NCERT Solutions for Class 10 Hindi Sparsh Chapter 15

NCERT Solutions for Class 10 Hindi Sparsh Chapter 15 अब कहाँ दूसरे के दुख से दुखी होने वाले

These Solutions are part of NCERT Solutions for Class 10 Hindi. Here we have given NCERT Solutions for Class 10 Hindi Sparsh Chapter 15 अब कहाँ दूसरे के दुख से दुखी होने वाले.

पाठ्य पुस्तक प्रश्न

मौखिक

(क) निम्नलिखित प्रश्नों के उत्तर एक-दो पंक्तियों में दीजिए-

प्रश्न 1.
बड़े-बड़े बिल्डर समुद्र को पीछे क्यों धकेल रहे थे?
उत्तर:
बिल्डर निर्माण कार्य हेतु उसे धीरे-धीरे पीछे धकेलते जा रहे थे।

प्रश्न 2.
लेखक का घर किस शहर में था?
उत्तर:
लेखक का घर वर्सावा शहर में था।

प्रश्न 3.
जीवन कैसे घरों में सिमटने लगा है?
उत्तर:
जीवन छोटे-छोटे डिब्बे जैसे घरों में सिमटने लगा है।

प्रश्न 4.
कबूतर परेशानी में इधर-उधर क्यों फड़फड़ा रहे थे?
उत्तर:
कबूतर परेशानी में इधर-उधर इसलिए फड़फड़ा रहे थे, क्योंकि उनका एक अंडा बिल्ली ने तोड़ दिया और दूसरा अंडा लेखक की माँ की असावधानी के कारण टूट गया। इस तरह दोनों अंडों के टूट जाने पर अर्थात् अपने बच्चों का विनाश देखकर कबूतर परेशानी में इधर-उधर फड़फड़ा रहे थे।

लिखित

(क) निम्नलिखित प्रश्नों के उत्तर (25-30 शब्दों में) लिखिए-

प्रश्न 1.
अरब में लशकर को ‘नूह’ के नाम से क्यों याद करते हैं?
उत्तर:
अरब में लशकर को ‘नूह’ के नाम से लोग इसलिए याद करते हैं, क्योंकि ‘नूह’ नामक पैगंबर का असली नाम ‘लशकर’ ही था।

प्रश्न 2.
लेखक की माँ किस समय पेड़ों के पत्ते तोड़ने के लिए मना करती थीं और क्यों?
उत्तर:
लेखक की माँ सूरज ढलने के समय आँगन के पेड़ों से पत्ते तोड़ने के लिए मना करती थीं, क्योंकि उनकी कहना था कि इस समय पत्ते तोड़ने से वे रोते हैं और किसी को भी दुख पहुँचाना ठीक नहीं है।

प्रश्न 3.
प्रकृति में आए असंतुलन का क्या परिणाम हुआ है?
उत्तर:
प्रकृति में आए असंतुलन का भयंकर परिणाम हुआ है। जैसे-अब गरमी के मौसम में गरमी की अति, बेवक्त की बरसातें, ज़लज़ले, सैलाब, तूफ़ान और नित नए रोगों का उत्पन्न होना आदि अर्थात् अतिवृष्टि, अल्पवृष्टि तथा अनावृष्टि का होना इसी का परिणाम है।

प्रश्न 4.
लेखक की माँ ने पूरे दिन का रोज़ा क्यों रखा?
उत्तर:
लेखक की माँ ने पूरे दिन का दोज़ा पश्चाताप हेतु रखा। दरअसल उनके मकान के दालान में दो रोशनदान थे, जिसमें कबूतर के जोड़े ने घोंसला बना लिया था। घोंसले में दिए दो अंडों में से एक को बिल्ली ने उचक कर एक तोड़ दिया। यह देखकर माँ बहुत दुखी हुईं। उन्होंने दूसरे अंडे को बचाने के लिए प्रयास किया, लेकिन ऐसा करने से दूसरा अंडा भी टूट गया। इस गुनाह के लिए ही उन्होंने एक दिन का रोज़ा रखा।

प्रश्न 5.
लेखक ने ग्वालियर से बंबई तक किन बदलावों को महसूस किया? पाठ के आधार पर स्पष्ट कीजिए।
उत्तर:
लेखक ने ग्वालियर से बंबई तक कई बदलावों को महसूस किया। जैसे–लेखक का वर्सावा में आज जो घर है, वहाँ पहले दूर तक जंगल था। पेड़ थे, परिंदे थे तथा दूसरे जानवर थे और अब यहाँ समंदर के किनारे लंबी-चौड़ी बस्ती बन गई है।

प्रश्न 6.
“डेरा डालने से आप क्या समझते हैं? स्पष्ट कीजिए।
उत्तर:
‘डेरा डालने से आशय है कि अपने रहने के लिए अस्थायी घर बनाना। पाठ में बताया गया है कि समुद्र के किनारे बस्तियाँ बन जाने से कई पेड़ कटे, कई पक्षी पलायन कर गए और जो न जा सके, उन्होंने जहाँ-तहाँ डेरी डाल लिया।

प्रश्न 7.
शेख अयाज़ के पिता अपनी बाजू पर काला च्योंटा रेंगता देखे भोजन छोड़कर क्यों उठ खड़े हुए?
उत्तर:
शेख आयाज़ के पिता अपनी बाजू पर च्योंटा रेंगता देख, भोजन छोड़कर इसलिए उठ खड़े हुए, क्योंकि उन्हें लगा कि उन्होंने एक च्योंटे को बेघर कर दिया है। वह च्योंटा कुएँ पर रहता था। इसलिए वे उसे छोड़ने कुएँ पर गए।

(ख) निम्नलिखित प्रश्नों के उत्तर (50-60 शब्दों में) लिखिए-

प्रश्न 1.
बढ़ती हुई आबादी का पर्यावरण पर क्या प्रभाव पड़ा है?
उत्तर:
बढ़ती हुई आबादी का पर्यावरण पर गहरा प्रभाव पड़ा है। बढ़ती हुई आबादी के कारण समुद्र ने पीछे सरकना शुरू कर दिया है, पेड़ों को काट-काटकर उन्हें रास्तों से हटाया जा रहा है तथा फैलते हुए प्रदूषण ने पंछियों को बस्तियों से भगाना शुरू कर दिया है। बारूद की विनाशलीला ने वातावरण को प्रदूषित करना शुरू कर दिया है। गर्मी की अति, सर्दी की अति, बरसात की अति, ज़लज़ले, सैलाब, तूफ़ान और नित नए-नए रोगों का उत्पन्न होना आदि का बढ़ती हुई आबादी के कारण ही तो सामना करना पड़ रहा है।

