi Class 6.

FAQs on NCERT Solutions for Class 6 Hindi Chapter 5: Rahim Ke Dohe (Malhar)

1. What are the main themes explored in Rahim's dohe featured in Chapter 5?

The main themes include wisdom, humility, the nature of time, and moral conduct. Rahim’s dohe often provide insights into ethical behaviour and practical life lessons.

2. Who was Rahim, and why is he significant in Hindi literature?

Rahim, also known as Rahimdas, was a prominent poet and scholar in Hindi literature. He is known for his dohe (couplets) which convey deep philosophical thoughts and moral teachings.

3. What is a doha, and how is it used in Rahim’s poetry?

A Doha is a two-line couplet with a specific rhythmic pattern and meaning. Rahim used dohas to express profound ideas and moral lessons in a concise and impactful manner.

4. Can you provide a summary of one of the dohe from the Class 6 Hindi Chapter 5 Rahim Ke Dohe?

For example, one of Rahim's dohas discusses the value of being humble and how one's actions should reflect inner virtue rather than outward show.

5. How does Rahim's use of language contribute to the effectiveness of his dohe?

Rahim’s use of simple yet profound language makes his dohe accessible and memorable. The use of metaphor and allegory adds depth to his teachings.

6. What lessons can students learn from Rahim's dohe in this chapter from Class 6 Hindi?

Students can learn about the importance of humility, the value of wisdom, and the significance of ethical behaviour in daily life.

7. How does Rahim’s poetry reflect the cultural and historical context of his time?

Rahim’s poetry reflects the socio-cultural and moral values of his time, offering insights into the ethical norms and philosophical beliefs prevalent in his era.

8. What are some examples of Rahim's dohe included in the Class 6 Hindi Chapter 5 Rahim Ke Dohe?

Examples include couplets on the nature of time, the value of learning from others, and the importance of not boasting.

9. How is Rahim's dohe relevant to modern readers?

Rahim’s dohe offers timeless wisdom and practical advice that remain relevant for personal growth and ethical conduct in contemporary times.

10. What role do illustrations or explanations play in understanding Rahim’s dohe?

Illustrations and explanations help clarify the meaning and context of the dohe, making it easier for readers to grasp the underlying message.

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CBSE Class 10 Science

Starting to study CBSE Class 10 Science can be both exciting and challenging for students. It's a core subject that plays a vital role in academics and future career choices. Covering a variety of topics in Physics, Chemistry, and Biology, mastering this subject requires commitment, careful planning, and a better understanding of the syllabus, exam pattern, and marking scheme. This blog aims to provide students with useful information on chapter-wise weightage, and effective preparation tips to help them succeed in the CBSE Class 10 Science exam for 2025.

• 1.0 CBSE Class 10 Science Syllabus

The syllabus of CBSE class 10 science is given below:

• 2.0 CBSE Class 10 Science Books

NCERT CBSE Class 10 Science textbooks are essential for student learning. They provide clear explanations and in-depth understanding of concepts, helping students grasp the subject better. For Class 10 CBSE students, NCERT books are the primary resource because they offer straightforward explanations, solved examples, and closely follow the syllabus. These books provide a structured approach to learning, making them crucial for preparing for CBSE exams.

• 3.0 CBSE Class 10 Science Chapter-wise Weightage

• 4.0 Class 10 Mock Test

Mock tests are crucial in a student's academic journey. Regular practice with mock tests helps students understand the exam pattern, question types, and time management strategies. They also highlight areas where students need improvement and offer a chance to sharpen problem-solving skills, leading to better exam performance. Moreover, mock tests help reduce exam anxiety by making students more familiar with the exam format.

• 5.0 Class 10 Science Preparation Tips

Grasp the Fundamentals: Building a strong foundation with thorough understanding helps achieve high marks in any exam. For CBSE Class 10 Board exams, NCERT books are essential for success.

Familiarize Yourself with the Syllabus & Exam Pattern: Before taking any exam, it's crucial to know the syllabus and exam pattern. This knowledge will help you plan your studies effectively.

Create a Study Plan: Once you're familiar with the exam pattern and syllabus, create a study timetable that suits you. Allocate equal time to Physics, Chemistry, and Biology, and include time for practice and solving questions.

Visualize Your Learning: Enhance your understanding by using diagrams, visuals, flow charts, and mind maps.

Learn Theorems, Derivations, and Formulas: To score well in science, it's important to learn key theorems, derivations, and formulas. Maintain a separate notebook for important formulas and derivations, making it easier to revise them closer to the exam.

Practice with CBSE Class 10 Science Sample Papers: Solving sample papers will help you understand the exam format and the types of questions that are likely to appear, aiding in better preparation. Students can visit this link for sample questions.

Solve Previous Year Question Papers: Practicing previous year question paper cbse class 10 science helps students become familiar with the exam pattern and types of questions. By solving these papers, scoring full marks becomes more achievable and realistic. Students can easily download cbse class 10 science question paper from this link .

Table of Contents

You can find subject-wise sample papers on the official CBSE website.

There has been no update regarding any changes to the CBSE Class 10 syllabus for the 2024-25 academic year.

The CBSE Class 10 examinations are worth 80 marks, with an additional 20 marks allocated for internal assessment.

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• CBSE Class 9 Mathematics...

CBSE Class 9 Mathematics Case Study Questions

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If you’re looking for a comprehensive and reliable study resource and case study questions for class 9 CBSE, myCBSEguide is the perfect door to enter. With over 10,000 study notes, solved sample papers and practice questions, it’s got everything you need to ace your exams. Plus, it’s updated regularly to keep you aligned with the latest CBSE syllabus . So why wait? Start your journey to success with myCBSEguide today!

Significance of Mathematics in Class 9

Mathematics is an important subject for students of all ages. It helps students to develop problem-solving and critical-thinking skills, and to think logically and creatively. In addition, mathematics is essential for understanding and using many other subjects, such as science, engineering, and finance.

CBSE Class 9 is an important year for students, as it is the foundation year for the Class 10 board exams. In Class 9, students learn many important concepts in mathematics that will help them to succeed in their board exams and in their future studies. Therefore, it is essential for students to understand and master the concepts taught in Class 9 Mathematics .

Case studies in Class 9 Mathematics

A case study in mathematics is a detailed analysis of a particular mathematical problem or situation. Case studies are often used to examine the relationship between theory and practice, and to explore the connections between different areas of mathematics. Often, a case study will focus on a single problem or situation and will use a variety of methods to examine it. These methods may include algebraic, geometric, and/or statistical analysis.

Example of Case study questions in Class 9 Mathematics

The Central Board of Secondary Education (CBSE) has included case study questions in the Class 9 Mathematics paper. This means that Class 9 Mathematics students will have to solve questions based on real-life scenarios. This is a departure from the usual theoretical questions that are asked in Class 9 Mathematics exams.

The following are some examples of case study questions from Class 9 Mathematics:

Class 9 Mathematics Case study question 1

There is a square park ABCD in the middle of Saket colony in Delhi. Four children Deepak, Ashok, Arjun and Deepa went to play with their balls. The colour of the ball of Ashok, Deepak,  Arjun and Deepa are red, blue, yellow and green respectively. All four children roll their ball from centre point O in the direction of   XOY, X’OY, X’OY’ and XOY’ . Their balls stopped as shown in the above image.

Answer the following questions:

Answer Key:

Class 9 Mathematics Case study question 2

• Now he told Raju to draw another line CD as in the figure
• The teacher told Ajay to mark  ∠ AOD  as 2z
• Suraj was told to mark  ∠ AOC as 4y
• Clive Made and angle  ∠ COE = 60°
• Peter marked  ∠ BOE and  ∠ BOD as y and x respectively

Now answer the following questions:

• 2y + z = 90°
• 2y + z = 180°
• 4y + 2z = 120°
• (a) 2y + z = 90°

Class 9 Mathematics Case study question 3

• (a) 31.6 m²
• (c) 513.3 m³
• (b) 422.4 m²

Class 9 Mathematics Case study question 4

How to Answer Class 9 Mathematics Case study questions

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Students need to be careful while solving the Class 9 Mathematics case study questions. They should not make any assumptions and should always check their answers. If they are stuck on a question, they should take a break and come back to it later. With some practice, the Class 9 Mathematics students will be able to crack case study questions with ease.

Class 9 Mathematics Curriculum at Glance

At the secondary level, the curriculum focuses on improving students’ ability to use Mathematics to solve real-world problems and to study the subject as a separate discipline. Students are expected to learn how to solve issues using algebraic approaches and how to apply their understanding of simple trigonometry to height and distance problems. Experimenting with numbers and geometric forms, making hypotheses, and validating them with more observations are all part of Math learning at this level.