प्रश्न 2.
लेखक की पत्नी को खिड़की में जाली क्यों लगवानी पड़ी?
उत्तर:
लेखक की पत्नी को खिड़की में जाली इसलिए लगवानी पड़ी, क्योंकि लेखक के घर में कबूतरों के एक जोड़े ने अपना घोंसला बना लिया था, जिससे उन कबूतरों का आना-जाना शुरू हो गया। जब वे कबूतर अपने बच्चों को दाना खिलाने आते, तो घर की किसी-न-किसी चीज़ से टकराकर उसे गिरा देते तथा कभी पुस्तकालय में घुसकर किताबें खराब कर देते थे।

प्रश्न 3.
समुद्र के गुस्से की क्या वजह थी? उसने अपना गुस्सा कैसे निकाला?
उत्तर:
समुद्र के गुस्से की वज़ह थी कि उसे निरंतर सिमटते जाना पड़ रहा था। बिल्डर निर्माण कार्य हेतु उसे धीरे-धीरे पीछे धकेलते जा रहे थे। उसने स्वयं को काफी सिकोड़ा, पर जब उसकी सहन-शक्ति समाप्त हो गई, तो उसे गुस्सा आ गया। उसने अपना गुस्सा प्रकट करने के लिए अपनी लहरों पर दौड़ते हुए तीन जहाज़ों को उठाकर बच्चों की गेंद की तरह तीनों दिशाओं में फेंक दिया। एक जहाज़ वर्ली के समुद्र के किनारे पर जा गिरा, तो दूसरा जहाज़ बांद्रा में कार्टर के सामने औंधे मुँह गिरा और तीसरा गेट-वे ऑफ इंडिया पर टूटकर सैलानियों का नज़ारा बना। ये तीनों जहाज़ कोशिश करने के बावजूद भी दुबारा चलने-फिरने योग्य न रहे।

प्रश्न 4.
‘मट्टी से मटूटी मिले, खो के सभी निशान। किसमें कितना कौन है, कैसे हो पहचान – इन पंक्तियों के माध्यम से लेखक क्या कहना चाहता है? स्पष्ट कीजिए।
उत्तर:
लेखक इन पंक्तियों के माध्यम से दो बातें कहना चाहता है। पहली-मृत्यु होने पर सभी प्राणियों के शरीर मिट्टी में मिलकर मिट्टी ही बन जाते हैं। अर्थात् फिर कोई पहचान नहीं रहती कि किस मिट्टी में किसका शरीर मिला है। दूसरी-आध्यात्मिकता की दृष्टि से जिस प्रकार समुद्र में बूंद के मिलने पर उसकी पहचान खो जाती है, उसी प्रकार सभी आत्माएँ मृत्यु के बाद कहाँ जाती हैं, मुक्त होती हैं अथवा परमात्मा में मिल जाती हैं; यह पहचानना भी अति दुष्कर है। संसार का रचयिता अपने अनुसार सबको रचता है, इसमें किसी की चाहत या इच्छा कार्य नहीं करती। जीवन के अंत में सभी इस मिट्टी में मिल जाते हैं। सबकी एक ही गति होती है।

(ग) निम्नलिखित के आशय स्पष्ट कीजिए-

प्रश्न 1.
नेचर की सहनशक्ति की एक सीमा होती है। नेचर के गुस्से का एक नमूना कुछ साल पहले बंबई में देखने को मिला था।
उत्तर:
इसका आशय है कि जैसे प्रत्येक मनुष्य की सहन-शक्ति की एक सीमा होती है, ठीक उसी तरह नेचर (प्रकृति) की सहन-शक्ति की भी सीमा होती है। जब हम प्रकृति के साथ छेड़खानी करते हैं, खिलवाड़ करते हैं, तो कुछ समय बाद उसकी सहन-शक्ति समाप्त हो जाती है। और तब वह बदला लेती है, आंदोलन करती है। ऐसा ही एक नमूना कुछ समय पहले बंबईवासियों को झेलना पड़ा था। उस समय भूकंप ने बंबईवासियों को इतना डरा दिया था कि उन्हें पूजा-स्थलों की शरण में जाना पड़ा था।

प्रश्न 2.
जो जितना बड़ा होता है, उसे उतना ही कम गुस्सा आता है।
उत्तर:
इसका आशय है कि जिस प्रकार धीर पुरुष छोटी-मोटी बातों से तथा दुख व अपमान आदि से नहीं घबराते, लेकिन अत्याचारों की अति होने पर वे अत्यधिक गंभीर परिणाम भुगतने के लिए अत्याचारी को बाध्य कर देते हैं उसी प्रकार समुद्र पर बिल्डरों और मछुआरों का कोप, धरती पर शिकारियों तथा भूमाफियों का अतिक्रमण तथा वायु में कल-कारखानों से निकलती ज़हरीली गैसों के कारण प्रकृति सभी को गंभीर परिणाम भुगतने के लिए बाध्य कर रही है, क्योंकि प्रकृति सबसे अधिक शक्तिशाली है।

प्रश्न 3.
इस बस्ती ने न जाने कितने परिंदों-चरिंदों से उनका घर छीन लिया है। इनमें से कुछ शहर छोड़कर चले गए हैं। जो नहीं जा सके हैं, उन्होंने यहाँ-वहाँ डेरा डाल लिया है।
उत्तर:
इसका आशय है कि मानव की बढ़ती महत्त्वाकांक्षाएँ, कभी न समाप्त होने वाली इच्छाएँ, बढ़ती हुई आबादी की लोलुपता की पूर्ति हेतु पेड़ों की कटाई, जंगलों का सफ़ाया, जंगली-जानवरों का शिकार तथा प्रकृति के साथ अनावश्यक छेड़खानी इत्यादि हरकतें पशु-पक्षियों को बेघर कर देती हैं, जिससे मनुष्य अपने दुर्भाग्य को ही निमंत्रण देता है। अतः मानव इस प्रकार के प्रकृति को असंतुलित करने वाले कुकृत्यों से बाज आना चाहिए।

प्रश्न 4.
शेख अयाज़ के पिता बोले ‘नहीं, यह बात नहीं है। मैंने एक घरवाले को बेघर कर दिया है। उस बेघर को कुएँ पर उसके घर छोड़ने जा रहा हूँ।’ इन पंक्तियों में छिपी हुई उनकी भावना को स्पष्ट कीजिए।
उत्तर:
इन पंक्तियों में शेख अयाज़ के पिता की परोपकार की भावना प्रकट हो रही है। वे एक परोपकारी व्यक्ति थे। समस्त जीवों के प्रति इन पंक्तियों के माध्यम से उनकी प्रेम-भावना उजागर हो रही है। दरअसल वे एक सरल हृदय, निष्कपट, दयालु और नेक इंसान थे। वे पशु-पक्षियों तथा कीड़े-मकोड़ों के दर्द और दुख-तकलीफ़ को भी समझते थे। उनकी भावनाएँ सभी के लिए समान थीं।

भाषा अध्ययन

प्रश्न 1.
उदाहरण के अनुसार निम्नलिखित वाक्यों में कारक चिह्नों को पहचानकर रेखांकित कीजिए और उनके नाम रिक्त स्थानों में लिखिए-
जैसे-
(क) माँ ने भोजन परोसा।                                               कर्ता
(ख) मैं किसी के लिए मुसीबत नहीं हूँ।                           ………………..
(ग) मैंने एक घर वाले को बेघर कर दिया।                      ………………..
(घ) कबूतर परेशानी में इधर-उधर फड़फड़ो रहे थे।         ………………..
(ङ) दरिया पर जाओ, तो उसे सलाम किया करो।            ………………..