The suggested curriculum covers number systems, algebra, geometry, trigonometry, mensuration, statistics, graphing, and coordinate geometry, among other topics. Math should be taught through activities that include the use of concrete materials, models, patterns, charts, photographs, posters, and other visual aids.

CBSE Class 9 Mathematics (Code No. 041)

Class 9 Mathematics question paper design

The CBSE Class 9 mathematics question paper design is intended to measure students’ grasp of the subject’s fundamental ideas. The paper will put their problem-solving and analytical skills to the test. Class 9 mathematics students are advised to go through the question paper pattern thoroughly before they start preparing for their examinations. This will help them understand the paper better and enable them to score maximum marks. Refer to the given Class 9 Mathematics question paper design.

QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)

myCBSEguide: Blessing in disguise

Class 9 is an important milestone in a student’s life. It is the last year of high school and the last chance to score well in the CBSE board exams. myCBSEguide is the perfect platform for students to get started on their preparations for Class 9 Mathematics. myCBSEguide provides comprehensive study material for all subjects, including practice questions, sample papers, case study questions and mock tests. It also offers tips and tricks on how to score well in exams. myCBSEguide is the perfect door to enter for class 9 CBSE preparations.

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16 thoughts on “CBSE Class 9 Mathematics Case Study Questions”

This method is not easy for me

aarti and rashika are two classmates. due to exams approaching in some days both decided to study together. during revision hour both find difficulties and they solved each other’s problems. aarti explains simplification of 2+ ?2 by rationalising the denominator and rashika explains 4+ ?2 simplification of (v10-?5)(v10+ ?5) by using the identity (a – b)(a+b). based on above information, answer the following questions: 1) what is the rationalising factor of the denominator of 2+ ?2 a) 2-?2 b) 2?2 c) 2+ ?2 by rationalising the denominator of aarti got the answer d) a) 4+3?2 b) 3+?2 c) 3-?2 4+ ?2 2+ ?2 d) 2-?3 the identity applied to solve (?10-?5) (v10+ ?5) is a) (a+b)(a – b) = (a – b)² c) (a – b)(a+b) = a² – b² d) (a-b)(a+b)=2(a² + b²) ii) b) (a+b)(a – b) = (a + b

MATHS PAAGAL HAI

All questions was easy but search ? hard questions. These questions was not comparable with cbse. It was totally wastage of time.

Where is search ? bar

maths is love

Can I have more questions without downloading the app.

I love math

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I am Riddhi Shrivastava… These questions was very good.. That’s it.. ..

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CBSE Class 9 Maths Most Important Case Study Based Questions With Solution

Cbse class 9 mathematics case study questions.

In this post I have provided CBSE Class 9 Maths Case Study Based Questions With Solution. These questions are very important for those students who are preparing for their final class 9 maths exam.

All these questions provided in this article are with solution which will help students for solving the problems. Dear students need to practice all these questions carefully with the help of given solutions.

As you know CBSE Class 9 Maths exam will have a set of cased study based questions in the form of MCQs. CBSE Class 9 Maths Question Bank given in this article can be very helpful in understanding the new format of questions for new session.

All Of You Can Also Read

Case studies in class 9 mathematics.

The Central Board of Secondary Education (CBSE) has included case study based questions in the Class 9 Mathematics paper in current session. According to new pattern CBSE Class 9 Mathematics students will have to solve case based questions. This is a departure from the usual theoretical conceptual questions that are asked in Class 9 Maths exam in this year.

Each question provided in this post has five sub-questions, each followed by four options and one correct answer. All CBSE Class 9th Maths Students can easily download these questions in PDF form with the help of given download Links and refer for exam preparation.

There is many more free study materials are available at Maths And Physics With Pandey Sir website. For many more books and free study material all of you can visit at this website.

Given Below Are CBSE Class 9th Maths Case Based Questions With Their Respective Download Links.

CBSE Case Study Questions for Class 9 Maths - Pdf PDF Download

CBSE Case Study Questions for Class  9 Maths

CBSE Case Study Questions for Class 9 Maths are a type of assessment where students are given a real-world scenario or situation and they need to apply mathematical concepts to solve the problem. These types of questions help students to develop their problem-solving skills and apply their knowledge of mathematics to real-life situations.

Chapter Wise Case Based Questions for Class 9 Maths

The CBSE Class 9 Case Based Questions can be accessed from Chapetrwise Links provided below:

Chapter-wise case-based questions for Class 9 Maths are a set of questions based on specific chapters or topics covered in the maths textbook. These questions are designed to help students apply their understanding of mathematical concepts to real-world situations and events.

Chapter 1: Number System

• Case Based Questions: Number System

Chapter 2: Polynomial

• Case Based Questions: Polynomial

Chapter 3: Coordinate Geometry

• Case Based Questions: Coordinate Geometry

Chapter 4: Linear Equations

• Case Based Questions: Linear Equations - 1
• Case Based Questions: Linear Equations -2

Chapter 5: Introduction to Euclid’s Geometry

• Case Based Questions: Lines and Angles

Chapter 7: Triangles

• Case Based Questions: Triangles

Chapter 8: Quadrilaterals

• Case Based Questions: Quadrilaterals - 1
• Case Based Questions: Quadrilaterals - 2

Chapter 9: Areas of Parallelograms

• Case Based Questions: Circles

Chapter 11: Constructions

• Case Based Questions: Constructions

Chapter 12: Heron’s Formula

• Case Based Questions: Heron’s Formula

Chapter 13: Surface Areas and Volumes

• Case Based Questions: Surface Areas and Volumes

Chapter 14: Statistics

• Case Based Questions: Statistics

Chapter 15: Probability

• Case Based Questions: Probability

Weightage of Case Based Questions in Class 9 Maths

Why are Case Study Questions important in Maths Class  9?

• Enhance critical thinking:  Case study questions require students to analyze a real-life scenario and think critically to identify the problem and come up with possible solutions. This enhances their critical thinking and problem-solving skills.
• Apply theoretical concepts:  Case study questions allow students to apply theoretical concepts that they have learned in the classroom to real-life situations. This helps them to understand the practical application of the concepts and reinforces their learning.
• Develop decision-making skills:  Case study questions challenge students to make decisions based on the information provided in the scenario. This helps them to develop their decision-making skills and learn how to make informed decisions.
• Improve communication skills:  Case study questions often require students to present their findings and recommendations in written or oral form. This helps them to improve their communication skills and learn how to present their ideas effectively.
• Enhance teamwork skills:  Case study questions can also be done in groups, which helps students to develop teamwork skills and learn how to work collaboratively to solve problems.

In summary, case study questions are important in Class 9 because they enhance critical thinking, apply theoretical concepts, develop decision-making skills, improve communication skills, and enhance teamwork skills. They provide a practical and engaging way for students to learn and apply their knowledge and skills to real-life situations.

Class 9 Maths Curriculum at Glance

The Class 9 Maths curriculum in India covers a wide range of topics and concepts. Here is a brief overview of the Maths curriculum at a glance:

• Number Systems:  Students learn about the real number system, irrational numbers, rational numbers, decimal representation of rational numbers, and their properties.
• Algebra:  The Algebra section includes topics such as polynomials, linear equations in two variables, quadratic equations, and their solutions.
• Coordinate Geometry:  Students learn about the coordinate plane, distance formula, section formula, and slope of a line.
• Geometry:  This section includes topics such as Euclid’s geometry, lines and angles, triangles, and circles.
• Trigonometry: Students learn about trigonometric ratios, trigonometric identities, and their applications.
• Mensuration: This section includes topics such as area, volume, surface area, and their applications.
• Statistics and Probability:  Students learn about measures of central tendency, graphical representation of data, and probability.

The Class 9 Maths curriculum is designed to provide a strong foundation in mathematics and prepare students for higher education in the field. The curriculum is structured to develop critical thinking, problem-solving, and analytical skills, and to promote the application of mathematical concepts in real-life situations. The curriculum is also designed to help students prepare for competitive exams and develop a strong mathematical base for future academic and professional pursuits.

Students can also access Case Based Questions of all subjects of CBSE Class 9

• Case Based Questions for Class 9 Science
• Case Based Questions for Class 9 Social Science
• Case Based Questions for Class 9 English
• Case Based Questions for Class 9 Hindi
• Case Based Questions for Class 9 Sanskrit

Frequently Asked Questions (FAQs) on Case Based Questions for Class 9 Maths

What is case-based questions.

Case-Based Questions (CBQs) are open-ended problem solving tasks that require students to draw upon their knowledge of Maths concepts and processes to solve a novel problem. CBQs are often used as formative or summative assessments, as they can provide insights into how students reason through and apply mathematical principles in real-world problems.

What are case-based questions in Maths?

Case-based questions in Maths are problem-solving tasks that require students to apply their mathematical knowledge and skills to real-world situations or scenarios.

What are some common types of case-based questions in class 9 Maths?

Common types of case-based questions in class 9 Maths include word problems, real-world scenarios, and mathematical modeling tasks.

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Cbse case study questions for class 9 maths - pdf, objective type questions, past year papers, semester notes, video lectures, extra questions, sample paper, shortcuts and tricks, study material, previous year questions with solutions, important questions, practice quizzes, mock tests for examination.

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Importance of cbse case study questions for class 9 maths - pdf, cbse case study questions for class 9 maths - pdf notes, cbse case study questions for class 9 maths - pdf class 9, study cbse case study questions for class 9 maths - pdf on the app.

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Case Study Questions for Class 9 Maths Chapter 12 Herons Formula

Case study questions for class 9 maths chapter 9 areas of parallelograms and triangles, case study questions for class 9 maths chapter 6 lines and angles, case study questions for class 9 maths chapter 7 triangles, case study questions for class 9 maths chapter 5 introduction to euclid’s geometry, case study and passage based questions for class 9 maths chapter 14 statistics, case study questions for class 9 maths chapter 1 real numbers, case study questions for class 9 maths chapter 4 linear equations in two variables, case study questions for class 9 maths chapter 3 coordinate geometry, case study questions for class 9 maths chapter 15 probability, case study questions for class 9 maths chapter 13 surface area and volume, case study questions for class 9 maths chapter 10 circles, case study questions for class 9 maths chapter 9 quadrilaterals, case study questions for class 9 maths chapter 2 polynomials.

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CBSE Case Study Questions for Class 9 Maths Circles Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 9 Maths Circles  in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

CBSE Case Study Questions for Class 9 Maths Circles PDF

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• Chapter 8 Quadrilaterals Case Study Questions
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• Chapter 12 Heron’s Formula Case Study Questions

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CBSE Class 10 Maths Case Study Questions for Chapter 9 - Some Applications of Trigonometry (Published By CBSE)

Check case study questions for cbse class 10 maths chapter 9 - some applications of trigonometry. these questions are published by the cbse itself for class 10 students..

Case study based questions are new for class 10 students. Therefore, it is quite essential that students practice with more of such questions so that they do not have problem in solving them in their Maths board exam. We have provided here the case study questions for CBSE Class 10 Maths Chapter 9 - Some Applications of Trigonometry. All these questions have been published by the Central Board of Secondary Education (CBSE) for the class 10 students. Therefore, students must solve all the questions seriously so that they may score the desired marks in their Maths exam.

Check Case Study Questions for Class 10 Maths Chapter 9:

CASE STUDY 1:

A group of students of class X visited India Gate on an education trip. The teacher and students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in New Delhi, dedicated to the troops of British India who died in wars fought between 1914 and 1919. The teacher also said that India Gate, which is located at the eastern end of the Rajpath (formerly called the Kingsway), is about 138 feet (42 metres) in height.

1. What is the angle of elevation if they are standing at a distance of 42m away from the monument?

Answer: b) 45 o

2. They want to see the tower at an angle of 60 o . So, they want to know the distance where they should stand and hence find the distance.

Answer: a) 25.24 m

3. If the altitude of the Sun is at 60 o , then the height of the vertical tower that will cast a shadow of length 20 m is

a) 20√3 m

b) 20/ √3 m

c) 15/ √3 m

d) 15√3 m

Answer: a) 20√3 m

4. The ratio of the length of a rod and its shadow is 1:1. The angle of elevation of the Sun is

5. The angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level is

a) corresponding angle

b) angle of elevation

c) angle of depression

d) complete angle

Answer: a) corresponding angle

CASE STUDY 2:

A Satellite flying at height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka, them being Nanda Devi(height 7,816m) and Mullayanagiri (height 1,930 m). The angles of depression from the satellite, to the top of Nanda Devi and Mullayanagiri are 30° and 60° respectively. If the distance between the peaks of the two mountains is 1937 km, and the satellite is vertically above the midpoint of the distance between the two mountains.

1. The distance of the satellite from the top of Nanda Devi is

a) 1139.4 km

b) 577.52 km

d) 1025.36 km

Answer: a) 1139.4 km

2. The distance of the satellite from the top of Mullayanagiri is

Answer: c) 1937 km

3. The distance of the satellite from the ground is

Answer: b) 577.52 km

4. What is the angle of elevation if a man is standing at a distance of 7816m from Nanda Devi?

5.If a mile stone very far away from, makes 45 o to the top of Mullanyangiri mountain. So, find the distance of this mile stone from the mountain.

a) 1118.327 km

b) 566.976 km

Also Check:

Case Study Questions for All Chapters of CBSE Class 10 Maths

Tips to Solve Case Study Based Questions Accurately

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Class 9 Maths Case Study Questions of Chapter 2 Polynomials PDF Download

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Case study Questions in Class 9 Mathematics Chapter 2  are very important to solve for your exam. Class 9 Maths Chapter 2 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving  Class 9 Maths Case Study Questions  Chapter 2 Polynomials

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

These case study questions challenge students to apply their knowledge of polynomials in real-life scenarios, enhancing their problem-solving abilities. This article provides for the Class 9 Maths Case Study Questions of Chapter 2: Polynomials , enabling students to practice and excel in their examinations.

Polynomials Case Study Questions With Answers

Here, we have provided case-based/passage-based questions for Class 9 Maths Chapter 2 Polynomials

Case Study/Passage Based Questions

Ankur and Ranjan start a new business together. The amount invested by both partners together is given by the polynomial p(x) = 4x 2 + 12x + 5, which is the product of their individual shares.

Coefficient of x 2 in the given polynomial is (a) 2 (b) 3 (c) 4 (d) 12

Answer: (c) 4

Total amount invested by both, if x = 1000 is (a) 301506 (b)370561 (c) 4012005 (d)490621

Answer: (c) 4012005

The shares of Ankur and Ranjan invested individually are (a) (2x + 1),(2x + 5)(b) (2x + 3),(x + 1) (c) (x + 1),(x + 3) (d) None of these

Answer: (a) (2x + 1),(2x + 5)

Name the polynomial of amounts invested by each partner. (a) Cubic (b) Quadratic (c) Linear (d) None of these

Answer: (c) Linear

Find the value of x, if the total amount invested is equal to 0. (a) –1/2 (b) –5/2 (c) Both (a) and (b) (d) None of these

Answer: (c) Both (a) and (b)

One day, the principal of a particular school visited the classroom. The class teacher was teaching the concept of a polynomial to students. He was very much impressed by her way of teaching. To check, whether the students also understand the concept taught by her or not, he asked various questions to students. Some of them are given below. Answer them

Which one of the following is not a polynomial? (a) 4x 2 + 2x – 1 (b) y+3/y (c) x 3 – 1 (d) y 2 + 5y + 1

Answer: (b) y+3/y

The polynomial of the type ax 2 + bx + c, a = 0 is called (a) Linear polynomial (b) Quadratic polynomial (c) Cubic polynomial (d) Biquadratic polynomial

Answer: (a) Linear polynomial

The value of k, if (x – 1) is a factor of 4x 3 + 3x 2 – 4x + k, is (a) 1 (b) –2 (c) –3 (d) 3

Answer: (c) –3

If x + 2 is the factor of x 3 – 2ax 2 + 16, then value of a is (a) –7 (b) 1 (c) –1 (d) 7

Answer: (b) 1

The number of zeroes of the polynomial x 2 + 4x + 2 is (a) 1 (b) 2 (c) 3 (d) 4

Answer: (b) 2

Case Study/Passage-Based Questions

Case Study 3. Amit and Rahul are friends who love collecting stamps. They decide to start a stamp collection club and contribute funds to purchase new stamps. They both invest a certain amount of money in the club. Let’s represent Amit’s investment by the polynomial A(x) = 3x^2 + 2x + 1 and Rahul’s investment by the polynomial R(x) = 2x^2 – 5x + 3. The sum of their investments is represented by the polynomial S(x), which is the sum of A(x) and R(x).