उत्तर:
(ख) मैं किसी के लिए मुसीबत नहीं हूँ।                           संप्रदान
(ग) मैंने एक घर वाले को बेघर कर दिया।                      कर्म
(घ) कबूतर परेशानी में इधर-उधर फड़फड़ा रहे थे।        अधिकरण
(ङ) दरिया पर जाओ तो उसे सलाम किया करो।            अधिकरण

प्रश्न 2.
नीचे दिए गए शब्दों के बहुवचन रूप लिखिए-

  1. चींटी,
  2. घोड़ा,
  3. आवाज़,
  4. बिल,
  5. फौज,
  6. रोटी,
  7. बिंदु,
  8. दीवार,
  9. टुकड़ा।

उत्तर:
        एकवचन                बहुवचन

  1. चींटी                       चींटियाँ
  2. घोड़ा                       घोड़े
  3. आवाज़                    आवाजें
  4. बिल                        बिलों
  5. बिंदु                        बिंदुओं
  6. फौज                       फौजें
  7. रोटी                        रोटियाँ
  8. दीवार                      दीवारें
  9. टुकड़ा                     टुकड़े

प्रश्न 3.
ध्यान दीजिए नुक्ता लगाने से शब्द के अर्थ में परिवर्तन हो जाता है। पाठ में ‘दफा’ शब्द का प्रयोग हुआ है, जिसका अर्थ होता है-बार (गणना संबंधी), कानून संबंधी। यदि इस शब्द में नुक्ता लगा दिया जाए, तो शब्द बनेगा ‘दफा जिसका अर्थ होता है-दूर करना, हटाना। यहाँ नीचे कुछ नुक्तायुक्त और नुक्तारहित शब्द दिए जा रहे हैं, उन्हें ध्यान से देखिए और अर्थगत अंतर को समझिए।

  1. सजा – सज़ा
  2. नाज – नाज़
  3. जरा – ज़रा
  4. तेज – तेज़

निम्नलिखित वाक्यों में उचित शब्द भरकर वाक्य पूरे कीजिए-

  1. आजकल ……………………… बहुत खराब है। (जमाना/ज़माना)
  2. पूरे कमरे को ……………………… दो। (सजा/सज़ा)
  3. ……………………… चीनी तो देना। (जरा जरा)
  4. माँ दही …………………….. भूल गई। (जमाना/ज़माना)
  5. दोषी को ………………………….: दी गई। (सजा/सज़ा)
  6. महात्मा के चेहरे पर ………………… था। (तेज/तेज़)

उत्तर:

  1. ज़माना
  2. सजा
  3. ज़रा
  4. जमाना
  5. सज़ा
  6. तेज

योग्यता विस्तार

प्रश्न 1.
पशु-पक्षी एवं वन्य संरक्षण केंद्रों में जाकर पशु-पक्षियों की सेवा-सुश्रूषा के संबंध में जानकारी प्राप्त कीजिए।
उत्तर:
छात्र स्वयं करें।

परियोजना कार्य

प्रश्न 1.
अपने आसपास प्रतिवर्ष एक पौधा लगाइए और उसकी समुचित देखभाल कर पर्यावरण में आए असंतुलन को रोकने में अपना योगदान दीजिए।
उत्तर:
छात्र स्वयं करें।

प्रश्न 2.
किसी ऐसी घटना का वर्णन कीजिए, जब अपने मनोरंजन के लिए मानव द्वारा पशु-पक्षियों का उपयोग किया गया हो।
उत्तर:
छात्र स्वयं करें।

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NCERT Solutions for Class 10 Hindi Sparsh Chapter 14

NCERT Solutions for Class 10 Hindi Sparsh Chapter 14 गिरगिट

These Solutions are part of NCERT Solutions for Class 10 Hindi. Here we have given NCERT Solutions for Class 10 Hindi Sparsh Chapter 14 गिरगिट.

पाठ्य पुस्तक प्रश्न

मौखिक

(क) निम्नलिखित प्रश्नों के उत्तर एक-दो पंक्तियों में दीजिए-

प्रश्न 1.
काठगोदाम के पास भीड़ क्यों इकट्ठी हो गई थी?
उत्तर:
किसी कुत्ते ने ख्यूक्रिन की उँगली काट ली थी। उसने कुत्ते को पकड़ लिया था जिससे कुत्ता डर और भय से चिल्ला रहा था। ख्यूक्रिन और कुत्ते की आवाजें सुनकर काठगोदाम के पास भीड़ इकट्ठी हो गई थी।

प्रश्न 2.
उँगली ठीक न होने की स्थिति में ख्यूक्रिन का नुकसान क्यों होता?
उत्तर:
ख्यूक्रिन का नुकसान इसलिए होता, क्योंकि ख्यूक्रिन एक कामकाजी आदमी था। वह सुनार था और उसका काम पेचीदा था। उसने खुद कहा था कि एक हफ्ते तक उँगली काम करने लायक नहीं हो सकेगी। इसीलिए तो उसने कुत्ते के मालिक से हरज़ाना दिलवाने की बात भी की।

प्रश्न 3.
कुत्ता क्यों किकिया रहा था?
उत्तर:
कुत्ता इसलिए किकिया रहा था क्योंकि उँगली में काट लेने के कारण ख्यूक्रिन उसे मार-पीट रहा था। उसने कुत्ते की टाँग पकड़ रखी थी जिससे कुत्ता आतंकित हो रहा था।

प्रश्न 4.
बाजार के चौराहे पर खामोशी क्यों थी?
उत्तर:
बाज़ार के चौराहे पर किसी आदमी का न होना यह संकेत करता है कि चौराहे पर मात्र बेजान वस्तुएँ थीं। खरीदने-बेचने का काम पूरी तरह बंद था अर्थात् बाज़ार में होने वाली हलचल पूरी तरह बंद थीं। बाजार की दुकानों के सामने कोई भिखारी तक नहीं था। उपरिलिखित कारणों से बाजार के चौराहे पर खामोशी थी।

प्रश्न 5.
जनरल साहब के बावर्ची ने कुत्ते के बारे में क्या बताया?
उत्तर:
जनरल साहब के बावर्ची ने कुत्ते के बारे में बताया कि यह हमारा नहीं है। यह तो जनरल साहब का है, जो कुछ देर पहले यहाँ आए हैं। उन्हें यही नस्ल पसंद है।