Q1. What is the coefficient of x^2 in Amit’s investment polynomial A(x)? (a) 3 (b) 2 (c) 1 (d) 0

Answer: (a) 3

Q2. What is the constant term in Rahul’s investment polynomial R(x)? (a) 2 (b) -5 (c) 3 (d) 0

Answer: (c) 6

Q3. What is the degree of the polynomial S(x), representing the sum of their investments? (a) 4 (b) 3 (c) 2 (d) 1

Answer: (c) 2

Q4. What is the coefficient of x in the polynomial S(x)? (a) 7 (b) -3 (c) 0 (d) 5

Answer: (b) -3

Q5. What is the sum of their investments, represented by the polynomial S(x)? (a) 5x^2 + 7x + 4 (b) 5x^2 – 3x + 4 (c) 5x^2 – 3x + 5 (d) 5x^2 + 7x + 5

Answer: (b) 5x^2 – 3x + 4

Case Study 4. A school is organizing a fundraising event to support a local charity. The students are divided into three groups: Group A, Group B, and Group C. Each group is responsible for collecting donations from different areas of the town.

Group A consists of 30 students and each student is expected to collect ‘x’ amount of money. The polynomial representing the total amount collected by Group A is given as A(x) = 2x^2 + 5x + 10.

Group B consists of 20 students and each student is expected to collect ‘y’ amount of money. The polynomial representing the total amount collected by Group B is given as B(y) = 3y^2 – 4y + 7.

Group C consists of 40 students and each student is expected to collect ‘z’ amount of money. The polynomial representing the total amount collected by Group C is given as C(z) = 4z^2 + 3z – 2.

Q1. What is the coefficient of x in the polynomial A(x)? (a) 2 (b) 5 (c) 10 (d) 0

Answer: (b) 5

Q2. What is the degree of the polynomial B(y)? (a) 2 (b) 3 (c) 4 (d) 1

Answer: (b) 3

Q3. What is the constant term in the polynomial C(z)? (a) 4 (b) 3 (c) -2 (d) 0

Answer: (c) -2

Q4. What is the sum of the coefficients of the polynomial A(x)? (a) 2 (b) 5 (c) 10 (d) 17

Answer: (c) 10

Q5. What is the total number of students in all three groups combined? (a) 30 (b) 20 (c) 40 (d) 90

Answer: (c) 40

The Class 9 Maths Case Study Questions of Chapter 2: Polynomials serve as a valuable resource for students seeking to enhance their understanding of polynomial concepts and problem-solving skills. By practicing these case studies, students can strengthen their grasp of polynomials and their applications in real-life scenarios. Embrace the opportunity to engage with practical problems and excel in your mathematical journey.

By Team Study Rate

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Mcq questions of class 9 maths chapter 2 polynomials with answers, mcq questions of class 9 social science civics chapter 3 constitutional design with answers, class 9 maths case study questions chapter 4 linear equations in two variables, this post has 2 comments.

I didnt understand last question

The number of zeroes of the polynomial x2 + 4x + 2 is The answer is too easy, i.e. 2

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Class 9th Maths - Coordinate Geometry Case Study Questions and Answers 2022 - 2023

By QB365 on 08 Sep, 2022

QB365 provides a detailed and simple solution for every Possible Case Study Questions in Class 9th Maths Subject - Coordinate Geometry, CBSE. It will help Students to get more practice questions, Students can Practice these question papers in addition to score best marks.

QB365 - Question Bank Software

Coordinate geometry case study questions with answer key.

9th Standard CBSE

Final Semester - June 2015

Mathematics

(b) What are the coordinates of C and D respectively?

(c) What is the distance between B and D?

(d) What is the distance between A and C?

(e) What are the coordinates of the point of intersection of AC and BD?

(ii) What are the coordinates of Police Station?

(iii) Distance between school and police station:

(iv) What are the coordinates of Library?

(v) In which quadrant the point (-1, 4) lies?  

(b) What are the coordinates of A and B respectively?

(c) The coordinates of point O in the sketch -2 is

(d) The point on the y-axis ( in sketch 2) which is equidistant from the points B and C is 

(e) The point on the x-axis ( in sketch 2) which is equidistant from the points C and D is

(b) What are the coordinates of R, taking A as origin?

(c) Side of lawn is :

(d) Shape of lawn is :

(e) Area of lawn is :

(ii) What are the coordinates of position 'D'?

(iii) What are the coordinates of position 'H'?

(iv) In which quadrant, the point 'C' lie?

(v) Find the perpendicular distance of the point E from the y-axis.

*****************************************

Coordinate geometry case study questions with answer key answer keys.

(a) (iii) A(3, 5); B(7, 9) (b) (i) C(11, 5); D(7, 1) (c) (iii) 8 units (d) (iii) 8 units (e) (i) (7, 5)

(i) (b) (2, 3) (ii) (a) (2, -1) (iii) (a) 4 (iv) (d) (6, 2) (v) (b) II

(a) (ii) A(13, 10); B(19, 10) (b) (iv) A(19, 6); B(13, 6) (c) (ii) (16, 8) (d) (i) (0, 8) (e)  (ii) (16, 0)

(a) (iv) C(10, 6) (b) (iii) R(5, 6) (c) (ii)   \(\sqrt{34}\)  units PS 2 = AS 2 + AP 2 = 5 2 + 3 2 = 25 + 9 = 34 ⇒ PS =  \(\sqrt{34}\) (d) (iv) Rhombus (e) (i) 30 sq. units Area of rhombus =  \(1 / 2\)  x product of diagonals =  \(1 / 2\)  x 6 x 10  = 30 sq. units

(i) (d) (-4, 3)  (ii) (b) (-3, -2) (iii) (b) (8, 4.5) (iv) (d) IV (v) (b) 10 units

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• NCERT Class 9
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• Chapter 10: Circles

NCERT Solutions for Class 9 Maths Chapter 10 Circles

* According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 9.

NCERT Solutions for Class 9 Maths Chapter 10 Circles are provided here in PDF format, which can be downloaded for free. The NCERT Solutions for the chapter Circles are included as per the latest update of the CBSE curriculum (2023-24) and have been designed by our expert teachers.

Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 10 Circles

Download most important questions for class 9 maths chapter – 10 circles.

All the solved questions of Chapter 10 Circles are with respect to the CBSE syllabus and guidelines to help students solve each exercise question present in the book and prepare for the exam. These serve as reference tools for the students to do homework and also support them in scoring good marks. Students can also get the solutions for Class 9th Maths all chapters exercise-wise and practise well for the exams.

• Chapter 1 Number System
• Chapter 2 Polynomials
• Chapter 3 Coordinate Geometry
• Chapter 4 Linear Equations in Two Variables
• Chapter 5 Introduction to Euclids Geometry
• Chapter 6 Lines and Angles
• Chapter 7 Triangles
• Chapter 8 Quadrilaterals
• Chapter 9 Areas of Parallelograms and Triangles
• Chapter 10 Circles
• Chapter 11 Constructions
• Chapter 12 Heron’s Formula
• Chapter 13 Surface Areas and Volumes
• Chapter 14 Statistics
• Chapter 15 Introduction to Probability

NCERT Solutions for Class 9 Maths Chapter 10 – Circles

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List of Exercises in Class 9 Maths Chapter 10 Exercise 10.1 Solutions 2 Questions (2 Short) Exercise 10.2 Solutions 2 Questions (2 long) Exercise 10.3 Solutions 3 Questions (3 long) Exercise 10.4 Solutions 6 Questions (6 long) Exercise 10.5 Solutions 12 Questions (12 long) Exercise 10.6 Solutions 10 Questions (10 long)

Access Answers of Maths NCERT Class 9 Chapter 10 Circles

Exercise: 10.1 (Page No: 171)

1. Fill in the blanks.

(i) The centre of a circle lies in ____________ of the circle. (exterior/ interior)

(ii) A point whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior/ interior)

(iii) The longest chord of a circle is a _____________ of the circle.

(iv) An arc is a ___________ when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and _____________ of the circle.

(vi) A circle divides the plane, on which it lies, in _____________ parts.

(i) The centre of a circle lies in interior of the circle.

(ii) A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the circle.

(iii) The longest chord of a circle is a diameter of the circle.

(iv) An arc is a semicircle when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and chord of the circle.

(vi) A circle divides the plane, on which it lies, in 3 (three) parts.

2. Write True or False. Give reasons for your solutions.

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

(ii) A circle has only a finite number of equal chords.

(iii) If a circle is divided into three equal arcs, each is a major arc.

(iv) A chord of a circle, which is twice as long as its radius, is the diameter of the circle.

(v) Sector is the region between the chord and its corresponding arc.

(vi) A circle is a plane figure.

(i) True. Any line segment drawn from the centre of the circle to any point on it is the radius of the circle and will be of equal length.

(ii) False. There can be infinite numbers of equal chords in a circle.

(iii) False. For unequal arcs, there can be major and minor arcs. So, equal arcs on a circle cannot be said to be major arcs or minor arcs.

(iv) True. Any chord whose length is twice as long as the radius of the circle always passes through the centre of the circle, and thus, it is known as the diameter of the circle.

(v) False. A sector is a region of a circle between the arc and the two radii of the circle.

(vi) True. A circle is a 2d figure, and it can be drawn on a plane.

Exercise: 10.2 (Page No: 173)

1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two circles can be congruent only when the distance of every point of both circles is equal from the centre.

For the second part of the question, it is given that AB = CD, i.e., two equal chords.

Now, it is to be proven that angle AOB is equal to angle COD.

Consider the triangles ΔAOB and ΔCOD.

OA = OC and OB = OD (Since they are the radii of the circle.)

AB = CD (As given in the question.)

So, by SSS congruency, ΔAOB ≅  ΔCOD

∴ By CPCT, we have,

∠ AOB = ∠ COD (Hence, proved).

2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Consider the following diagram.

Here, it is given that ∠ AOB = ∠ COD, i.e., they are equal angles.

Now, we will have to prove that the line segments AB and CD are equal, i.e., AB = CD.

In triangles AOB and COD,

∠ AOB = ∠ COD (As given in the question.)

OA = OC and OB = OD (These are the radii of the circle.)

So, by SAS congruency, ΔAOB ≅  ΔCOD

∴ By the rule of CPCT, we have,

AB = CD (Hence, proved.)

Exercise: 10.3 (Page No: 176)

1. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

In these two circles, no point is common.

Here, only one point, ‘P’, is common.

Even here, P is the common point.

Here, two points are common, which are P and Q.

No point is common in the above circle.

2. Suppose you are given a circle. Give a construction to find its centre.

The construction steps to find the centre of the circle is:

Step I: Draw a circle first.

Step II: Draw 2 chords, AB and CD, in the circle.

Step III: Draw the perpendicular bisectors of AB and CD.

Step IV: Connect the two perpendicular bisectors at a point. This intersection point of the two perpendicular bisectors is the centre of the circle.

3. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

It is given that two circles intersect each other at P and Q.

OO’ is perpendicular bisector of PQ.

(i) PR = RQ

(ii) ∠PRO = ∠PRO’ = ∠QRO = ∠QRO’ = 90 0

In triangles ΔPOO’ and ΔQOO’,

OP = OQ and O’P = O’Q (Since they are also the radii.)

OO’ = OO’ (It is the common side.)

So, it can be said that ΔPOO’ ≅  ΔQOO’ (SSS Congruence rule)

∴ ∠ POO’ = ∠ QOO’ (c.p.c.t)— (i)

Even triangles ΔPOR and ΔQOR are similar by SAS congruency.

OP = OQ (Radii)

∠ POR = ∠ QOR (As ∠ POO’ = ∠ QOO’)

OR = OR (Common arm)

So, ΔOPO’ ≅  ΔOQO’ (SAS Congruence rule)

∴ PR = QR and ∠ PRO = ∠ QRO (c.p.c.t) …. (ii)

As PQ is a line

∠ PRO + ∠ QRO = 180°

∠ PRO + ∠ PRO = 180° (Using (ii))

2 ∠ PRO = 180°

∠ PRO = 90°

So ∠ QRO = ∠ PRO = 90°

∠ PRO’ = ∠QRO = 90° and ∠QRO’ = ∠PRO = 90° (Vertically opposite angles)

∠ PRO = ∠ QRO = ∠ PRO’ = ∠QRO’ = 90°

So, OO’ is the perpendicular bisector of PQ.

Exercise: 10.4 (Page No: 179)

1. Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centres is 4 cm. Find the length of the common chord.

The perpendicular bisector of the common chord passes through the centres of both circles.

As the circles intersect at two points, we can construct the above figure.

Consider AB as the common chord and O and O’ as the centres of the circles.

As the radius of the bigger circle is more than the distance between the two centres, we know that the centre of the smaller circle lies inside the bigger circle.

The perpendicular bisector of AB is OO’.

OA = OB = 3 cm

As O is the midpoint of AB

AB = 3 cm + 3 cm = 6 cm

The length of the common chord is 6 cm.

It is clear that the common chord is the diameter of the smaller circle.

2. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Let AB and CD be two equal cords (i.e., AB = CD). In the above question, it is given that AB and CD intersect at a point, say, E.

It is now to be proven that the line segments AE = DE and CE = BE

Construction Steps

Step 1: From the centre of the circle, draw a perpendicular to AB, i.e., OM ⊥ AB.

Step 2: Similarly, draw ON ⊥ CD.

Step 3: Join OE.

Now, the diagram is as follows:

From the diagram, it is seen that OM bisects AB, and so OM ⊥ AB

Similarly, ON bisects CD, and so ON ⊥ CD.

It is known that AB = CD. So,

AM = ND — (i)

and MB = CN — (ii)

Now, triangles ΔOME and ΔONE are similar by RHS congruency, since

∠ OME = ∠ ONE (They are perpendiculars.)

OE = OE (It is the common side.)

OM = ON (AB and CD are equal, and so they are equidistant from the centre.)

∴ ΔOME ≅  ΔONE

ME = EN (by CPCT) — (iii)

Now, from equations (i) and (ii), we get

AM+ME = ND+EN

So, AE = ED

Now from equations (ii) and (iii), we get

MB-ME = CN-EN

So, EB = CE (Hence, proved)

3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

From the question, we know the following:

(i) AB and CD are 2 chords which are intersecting at point E.

(ii) PQ is the diameter of the circle.

(iii) AB = CD.

Now, we will have to prove that ∠ BEQ = ∠ CEQ

For this, the following construction has to be done.

Construction:

Draw two perpendiculars are drawn as OM ⊥ AB and ON ⊥ D. Now, join OE. The constructed diagram will look as follows:

Now, consider the triangles ΔOEM and ΔOEN.

So, by RHS congruency criterion, ΔOEM ≅ ΔOEN.

Hence, by the CPCT rule, ∠ MEO = ∠ NEO

∴ ∠ BEQ = ∠ CEQ (Hence, proved)

4. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 10.25).

The given image is as follows:

First, draw a line segment from O to AD, such that OM ⊥ AD.

So, now OM is bisecting AD since OM ⊥ AD.

Therefore, AM = MD — (i)

Also, since OM ⊥ BC, OM bisects BC.

Therefore, BM = MC — (ii)

From equation (i) and equation (ii),

AM-BM = MD-MC

5. Three girls, Reshma, Salma and Mandip, are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, and Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Let the positions of Reshma, Salma and Mandip be represented as A, B and C, respectively.

From the question, we know that AB = BC = 6cm

So, the radius of the circle, i.e., OA = 5cm

Now, draw a perpendicular BM ⊥ AC.

Since AB = BC, ABC can be considered an isosceles triangle. M is the mid-point of AC. BM is the perpendicular bisector of AC, and thus it passes through the centre of the circle.

let AM = y and

So, BM will be = (5-x).

By applying the Pythagorean theorem in ΔOAM, we get

OA 2 = OM 2 +AM 2

⇒ 5 2 = x 2 +y 2 — (i)

Again, by applying the Pythagorean theorem in ΔAMB,

AB 2 = BM 2 +AM 2

⇒ 6 2 = (5-x) 2 +y 2 — (ii)

Subtracting equation (i) from equation (ii), we get

36-25 = (5-x) 2 +y 2  -x 2 -y 2

Now, solving this equation, we get the value of x as

Substituting the value of x in equation (i), we get

y 2 +(49/25) = 25

⇒ y 2 = 25 – (49/25)

Solving it, we get the value of y as

= 2×(24/5) m

So, the distance between Reshma and Mandip is 9.6 m.

6. A circular park of radius 20m is situated in a colony. Three boys, Ankur, Syed and David, are sitting at equal distances on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

First, draw a diagram according to the given statements. The diagram will look as follows:

Here, the positions of Ankur, Syed and David are represented as A, B and C, respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle.

AD ⊥ BC is drawn. Now, AD is the median of ΔABC, and it passes through the centre O.