लिखित

(क) निम्नलिखित प्रश्नों के उत्तर (25-30 शब्दों में) लिखिए-

प्रश्न 1.
ख्यूक्रिन ने मुआवज़ा पाने की क्या दलील दी?
उत्तर:
ख्यूक्रिन ने मुआवजा पाने की यह दलील दी कि वह कामकाजी आदमी है तथा उसका काम पेंचीदा किस्म का है। कुत्ते द्वारा उँगली काट खाने से अब वह हफ्ते भर तक काम नहीं कर सकता है। इससे उसका काफ़ी नुकसान होगा।

प्रश्न 2.
ख्यूक्रिन ने ओचुमेलॉव को उँगली ऊपर उठाने का क्या कारण बताया ?
उत्तर:
ख्यूक्रिन ने ओचुमेलॉव को उँगली ऊपर उठाने का यह कारण बताया कि वह! तो मेढ़े की तरह चुपचाप जा रहा था, मुझे मित्री मित्रिच से लकड़ी लेकर कुछ काम निपटाना था, तब अचानक इस कंबख्त कुत्ते ने मेरी उँगली काट खाई। उसने साथ में यह भी कहा कि वह एक कामकाजी आदमी है और उसका काम पेचीदा किस्म का है, इसलिए वह सप्ताह तक काम करने की स्थिति में नहीं है।

प्रश्न 3.
येल्दीरीन ने ख्यूक्रिन को दोषी ठहराते हुए क्या कहा?
उत्तर:
येल्दीरीन ने ख्यूक्रिन को दोषी ठहराते हुए इंसपेक्टर से कहा कि ख्यूक्रिन हमेशा कोई-न-कोई शरारत करता रहता है। जरूर ही इसने अपनी जलती सिगरेट से इसकी नाक जला दी होगी, जिससे कुत्ते ने इसे काटा है। इसमें सारा दोष ख्यूक्रिन का ही है।

प्रश्न 4.
ओचुमेलॉव ने जनरल साहब के पास यह संदेश क्यों भिजवाया होगा कि उनसे कहना कि यह मुझे मिला और मैंने इसे वापस उनके पास भेजा है?
उत्तर:
ओचुमेलॉव एक चापलूस तथा भाई-भतीजावाद में विश्वास रखने वाला पुलिस इंस्पेक्टर था। वह जनरल साहब पर अपनी कर्तव्यनिष्ठा व वफ़ादारी की छाप छोड़ना चाहता था। इसलिए उसने जनरल साहब को यह संदेश भिजवाया कि यह कुत्ता उसे गली में मिला है और वह उसे वापिस भेज रहा है, क्योंकि उसे जनरल साहब के कुत्ते का बहुत खयाल है। कुत्ता काफ़ी महँगा है और इसे कोई भी नुकसान पहुँचा सकता है।

प्रश्न 5.
भीड़ ख्यूक्रिन पर क्यों हँसने लगती है?
उत्तर:
कुत्ते द्वारा काटे जाने से काम करने में असमर्थ ख्यूक्रिन कुत्ते के मालिक से मुआवजा चाहता था, परंतु जब इंसपेक्टर को ज्ञात होता है कि वह कुत्ता जनरल के भाई साहब का है तो वह ख्यूक्रिन को धमकाने और कुत्ते को पुचकारने लगता है। इससे ख्यूक्रिन की स्थिति कुत्ते से भी बदतर हो जाती है और भीड़ हँसने लगती है।

(ख) निम्नलिखित प्रश्नों के उत्तर (50-60 शब्दों में) लिखिए-

प्रश्न 1.
किसी कील-वील से उँगली छील ली होगी’-ऐसा ओचुमेलॉव ने क्यों कहा?
उत्तर:
ओचुमेलॉव दोहरी भूमिका जीने वाला चाटुकार, अवसरवादी इंसपेक्टर था। पहले तो वह ख्यूक्रिन को काटने वाले कुत्ते के मालिक को सबक सिखाने, उसे खुला छोड़ देने के लिए दंड देना चाहता था। वह जानना चाहता था कि आखिर यह कुत्ता है किसका, पर जैसे ही उसे पता चलता है कि यह कुत्ता जनरल साहब का है, वह अपना रंग-ढंग और बात बदल लेता है। वह कहता है ख्यूक्रिन कुत्ते पर झूठा आरोप लगा रहा है, जबकि उसने कील-वील से उँगली छील ली होगी। वह ऐसा इसलिए कहता है ताकि वह जनरल साहब की कृपा का पात्र बनकर पदोन्नति पा सके।

प्रश्न 2.
ओचुमेलॉव के चरित्र की विशेषताओं को अपने शब्दों में लिखिए।
उत्तर:
ओचुमेलॉव के चरित्र की विशेषताएँ निम्नलिखित हैं

  1. ओचुमेलॉव एक अवसरवादी पुलिस इंस्पेक्टर है।
  2. वह समय, परिस्थिति, अवसर को देखकर हमेशा अपनी प्रतिक्रिया एवं कथनों को बदल लेता है।
  3. वह अपनी वाकू-पटुता की विशेषताओं का अच्छी तरह से लाभ उठाता है। तभी तो कभी ख्यूक्रिन को हरज़ाना दिलाने की बात करता है, तो कभी मरियल-से कुत्ते को ‘सुंदर डॉगी’ कहता है।
  4. वह उच्च वर्ग के प्रति विशेष पक्षपात का सहारा लेता है और कुत्ते को जनरल साहब के पास भेज देता है।
    अर्थात् ओचुमेलॉव अवसरवादी, वाक्-पटुता में माहिर तथा उच्च वर्ग का साथ देनेवाला व कहानी का मुख्य पात्र है।

प्रश्न 3.
यह जानने के बाद कि कुत्ता जनरल साहब के भाई का है-ओचुमेलॉव के विचारों में क्या परिवर्तन आया और क्यों?
उत्तर:
यह जानने के बाद कि कुत्ता जनरल साहब के भाई का है-ओचुमेलॉव के व्यवहार में जमीन-आसमान का अंतर आ जाता है। कुछ देर पहले तक जो इंसपेक्टर कुत्ते का काम खत्म करने की बात कर रहा था, वह यह बात जानने के बाद गिरगिट की भाँति रंग बदल लेता है और कुत्ते को पुचकारते हुए उसे अत्यंत सुंदर डॉगी और अत्यंत खूबसूरत पिल्ला कहने लगता है। वह उसे नन्हा शैतान कहकर जनरल के बावर्ची को सौंप देता है। अब वह पीड़ित ख्यूक्रिन को ही धमकाता है। ऐसा वह इसलिए करता है ताकि जनरल साहब से अपना स्वार्थ पूरा करा सके। वह उन्हें नाराज नहीं करना चाहता था तथा समय आने पर उनसे लाभ उठाने की फिराक में था।

प्रश्न 4.
ख्यूक्रिन का यह कथन कि “मेरा एक भाई भी पुलिस में है …।’ समाज की किस वास्तविकता की ओर संकेत करता है ?
उत्तर:
ख्यूक्रिन का यह कथन समाज में फैली अराजकता तथा भाई-भतीजावाद की वास्तविकता की ओर संकेत करता है। जब भी किसी व्यक्ति का कोई संबंधी पुलिस में होता है, तो वह उसके सहारे से न्याय प्राप्त करना चाहता है। जब पुलिस इंस्पेक्टर को यह पता चला कि यह कुत्ता जनरल साहब या उनके भाई का है तो उसने ख्यूक्रिन को ही दोषी ठहरा दिया, तब ख्यूक्रिन ने न्याय न मिलता देख तथा न्याय मिलने की इच्छा में ऐसा कहा कि उसका भाई भी पुलिस में है।

प्रश्न 5.
इस कहानी का शीर्षक ‘गिरगिट क्यों रखा होगा? क्या आप इस कहानी के लिए कोई अन्य शीर्षक सुझा सकते हैं?
अपने शीर्षक का आधार भी स्पष्ट कीजिए।
उत्तर:
इस कहानी का शीर्षक ‘गिरगिट’ इसलिए रखा गया होगा क्योंकि गिरगिट ऐसा जीव है जो परिस्थिति की माँग के अनुसार रंग बदलने में सिद्धहस्त है। इस कहानी का प्रमुख पात्र ओचुमेलॉव का चरित्र भी कुछ ऐसा ही है। वह अपने स्वार्थ एवं लाभ के लिए अपनी सोच, बात, व्यवहार और दृष्टिकोण को बार-बार बदलता रहता है। इतना ही नहीं वह सच को झूठ और झूठ को सच सिद्ध करने से भी नहीं चूकता है। वह भीड़ और जनसमूह को देखकर उनके जैसा बनने की कोशिश करते हुए उनसे दिखाता है परंतु जनरल और उनके भाई का नाम सुनते ही गिरगिट की भाँति रंग बदल लेता है। इंसपेक्टर का ऐसा व्यक्तित्व देख इसका शीर्षक ‘गिरगिट’ रखा गया होगा। इसका अन्य शीर्षक हो सकता है-स्वार्थी इंसपेक्टर या हरजाना, क्योंकि दोनों ही शीर्षक पाठ का प्रतिनिधित्व करते हैं।

प्रश्न 6.
‘गिरगिट’ कहानी के माध्यम से समाज की किन विसंगतियों पर व्यंग्य किया गया है? क्या आप ऐसी विसंगतियाँ अपने समाज में देखते हैं? स्पष्ट कीजिए।
उत्तर:
‘गिरगिट कहानी के माध्यम से लेखक ने समाज में व्याप्त कई विसंगतियों पर व्यंग्य किया है, जो निम्नलिखित हैं-

  1. पुलिस समाज में उच्च वर्ग का साथ देती है तथा आम जनता के साथ उसका व्यवहार उपेक्षापूर्ण होता है।
  2. सामान्य व्यक्ति को तो छोटे-से अपराध के लिए भी दंडित किया जाता है, लेकिन उच्च वर्ग के छोटे-मोटे अपराधों को अनदेखा कर दिया जाता है।
  3. समाज में भ्रष्टाचार चारों ओर व्याप्त है, छोटे से बड़े स्तर तक अधिकतर लोग बेईमानी का साथ देते हैं।
  4. भाई-भतीजावाद का बोलबाला है।

आज भी हमारे समाज में ऐसी विसंगतियाँ देखने को मिलती हैं। आज कमजोर व असहाय व्यक्ति की कोई सहायता नहीं करता। लोग पक्षपात पूर्ण व्यवहार के शिकार हो रहे हैं। हम देखते हैं कि पुलिस की मनमानी आज भी चल रही है। उच्चवर्ग व शासन-वर्ग अपराध कर भी दे, तो वह बच जाता है या उसे बचा दिया जाता है। आज सज्जन व्यक्ति को समाज में अधिक परेशानी उठानी पड़ती है। पुलिस का व्यवहार सामान्य आदमी के प्रति उपेक्षापूर्ण है। पुलिस जनता का नहीं अपना हित साधती है। आज समाज में जंगलराज है। जिसकी लाठी उसकी भैंस’ वाला सिद्धांत चल रहा है। पुलिस अधिकारियों के नाम अलग भले ही हों, पर काम एक जैसे ही हैं।

(ग) निम्नलिखित के आशय स्पष्ट कीजिए-

प्रश्न 1.
उसकी आँसुओं से सनी आँखों में संकट और आतंक की गहरी छाप थी।
उत्तर:
आशय यह है जिस कुत्ते ने ख्यूक्रिन की उँगली काट खाई थी, उसे ख्यूक्रिन ने पकड़ लिया था। वह जोर-जोर से किकिया रहा था। ख्यूक्रिन ने उसे मारा भी था, इसलिए वह काँप रहा था। पकड़ लिए जाने से कुत्ता भावी संकट का अनुमान कर और भी आतंकित हो रहा था। यह सब उसकी आँखों में साफ़ तौर पर देखा जा सकता था।

प्रश्न 2.
कानून सम्मत तो यही है … कि सब लोग अब बराबर हैं।
उत्तर:
इसका आशय है कि जब येल्दीरीन ख्यूक्रिन को दोषी ठहराता है, तो ख्यूक्रिन अपने बचाव के लिए कानून की दुहाई देता है और निष्पक्षता की बात करता है कि कानून की दृष्टि में समाज के सभी लोग समान हैं। कानून किसी के साथ भी भेदभाव नहीं करता अर्थात् वह ओचुमेलॉव से न्याय दिलाने की बात कहता है, जिसका वह हकदार है।

प्रश्न 3.
हुजूर! यह तो जनशांति भंग हो जाने जैसा कुछ दीख रहा है।
उत्तर:
ये सिपाही येल्दीरीन का कथन है जो देखता है कि काठगोदाम के पास भीड़ जमा है। एक आदमी चीखकर कहता है, “मत जाने दो।” एक कुत्ता किकिया रहा है और उसकी टाँग पकड़े ख्यूक्रिन चीख रहा है। यह दृश्य देख सिपाही अपने इंसपेक टर ओचुमेलॉव से कहता है कि कानून व्यवस्था भंग होने से लोग इस प्रकार इकट्ठे होकर शायद विद्रोह करना चाहते हैं।

भाषा अध्ययन

प्रश्न 1.
नीचे दिए गए वाक्यों में उचित विराम-चिह्न लगाइए-

  1. माँ ने पूछा बच्चो कहाँ जा रहे हो
  2. घर के बाहर सारा सामान बिखरा पड़ा था।
  3. हाय राम यह क्या हो गया।
  4. रीना सुहेल कविता और शेखर खेल रहे थे
  5. सिपाही ने कहा ठहर तुझे अभी मजा चखाता हूँ।

उत्तर:

  1. माँ ने पूछा- “बच्चो! कहाँ जा रहे हो?”
  2. घर के बाहर सारा सामान बिखरा पड़ा था।
  3. हाय राम! यह क्या हो गया?
  4. रीना, सुहेल, कविता और शेखर खेल रहे थे।
  5. सिपाही ने कहा- “ठहर, तुझे अभी मजा चखाता हूँ।”