Also, O is the centroid of the ΔABC. OA is the radius of the triangle.

OA = 2/3 AD

Let the side of a triangle a metres, then BD = a/2 m.

Applying Pythagoras’ theorem in ΔABD,

AB 2 = BD 2 +AD 2

⇒ AD 2 = AB 2 -BD 2

⇒ AD 2 = a 2 -(a/2) 2

⇒ AD 2 = 3a 2 /4

⇒ AD = √3a/2

20 m = 2/3 × √3a/2

So, the length of the string of the toy is 20√3 m.

Exercise: 10.5 (Page No: 184)

1. In Fig. 10.36, A, B and C are three points on a circle with centre O, such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.

It is given that,

∠ AOC = ∠ AOB+ ∠ BOC

So, ∠ AOC = 60°+30°

∴ ∠ AOC = 90°

It is known that an angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.

∠ ADC = (½) ∠ AOC

= (½)× 90° = 45°

2. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Here, the chord AB is equal to the radius of the circle. In the above diagram, OA and OB are the two radii of the circle.

Now, consider the ΔOAB. Here,

AB = OA = OB = radius of the circle

So, it can be said that ΔOAB has all equal sides, and thus, it is an equilateral triangle.

∴ ∠ AOC = 60°

And, ∠ ACB = ½ ∠ AOB

So, ∠ ACB = ½ × 60° = 30°

Now, since ACBD is a cyclic quadrilateral,

∠ ADB + ∠ ACB = 180° (They are the opposite angles of a cyclic quadrilateral)

So, ∠ ADB = 180°-30° = 150°

So, the angle subtended by the chord at a point on the minor arc and also at a point on the major arc is 150° and 30°, respectively.

3. In Fig. 10.37, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Since the angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.

So, the reflex ∠ POR = 2× ∠ PQR

We know the values of angle PQR as 100°.

So, ∠ POR = 2×100° = 200°

∴ ∠ POR = 360°-200° = 160°

Now, in ΔOPR,

OP and OR are the radii of the circle.

So, OP = OR

Also, ∠ OPR = ∠ ORP

Now, we know the sum of the angles in a triangle is equal to 180 degrees.

∠ POR+ ∠ OPR+ ∠ ORP = 180°

∠ OPR+ ∠ OPR = 180°-160°

As ∠ OPR = ∠ ORP

2 ∠ OPR = 20°

Thus, ∠ OPR = 10°

4. In Fig. 10.38, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

We know that angles in the segment of the circle are equal, so,

∠BAC = ∠BDC

Now. in the ΔABC, the sum of all the interior angles will be 180°.

So, ∠ABC+∠BAC+∠ACB = 180°

Now, by putting the values,

∠BAC = 180°-69°-31°

So, ∠BAC = 80°

∴ ∠BDC = 80°

5. In Fig. 10.39, A, B, C and D are four points on a circle. AC and BD intersect at a point E, such that ∠ BEC = 130° and ∠ ECD = 20°. Find BAC.

We know that the angles in the segment of the circle are equal.

∠ BAC = ∠ CDE

Now, by using the exterior angles property of the triangle,

In ΔCDE, we get

∠ CEB = ∠ CDE+∠ DCE

We know that ∠ DCE is equal to 20°.

So, ∠ CDE = 110°

∠ BAC and ∠ CDE are equal

∴ ∠ BAC = 110°

6. ABCD is a cyclic quadrilateral whose diagonals intersect at point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Consider the chord CD.

We know that angles in the same segment are equal.

So, ∠ CBD = ∠ CAD

∴ ∠ CAD = 70°

Now, ∠ BAD will be equal to the sum of angles BAC and CAD.

So, ∠ BAD = ∠ BAC+∠ CAD

∴ ∠ BAD = 100°

We know that the opposite angles of a cyclic quadrilateral sum up to 180 degrees.

∠ BCD+∠ BAD = 180°

It is known that ∠ BAD = 100°

So, ∠ BCD = 80°

Now, consider the ΔABC.

Here, it is given that AB = BC

Also, ∠ BCA = ∠ CAB (They are the angles opposite to equal sides of a triangle)

∠ BCA = 30°

also, ∠ BCD = 80°

∠ BCA +∠ ACD = 80°

Thus, ∠ ACD = 50° and ∠ ECD = 50°

7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Draw a cyclic quadrilateral ABCD inside a circle with centre O, such that its diagonal AC and BD are two diameters of the circle.

We know that the angles in the semi-circle are equal.

So, ∠ ABC = ∠ BCD = ∠ CDA = ∠ DAB = 90°

So, as each internal angle is 90°, it can be said that the quadrilateral ABCD is a rectangle.

8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

9. Two circles intersect at two points, B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q, respectively (see Fig. 10.40). Prove that ∠ ACP = ∠ QCD.

Join the chords AP and DQ.

For chord AP, we know that angles in the same segment are equal.

So, ∠ PBA = ∠ ACP — (i)

Similarly, for chord DQ,

∠ DBQ = ∠ QCD — (ii)

It is known that ABD and PBQ are two line segments which are intersecting at B.

At B, the vertically opposite angles will be equal.

∴ ∠ PBA = ∠ DBQ — (iii)

From equation (i), equation (ii) and equation (iii), we get

∠ ACP = ∠ QCD

10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.

First, draw a triangle ABC and then two circles having diameters of AB and AC, respectively.

We will have to now prove that D lies on BC and BDC is a straight line.

We know that angles in the semi-circle are equal.

So, ∠ ADB = ∠ ADC = 90°

Hence, ∠ ADB+∠ ADC = 180°

∴ ∠ BDC is a straight line.

So, it can be said that D lies on the line BC.

11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠CBD.

We know that AC is the common hypotenuse and ∠ B = ∠ D = 90°.

Now, it has to be proven that ∠ CAD = ∠ CBD

Since ∠ ABC and ∠ ADC are 90°, it can be said that they lie in a semi-circle.

So, triangles ABC and ADC are in the semi-circle, and the points A, B, C and D are concyclic.

Hence, CD is the chord of the circle with centre O.

We know that the angles which are in the same segment of the circle are equal.

∴ ∠ CAD = ∠ CBD

12. Prove that a cyclic parallelogram is a rectangle.

It is given that ABCD is a cyclic parallelogram, and we will have to prove that ABCD is a rectangle.

Thus, ABCD is a rectangle.

Exercise: 10.6 (Page No: 186)

1.  Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

In ΔPOO’ and ΔQOO’

OP = OQ          (Radius of circle 1)

O’P = O’Q        (Radius of circle 2)

OO’ = OO’        (Common arm)

So, by SSS congruency, ΔPOO’ ≅ ΔQOO’

Thus, ∠OPO’ = ∠OQO’ (proved).

2. Two chords AB and CD of lengths 5 cm and 11 cm, respectively, of a circle, are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6, find the radius of the circle.

Here, OM ⊥ AB and ON ⊥ CD are drawn, and OB and OD are joined.

We know that AB bisects BM as the perpendicular from the centre bisects the chord.

Since AB = 5 so,

BM = AB/2 = 5/2

Similarly, ND = CD/2 = 11/2

Now, let ON be x.

So, OM = 6−x.

Consider ΔMOB,

OB 2 = OM 2 +MB 2

Consider ΔNOD,

OD 2 = ON 2 + ND 2

We know, OB = OD (radii)

From equation 1 and equation 2, we get

Now, from equation (2), we have,

OD 2 = 1 2 +(121/4)

Or OD = (5/2)×√5 cm

3. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance 4 cm from the centre, what is the distance of the other chord from the centre?

Here, AB and CD are 2 parallel chords. Now, join OB and OD.

Distance of smaller chord AB from the centre of the circle = 4 cm

So, OM = 4 cm

MB = AB/2 = 3 cm

Consider ΔOMB.

Or, OB = 5cm

Now, consider ΔOND.

OB = OD = 5 (Since they are the radii.)

ND = CD/2 = 4 cm

Now, OD 2 = ON 2 +ND 2

Or, ON = 3 cm

4. Let the vertex of an angle ABC be located outside a circle, and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

Consider the diagram.

Here AD = CE

We know any exterior angle of a triangle is equal to the sum of interior opposite angles.

∠DAE = ∠ABC+∠AEC (in ΔBAE) ——————-(i)

DE subtends ∠DOE at the centre and ∠DAE in the remaining part of the circle.

∠DAE = (½)∠DOE ——————-(ii)

Similarly, ∠AEC = (½)∠AOC  ——————-(iii)

Now, from equations (i), (ii), and (iii), we get

(½)∠DOE = ∠ABC+(½)∠AOC

Or, ∠ABC = (½)[∠DOE-∠AOC]  (Hence, proved)

5. Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

To prove: A circle drawn with Q as the centre will pass through A, B and O (i.e., QA = QB = QO).

Since all sides of a rhombus are equal,

Now, multiply (½) on both sides.

(½)AB = (½)DC

So, AQ = DP

Since Q is the midpoint of AB,

Again, as PQ is drawn parallel to AD,

Now, as AQ = BQ and RA = QO, we get

QA = QB = QO (Hence, proved)

6. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

Here, ABCE is a cyclic quadrilateral. In a cyclic quadrilateral, the sum of the opposite angles is 180°.

So, ∠AEC+∠CBA = 180°

As ∠AEC and ∠AED are linear pairs,

∠AEC+∠AED = 180°

Or, ∠AED = ∠CBA … (1)

We know in a parallelogram, opposite angles are equal.

So, ∠ADE = ∠CBA … (2)

Now, from equations (1) and (2), we get

∠AED = ∠ADE

Now, AD and AE are angles opposite to equal sides of a triangle.

∴ AD = AE (proved)

7. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.

Here, chords AB and CD intersect each other at O.

Consider ΔAOB and ΔCOD.

∠AOB = ∠COD (They are vertically opposite angles.)

OB = OD (Given in the question.)

OA = OC (Given in the question.)

So, by SAS congruency, ΔAOB ≅ ΔCOD

Also, AB = CD (By CPCT)

Similarly, ΔAOD ≅ ΔCOB

Or, AD = CB (By CPCT)

In quadrilateral ACBD, opposite sides are equal.

So, ACBD is a parallelogram.

We know that opposite angles of a parallelogram are equal.

So, ∠A = ∠C

Also, as ABCD is a cyclic quadrilateral,

∠A+∠C = 180°

⇒∠A+∠A = 180°

Or, ∠A = 90°

As ACBD is a parallelogram and one of its interior angles is 90°, so, it is a rectangle.

∠A is the angle subtended by chord BD. And as ∠A = 90°, therefore, BD should be the diameter of the circle. Similarly, AC is the diameter of the circle.

8. Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F, respectively. Prove that the angles of the triangle DEF are 90°–(½)A, 90°–(½)B and 90°–(½)C.

Here, ABC is inscribed in a circle with centre O, and the bisectors of ∠A, ∠B and ∠C intersect the circumcircle at D, E and F, respectively.

Now, join DE, EF and FD.

As angles in the same segment are equal, so,

∠EDA = ∠FCA ————-(i)

∠FDA = ∠EBA ————-(i)

By adding equations (i) and (ii), we get

∠FDA+∠EDA = ∠FCA+∠EBA

Or, ∠FDE = ∠FCA+∠EBA = (½)∠C+(½)∠B

We know, ∠A +∠B+∠C = 180°

In a similar way,

∠FED = [90° -(∠B/2)] °

∠EFD = [90° -(∠C/2)] °

9. Two congruent circles intersect each other at points A and B. Through A, any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

The diagram will be

Here, ∠APB = ∠AQB (as AB is the common chord in both the congruent circles.)

Now, consider ΔBPQ.

∠APB = ∠AQB

So, the angles are opposite to equal sides of a triangle.

10. In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.

Consider this diagram.

Here, join BE and CE.

Now, since AE is the bisector of ∠BAC,

∠BAE = ∠CAE

∴arc BE = arc EC

This implies chord BE = chord EC

Now, consider triangles ΔBDE and ΔCDE.

DE = DE     (It is the common side)

BD = CD     (It is given in the question)

BE = CE      (Already proved)

So, by SSS congruency, ΔBDE ≅ ΔCDE.

Thus, ∴∠BDE = ∠CDE

We know, ∠BDE = ∠CDE = 180°

Or, ∠BDE = ∠CDE = 90°

∴ DE ⊥ BC (Hence, proved).

Chapter 10, Circles, of Grade 9, is one of the most important chapters, whose concepts will also be used in Class 10. The weightage of this chapter in the final exam is around 15 marks. Therefore, students are advised to read the chapter carefully and practise each and every question included in the textbook with the help of NCERT Solutions ,  along with examples, to have good practice.

Topics covered in Chapter 10 Circles, are

• Circles and the related terms
• Angle Subtended by a Chord at a Point
• Perpendicular from the Centre to a Chord
• Circle through Three Points
• Equal Chords and Their Distances from the Centre
• Angle Subtended by an Arc of a Circle
• Cyclic Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 10 – Circles are made available for students looking to solve all the problems of Ex-10.1. The methods by which problems have been solved, in a broad way, so that students find it easy to understand the fundamentals of circles. Some of the important points of this chapter are

• A circle is a simple closed geometrical shape equidistant from a central point. It is an important shape in the field of geometry.
• Every circle has its centre.
• The straight line from the centre to the circumference of a circle is called the radius of the circle.
• The length of the line through the centre that touches two points on the edge of the circle is called a diameter.
• The total distance around the circle is called the Circumference.
• The area of the circle can be calculated by applying the formula: A = π r 2 , where A is the Area, r is the radius, and the value of π is 3.14.

Key Features of NCERT Solutions for Class 9 Maths Chapter 10 – Circles

• The solutions for the chapter Circles work as a reference for the students.
• Students will be able to resolve all the problems of this chapter in a faster way.
• It is good learning material for exam preparation and to do the revision for Class 9 Maths Chapter 10.
• The questions of Circles are solved by our subject experts.
• The NCERT Solutions are given as per the latest update on the CBSE syllabus and guidelines.

Disclaimer:

Dropped Topics –  10.1 Introduction, 10.2 Circles and its related terms: Review and Circle through three points.

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CBSE Class 10 Maths Case Study

CBSE Board has introduced the case study questions for the ongoing academic session 2021-22. The board will ask the paper on the basis of a different exam pattern which has been introduced this year where 50% syllabus is occupied for MCQ for Term 1 exam. Selfstudys has provided below the chapter-wise questions for CBSE Class 10 Maths. Students must solve these case study based problems as soon as they are done with their syllabus. 

These case studies are in the form of Multiple Choice Questions where students need to answer them as asked in the exam. The MCQs are not that difficult but having a deep and thorough understanding of NCERT Maths textbooks are required to answer these. Furthermore, we have provided the PDF File of CBSE Class 10 maths case study 2021-2022.

Class 10 Maths (Formula, Case Based, MCQ, Assertion Reason Question with Solutions)

In order to score good marks in the term 1 exam students must be aware of the Important formulas, Case Based Questions, MCQ and Assertion Reasons with solutions. Solving these types of questions is important because the board will ask them in the Term 1 exam as per the changed exam pattern of CBSE Class 10th.

Important formulas should be necessarily learned by the students because the case studies are solved with the help of important formulas. Apart from that there are assertion reason based questions that are important too. 

Assertion Reasoning is a kind of question in which one statement (Assertion) is given and its reason is given (Explanation of statement). Students need to decide whether both the statement and reason are correct or not. If both are correct then they have to decide whether the given reason supports the statement or not. In such ways, assertion reasoning questions are being solved. However, for doing so and getting rid of confusions while solving. Students are advised to practice these as much as possible.

For doing so we have given the PDF that has a bunch of MCQs questions based on case based, assertion, important formulas, etc. All the Multiple Choice problems are given with detailed explanations.

CBSE Class 10th Case study Questions

Recently CBSE Board has the exam pattern and included case study questions to make the final paper a little easier. However, Many students are nervous after hearing about the case based questions. They should not be nervous because case study are easy and given in the board papers to ease the Class 10th board exam papers. However to answer them a thorough understanding of the basic concepts are important. For which students can refer to the NCERT textbook.

Basically, case study are the types of questions which are developed from the given data. In these types of problems, a paragraph or passage is given followed by the 5 questions that are given to answer . These types of problems are generally easy to answer because the data are given in the passage and students have to just analyse and find those data to answer the questions.

CBSE Class 10th Assertion Reasoning Questions

These types of questions are solved by reading the statement, and given reason. Sometimes these types of problems can make students confused. To understand the assertion and reason, students need to know that there will be one statement that is known as assertion and another one will be the reason, which is supposed to be the reason for the given statement. However, it is students duty to determine whether the statement and reason are correct or not. If both are correct then it becomes important to check, does reason support the statement? 

Moreover, to solve the problem they need to look at the given options and then answer them.