प्रश्न 2.
नीचे दिए गए वाक्यों में रेखांकित अंश पर ध्यान दीजिए-

  • मेरा एक भाई भी पुलिस में है।
  • यह तो अति सुंदर ‘डॉगी है।
  • कल ही मैंने बिलकुल इसी की तरह का एक कुत्ता उनके आँगन में देखा था।

वाक्य के रेखांकित अंश ‘निपात’ कहलाते हैं, जो वाक्य के मुख्य अर्थ पर बल देते हैं। वाक्य में इनसे पता चलता है। कि किस बात पर बल दिया जा रहा है और वाक्य क्या अर्थ दे रहा है। वाक्य में जो अव्यय किसी शब्द या पद के बाद लगकर उसके अर्थ में विशेष प्रकार का बल या भाव उत्पन्न करने में सहायता करते हैं, उन्हें निपात कहते हैं;
जैसे- ही, भी, तो, तक आदि।
ही, भी, तो, तक आदि निपातों का प्रयोग करते हुए पाँच वाक्य बनाइए-
उत्तर: 

  1. मैं आज ही अलीगढ़ जाऊँगा।
  2. अब तुम भी कुछ करो।
  3. मैं जा तो रहा हूँ।
  4. कश्मीर से कन्याकुमारी तक भारत एक है।
  5. तुम आज से ही पढ़ाई शुरू कर दो।

प्रश्न 3.
पाठ में आए मुहावरों में से पाँच मुहावरे छाँटकर उनका वाक्य में प्रयोग कीजिए।
उत्तर:

  1. उस भयानक जंगल में आदमी का नामो-निशान तक नहीं था।
  2. भारत में गरीबी के रेखा के नीचे जीने वाले नरक की जिंदगी जी रहे हैं
  3. हमें अपने बड़ों की बातें गाँठ बाँध लेनी चाहिए
  4. सिपाही ने चोर को पीटकर मजा चखाया
  5. लालबत्ती पर चौराहा पार करने वालों पर जुर्माना होना चाहिए।

प्रश्न 4.
नीचे दिए गए शब्दों में उचित उपसर्ग लगाकर शब्द बनाइए
………….. + भाव        = …………..
………….. + पसंद      =  …………..
………….. + धारण      = …………..
………….. + उपस्थित  = …………..
………….. + लायक    = …………..
………….. + विश्वास    = …………..
………….. + परवाह    = …………..
………….. + कारण    = …………..
उत्तर:
            उपसर्ग                शब्द:

  1. अ + भाव           =  अभाव
  2. ना + पसंद         =  नापसंद
  3. निर् + धारण       =  निर्धारण
  4. अन् + उपस्थित  =  अनुपस्थित
  5. ना + लायक       =  नालायक
  6. अ + विश्वास       =  अविश्वास
  7. ला + परवाह      =  लापरवाह
  8. अ + कारण       =  अकारण

प्रश्न 5.
नीचे दिए गए शब्दों में उचित प्रत्यय लगाकर शब्द बनाइए-

  1. मदद   +  ……………..  = ……………..
  2. बुधि    +  ……………..  = ……………..
  3. गंभीर  +  ……………..  = ……………..
  4. सभ्य   +  ……………..  = ……………..
  5. ठंड     + ……………..  = ……………..
  6. प्रदर्शन + ……………..  = ……………..

उत्तर:

  1. मदद + गार = मददगार
  2. बुधि + हीन = बुधिहीन
  3. गंभीर + ता = गंभीरता
  4. सभ्य + ता = सभ्यता
  5. ठंड + क = ठंडक
  6. प्रदर्शन + ई = प्रदर्शनी

प्रश्न 6.
नीचे दिए गए वाक्यों के रेखांकित पदबंध का प्रकार बताइए-

  1. दुकानों में ऊँघते हुए चेहरे बाहर झाँके।
  2. लाल बालोंवाली एक सिपाही चला आ रहा था।
  3. यह ख्यूक्रिन हमेशा कोई न कोई शरारत करता रहता है
  4. एक कुत्ता तीन टाँगों के बल रेंगता चला आ रहा है।

उत्तर:

  1. संज्ञा पदबंध
  2. विशेषण पदबंध
  3. क्रिया पदबंध
  4. क्रिया-विशेषण पदबंध

प्रश्न 7.
आपके मोहल्ले में लावारिस/आवारा कुत्तों की संख्या बहुत ज्यादा हो गई है, जिससे आने-जाने वाले लोगों को। असुविधा होती है। अतः लोगों की सुरक्षा को ध्यान में रखते हुए नगर निगम अधिकारी को एक पत्र लिखिए।
उत्तर:
सेवा में,
अधिकारी
दिल्ली नगर निगम
सेक्टर-22
रोहिणी दिल्ली-85

विषय : लावारिस कुत्तों से सुरक्षा के लिए पत्र
महोदय,
विनम्र अनुरोध यह है कि हमारे गाँव वेगमपुर में चार-पाँच कुत्तों ने आतंक मचा रखा है। वे किसी भी चलते-फिरते व्यक्ति को काट लेते हैं। इसका परिणाम यह होता है कि अपने प्राण बचाने के लिए लोगों को हॉस्पिटल जाकर इंजेकशन लगवाने पड़ते हैं। यह समस्या स्थाई-सी लग रही है। काफी दिनों से लोग कुत्तों को लेकर परेशान हैं। आपसे निवेदन है कि अतिशीघ्र कुत्तों के भय से जनता को भयमुक्त कराने के लिए प्रभावी प्रयास करें। विश्वास है कि आप हमारी सहायता करेंगे।

भवदीय
क०ख०ग
वेगमपुर
दिनांक…………..

योग्यता विस्तार

प्रश्न 1.
जिस प्रकार गिरगिट शत्रु से स्वयं को बचाने के लिए अपने आस-पास के परिवेश के अनुसार रंग बदल लेता है, उसी प्रकार कई व्यक्ति अपने स्वार्थ के लिए परिस्थितियों के अनुसार अपनी बात, व्यवहार, दृष्टिकोण, विचार को बदल लेते हैं। यही कारण है कि ऐसे व्यक्तियों को गिरगिट’ कहा जाता है।
उत्तर:
छात्र स्वयं करें ।

प्रश्न 2.
अवसर के अनुसार व्यावहारिकता का सहारा लेना आप कहाँ तक उचित समझते हैं? इस विषय पर कक्षा में चर्चा कीजिए।
उत्तर:
छात्र स्वयं करें ।

प्रश्न 3.
यहाँ आपने रूसी लेखक चेखव की कहानी पढ़ी है। अवसर मिले तो लियो ताल्स्ताय की कहानियाँ भी पढ़िए।
उत्तर:
छात्र स्वयं करें।

परियोजना कार्य

प्रश्न 1.
‘गिरगिट’ कहानी में आवारा पशुओं से जुड़े किस नियम की चर्चा हुई है? क्या आप इस नियम को उचित मानते हैं? तर्क सहित उत्तर दीजिए।
उत्तर:
‘गिरगिट’ कहानी में आवारा पशुओं से जुड़े इस नियम की चर्चा हुई है कि आवारा कुत्ते या पशु सार्वजनिक स्थान पर नहीं छोड़ने चाहिए, क्योंकि इनसे दुर्घटनाओं की संभावना बढ़ाती है। इनको छोड़ने वालों के खिलाफ कानूनी कार्यवाही करनी चाहिए। उन्हें दंडित किया जाए या करवाया जाए। दंड मिलने के भय से लोग आवारा कुत्तों या पशुओं को नहीं छोड़ेगें। आवारा कुत्तों को उनके मालिकों को सुपुर्द कर देना चाहिए। ये आवारा कुत्ते या पशु किसी भी निर्दोष व्यक्ति को काट या मार सकते हैं। इनसे जनता की सुरक्षा बहुत आवश्यक है। इनसे रक्षा के लिए कानून बनना चाहिए तथा कानून का उल्लंघन करने वालों को दंडित किया जाए। हम इस नियम से पूरी तरह सहमत हैं कि कुत्ते या पशुओं के आवारा छोड़ने वालों के खिलाफ सख्त से सख्त कानून बनाया जाए और नियम तोड़ने वालों को दंडित किया जाए ताकि आम आदमी की सुरक्षा की जा सके।

प्रश्न 2.
गिरगिट कहानी का कक्षा में या विद्यालय में मंचन कीजिए। मंचन के लिए आपको किस प्रकार की तैयारी और सामग्री की ज़रूरत होगी, उनकी एक सूची भी बनाइए।
उत्तर:
छात्र स्वयं करें।

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1

More Exercises

Question 1.
Using the given information, find the value of x in each of the following figures :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.2
Solution:
(i) ∠ADB and ∠ACB are in the same segment.
∠ADB = ∠ACB = 50°
Now in ∆ADB,
∠DAB + X + ∠ADB = 180°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q1.6

Question 2.
If O is the centre of the circle, find the value of x in each of the following figures (using the given information):
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.2
Solution:
(i) ∠ACB = ∠ADB
(Angles in the same segment of a circle)
But ∠ADB = x°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q2.7

Question 3.
(a) In the figure (i) given below, AD || BC. If ∠ACB = 35°. Find the measurement of ∠DBC.
(b) In the figure (ii) given below, it is given that O is the centre of the circle and ∠AOC = 130°. Find∠ABC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q3.1
Solution:
(a) Construction: Join AB
∠A = ∠C = 35° [∵ Alt angles]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q3.2

Question 4.
(a) In the figure (i) given below, calculate the values of x and y.
(b) In the figure (ii) given below, O is the centre of the circle. Calculate the values of x and y.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q4.1
Solution:
(a) ABCD is a cyclic quadrilateral
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q4.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q4.3

Question 5.
(a) In the figure (i) given below, M, A, B, N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON.
(b) In the figrue (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find
(i) ∠ACB
(ii) ∠OBC
(iii) ∠OAB
(iv) ∠CBA
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q5.1
Solution:
(a) ∠NYB = 50°, ∠YNB = 20°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q5.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q5.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q5.4

Question 6.
(a) In the figure (i) given below, O is the centre of the circle and ∠PBA = 42°. Calculate the value of ∠PQB
(b) In the figure (ii) given below, AB is a diameter of the circle whose centre is O. Given that ∠ECD = ∠EDC = 32°, calculate
(i) ∠CEF
(ii) ∠COF.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q6.1
Solution:
(a) In ∆APB,
∠APB = 90° (Angle in a semi-circle)
But ∠A + ∠APB + ∠ABP = 180° (Angles of a triangle)
∠A + 90° + 42°= 180°
∠A + 132° = 180°
⇒ ∠A = 180° – 132° = 48°
But ∠A = ∠PQB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q6.2

Question 7.
(a) In the figure (i) given below, AB is a diameter of the circle APBR. APQ and RBQ are straight lines, ∠A = 35°, ∠Q = 25°. Find (i) ∠PRB (ii) ∠PBR (iii) ∠BPR.
(b) In the figure (ii) given below, it is given that ∠ABC = 40° and AD is a diameter of the circle. Calculate ∠DAC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q7.1
Solution:
(a) (i) ∠PRB = ∠BAP
(Angles in the same segment of the circle)
∴ ∠PRB = 35° (∵ ∠BAP = 35° given)
(ii) In ∆PRQ,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q7.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q7.3

Question 8.
(a) In the figure given below, P and Q are centres of two circles intersecting at B and C. ACD is a st. line. Calculate the numerical value of x.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q8.1
(b) In the figure given below, O is the circumcentre of triangle ABC in which AC = BC. Given that ∠ACB = 56°, calculate
(i)∠CAB
(ii)∠OAC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q8.2
Solution:
Given that
(a) Arc AB subtends ∠APB at the centre
and ∠ACB at the remaining part of the circle
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q8.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q8.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q8.5

Question 9.
(a) In the figure (i) given below, chord ED is parallel to the diameter AC of the circle. Given ∠CBE = 65°, calculate ∠DEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q9.1
(b) In the figure (ii) given below, C is a point on the minor arc AB of the circle with centre O. Given ∠ACB = p°, ∠AOB = q°, express q in terms of p. Calculate p if OACB is a parallelogram.
Solution:
(a) ∠CBE = ∠CAE
(Angle in the same segment of a circle)
⇒ ∠CAE = 65°
∠AEC = 90° (Angle in a semi circle)
Now in ∆AEC
∠AEC + ∠CAE + ∠ACE = 180° (Angle of a triangle)
⇒ 90°+ 65° +∠ACE = 180°
⇒ 155° + ∠ACE = 180°
⇒ ∠ACE = 180° – 155° – 25°
∵AC || ED (given)
∴∠ACE = ∠DEC (alternate angles)
∴∠DEC = 25°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q9.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q9.3

Question 10.
(a) In the figure (i) given below, straight lines AB and CD pass through the centre O of a circle. If ∠OCE = 40° and ∠AOD = 75°, find the number of degrees in :
(i) ∠CDE
(ii) ∠OBE.
(b) In the figure (ii) given below, I is the incentre of ∆ABC. AI produced meets the circumcircle of ∆ABC at D. Given that ∠ABC = 55° and ∠ACB = 65°, calculate
(i) ∠BCD
(ii) ∠CBD
(iii) ∠DCI
(iv) ∠BIC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q10.1
Solution:
(a) (i) ∠CED = 90° (Angle in semi-circle)
In ∆CED
∠CED + ∠CDE + ∠DCE = 180°
⇒ 90° +∠CDE + 40° = 180°
⇒ 130° + ∠CDE = 180°
⇒ ∠CDE = 180° – 130° = 50°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q10.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q10.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q10.4

NCVT MIS 2019

Question 11.
O is the circumcentre of the triangle ABC and D is mid-point of the base BC. Prove that ∠BOD = ∠A.
Solution:
In the given figure, O is the centre of circumcentre of ∆ABC.
D is mid-point of BC. BO, CO and OD are joined.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q11.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q11.2

Question 12.
In the given figure, AB and CD are equal chords. AD and BC intersect at E. Prove that AE = CE and BE = DE.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q12.1
Solution:
In the given figure, AB and CD are two equal chords
AD and BC intersect each other at E.
To prove : AE = CE and BE = DE
Proof:
In ∆AEB and ∆CED
AB = CD (given)
∠A = ∠C (angles in the same segment)
∠B = ∠D (angles in the same segment)
∴ ∆AEB ≅ ∆CED (ASA axiom)
∴ AE = CE and BE = DE (c.p.c.t.)