CBSE Class 10 Maths Case Based MCQ

CBSE Class 10 Maths Case Based MCQ are either Multiple Choice Questions or assertion reasons. To solve such types of problems it is ideal to use elimination methods. Doing so will save time and answering the questions will be much easier. Students preparing for the board exams should definitely solve these types of problems on a daily basis.

Also, the CBSE Class 10 Maths MCQ Based Questions are provided to us to download in PDF file format. All are developed as per the latest syllabus of CBSE Class Xth.

Class 10th Mathematics Multiple Choice Questions

Class 10 Mathematics Multiple Choice Questions for all the chapters helps students to quickly revise their learnings, and complete their syllabus multiple times. MCQs are in the form of objective types of questions whose 4 different options are given and one of them is a true answer to that problem. Such types of problems also aid in self assessment.

Case Study Based Questions of class 10th Maths are in the form of passage. In these types of questions the paragraphs are given and students need to find out the given data from the paragraph to answer the questions. The problems are generally in Multiple Choice Questions.

The Best Class 10 Maths Case Study Questions are available on Selfstudys.com. Click here to download for free.

To solve Class 10 Maths Case Studies Questions you need to read the passage and questions very carefully. Once you are done with reading you can begin to solve the questions one by one. While solving the problems you have to look at the data and clues mentioned in the passage.

In Class 10 Mathematics the assertion and reasoning questions are a kind of Multiple Choice Questions where a statement is given and a reason is given for that individual statement. Now, to answer the questions you need to verify the statement (assertion) and reason too. If both are true then the last step is to see whether the given reason support=rts the statement or not.

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NCERT Solutions for Class 9 Maths Chapter 10 Circles

NCERT Solutions for Class 9 Maths Chapter 10 Circles are provided here. Our NCERT Maths solutions contain all the questions of the NCERT textbook that are solved and explained beautifully. Here you will get complete NCERT Solutions for Class 9 Maths Chapter 10 all exercises Exercise in one place. These solutions are prepared by the subject experts and as per the latest NCERT syllabus and guidelines. CBSE Class 9 Students who wish to score good marks in the maths exam must practice these questions regularly.

Class 9 Maths Chapter 10 Circles NCERT Solutions

Below we have provided the solutions of each exercise of the chapter. Go through the links to access the solutions of exercises you want. You should also check out our NCERT Class 9 Solutions for other subjects to score good marks in the exams.

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.1

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.2

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.3

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.4

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.5

NCERT Solutions for Class 9 Maths Chapter 10 Exercise 10.6

NCERT Solutions for Class 9 Maths Chapter 10 – Topic Discussion

Below we have listed the topics that have been discussed in this chapter.

• Circles and the related terms
• Angle Subtended by a Chord at a Point
• Perpendicular from the Centre to a Chord
• Circle through Three Points
• Equal Chords and Their Distances from the Centre
• Angle Subtended by an Arc of a Circle
• Cyclic Quadrilaterals

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CBSE Case Study Questions Class 9 Maths Chapter 2 Polynomials PDF Download

CBSE Case Study Questions Class 9 Maths Chapter 2 Polynomials PDF Download  are very important to solve for your exam. Class 9 Maths Chapter 2 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving Case Study Questions Class 9 Maths Chapter 2 Polynomials

• Checkout: Class 9 Science Case Study Questions
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Polynomials Case Study Questions With Answers

Case study questions class 9 maths chapter 2.

Case Study/Passage-Based Questions

Case Study 1. Ankur and Ranjan start a new business together. The amount invested by both partners together is given by the polynomial p(x) = 4x 2 + 12x + 5, which is the product of their individual shares.

Coefficient of x 2 in the given polynomial is (a) 2 (b) 3 (c) 4 (d) 12

Answer: (c) 4

Total amount invested by both, if x = 1000 is (a) 301506 (b)370561 (c) 4012005 (d)490621

Answer: (c) 4012005

The shares of Ankur and Ranjan invested individually are (a) (2x + 1),(2x + 5)(b) (2x + 3),(x + 1) (c) (x + 1),(x + 3) (d) None of these

Answer: (a) (2x + 1),(2x + 5)

Name the polynomial of amounts invested by each partner. (a) Cubic (b) Quadratic (c) Linear (d) None of these

Answer: (c) Linear

Find the value of x, if the total amount invested is equal to 0. (a) –1/2 (b) –5/2 (c) Both (a) and (b) (d) None of these

Answer: (c) Both (a) and (b)

Case Study 2. One day, the principal of a particular school visited the classroom. The class teacher was teaching the concept of a polynomial to students. He was very much impressed by her way of teaching. To check, whether the students also understand the concept taught by her or not, he asked various questions to students. Some of them are given below. Answer them

Which one of the following is not a polynomial? (a) 4x 2 + 2x – 1 (b) y+3/y (c) x 3 – 1 (d) y 2 + 5y + 1

Answer: (b) y+3/y

The polynomial of the type ax 2 + bx + c, a = 0 is called (a) Linear polynomial (b) Quadratic polynomial (c) Cubic polynomial (d) Biquadratic polynomial

Answer: (a) Linear polynomial

The value of k, if (x – 1) is a factor of 4x 3 + 3x 2 – 4x + k, is (a) 1 (b) –2 (c) –3 (d) 3

Answer: (c) –3

If x + 2 is the factor of x 3 – 2ax 2 + 16, then value of a is (a) –7 (b) 1 (c) –1 (d) 7

Answer: (b) 1

The number of zeroes of the polynomial x 2 + 4x + 2 is (a) 1 (b) 2 (c) 3 (d) 4

Answer: (b) 2

Case Study 3. Amit and Rahul are friends who love collecting stamps. They decide to start a stamp collection club and contribute funds to purchase new stamps. They both invest a certain amount of money in the club. Let’s represent Amit’s investment by the polynomial A(x) = 3x^2 + 2x + 1 and Rahul’s investment by the polynomial R(x) = 2x^2 – 5x + 3. The sum of their investments is represented by the polynomial S(x), which is the sum of A(x) and R(x).

Q1. What is the coefficient of x^2 in Amit’s investment polynomial A(x)? (a) 3 (b) 2 (c) 1 (d) 0

Answer: (a) 3

Q2. What is the constant term in Rahul’s investment polynomial R(x)? (a) 2 (b) -5 (c) 3 (d) 0

Answer: (c) 6

Q3. What is the degree of the polynomial S(x), representing the sum of their investments? (a) 4 (b) 3 (c) 2 (d) 1

Answer: (c) 2

Q4. What is the coefficient of x in the polynomial S(x)? (a) 7 (b) -3 (c) 0 (d) 5

Answer: (b) -3

Q5. What is the sum of their investments, represented by the polynomial S(x)? (a) 5x^2 + 7x + 4 (b) 5x^2 – 3x + 4 (c) 5x^2 – 3x + 5 (d) 5x^2 + 7x + 5

Answer: (b) 5x^2 – 3x + 4

Case Study 4. A school is organizing a fundraising event to support a local charity. The students are divided into three groups: Group A, Group B, and Group C. Each group is responsible for collecting donations from different areas of the town.

Group A consists of 30 students and each student is expected to collect ‘x’ amount of money. The polynomial representing the total amount collected by Group A is given as A(x) = 2x^2 + 5x + 10.

Group B consists of 20 students and each student is expected to collect ‘y’ amount of money. The polynomial representing the total amount collected by Group B is given as B(y) = 3y^2 – 4y + 7.

Group C consists of 40 students and each student is expected to collect ‘z’ amount of money. The polynomial representing the total amount collected by Group C is given as C(z) = 4z^2 + 3z – 2.

Q1. What is the coefficient of x in the polynomial A(x)? (a) 2 (b) 5 (c) 10 (d) 0

Answer: (b) 5

Q2. What is the degree of the polynomial B(y)? (a) 2 (b) 3 (c) 4 (d) 1

Answer: (b) 3

Q3. What is the constant term in the polynomial C(z)? (a) 4 (b) 3 (c) -2 (d) 0

Answer: (c) -2

Q4. What is the sum of the coefficients of the polynomial A(x)? (a) 2 (b) 5 (c) 10 (d) 17

Answer: (c) 10

Q5. What is the total number of students in all three groups combined? (a) 30 (b) 20 (c) 40 (d) 90

Answer: (c) 40

Hope the information shed above regarding Case Study and Passage Based Questions for Case Study Questions Class 9 Maths Chapter 2 Polynomials with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 9 Maths Polynomials Case Study and Passage-Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate

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