Question 13.
(a) In the figure (i) given below, AB is a diameter of a circle with centre O. AC and BD are perpendiculars on a line PQ. BD meets the circle at E. Prove that AC = ED.
(b) In the figure (ii) given below, O is the centre of a circle. Chord CD is parallel to the diameter AB. If ∠ABC = 25°, calculate ∠CED.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q13.1
Solution:
(a) Given: AB is the diameter of a circle with centre O.
AC and BD are perpendiculars on a line PQ,
such that BD meets the circle at E.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q13.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q13.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q13.4

Question 14.
In the adjoining figure, O is the centre of the given circle and OABC is a parallelogram. BC is produced to meet the circle at D.
Prove that ∠ABC = 2 ∠OAD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q14.1
Solution:
Given: In the figure,
OABC is a || gm and O is the centre of the circle.
BC is produced to meet the circle at D.
To Prove : ∠ABC = 2∠OAD.
Construction: Join AD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q14.2

Question 15.
(a) In the figure (i) given below, P is the point of intersection of the chords BC and AQ such that AB = AP. Prove that CP = CQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q15.1
(b) In the figure (i) given below, AB = AC = CD, ∠ADC = 38°. Calculate :
(i) ∠ABC (ii) ∠BEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q15.2
Solution:
(a) Given: Two chords AQ and BC intersect each other at P
inside the circle. AB and CQ are joined and AB = AP.
To Prove : CP = CQ
Construction : Join AC.
Proof: In ∆ABP and ∆CQP
∴ ∠B = ∠Q
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q15.4

Question 16.
(a) In the figure (i) given below, CP bisects ∠ACB. Prove that DP bisects ∠ADB.
(b) In the figure (ii) given below, BDbisects ∠ABC. Prove that \(\frac { AB }{ BD } =\frac { BE }{ BC } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q16.1
Solution:
(a)Given: In the figure, CP is the bisector of
∠ACB meeting the circle at P.
PD is joined
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q16.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q16.4

Question 17.
(a) In the figure (ii) given below, chords AB and CD of a circle intersect at E.
(i) Prove that triangles ADE and CBE are similar.
(ii) Given DC =12 cm, DE = 4 cm and AE = 16 cm, calculate the length of BE.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.1
(b) In the figure (ii) given below, AB and CD are two intersecting chords of a circle. Name two triangles which are similar. Hence, calculate CP given that AP = 6cm, PB = 4 cm, and CD = 14 cm (PC > PD).
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.2
Solution:
(a) Given: Two chords AB and CD intersect each other
at E inside the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q17.6

Question 18.
In the adjoining figure, AE and BC intersect each other at point D. If ∠CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm, find DE. (2008)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q18.1
Solution:
In the figure, AE and BC intersect each other at D.
AB is joined.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q18.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q18.3

Question 19.
(a) In the figure (i) given below, PR is a diameter of the circle, PQ = 7 cm, QR = 6 cm and RS = 2 cm. Calculate the perimeter of the cyclic quadrilateral PQRS.
(b) In the figure (ii) given below, the diagonals of a cyclic quadrilateral ABCD intersect in P and the area of the triangle APB is 24 cm². If AB = 8 cm and CD = 5 cm, calculate the area of ∆DPC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q19.1
Solution:
(a) PR is the diameter of the circle
PQ = 7 cm, QR = 6 cm, RS = 2 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q19.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q19.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q19.4

Question 20.
(a) In the figure (i) given below, QPX is the bisector of ∠YXZ of the triangle XYZ. Prove that XY : XQ = XP : XZ,
(b) In the figure (ii) given below, chords BA and DC of a circle meet at P. Prove that:
(i) ∠PAD = ∠PCB
(ii) PA. PB = PC . PD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q20.1
Solution:
(a) Given: ∆XYZ is inscribed in a circle.
Bisector of ∠YXZ meets the circle at Q.
QY is joined.
To Prove : XY : XQ = XP : XZ
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q20.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.1 Q20.4

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Chapter Test

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

More Exercises

Question 1.
Draw a straight line AB of length 8 cm. Draw the locus of all points which are equidistant from A and B. Prove your statement.
Solution:
(i) Draw a line segment AB = 8 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q1.1
(ii) Draw the perpendicular bisector of AB intersecting AB at D.
∴ Every point P on it will be equidistant from A and B.
(iii) Take a point P on the perpendicular bisector.
(iv) Join PA and PB.
Proof: In ∆PAD and ∆PBD
PD = PD (common)
AD = BD (D is mid-point of AB)
∠PDA = ∠PDB (each 90°)
∴ ∆ PAD ≅ ∆ PBD (SAS axiom of congruency)
∴PA = PB (c.p.c.t.)
Similarly, we can prove any other point on the
perpendicular bisector of AB is equidistant from A and B.
Hence Proved.

Question 2.
A point P is allowed to travel in space. State the locus of P so that it always remains at a constant distance from a fixed point C.
Solution:
The point P is moving in the space and
it is at a constant distance from a fixed point C.
∴ Its locus is a sphere.

Question 3.
Draw a line segment AB of length 7 cm. Construct the locus of a point P such that area of triangle PAB is 14 cm².
Solution:
Base of ∆PAB = 7 cm
and its area = 14 cm²
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q3.1
Now draw a line XY parallel to AB at a distance of 4 cm.
Now take any point P on XY
Join PA and PB
area of ∆PAB = 14 cm.
Hence locus of P is the line XY
which is parallel to AB at a distance of 4 cm.

Question 4.
Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is
(i) at a distance of 3 cm from AB.
(ii) at a distance of 5 cm from the point M. Mark the points P, Q, R, S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.
Solution:
Steps of Construction :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q4.1
(i) Take a line AB = 12 cm
(ii) Take M, the midpoint of AB.
(iii) Draw straight lines CD and EF parallel to AB at a distance of 3 cm.
(iv) With centre M and radius 5 cm,
draw areas which intersect CD at P and Q and EF at R and S.
(v) Join QR and PS.
PQRS is a rectangle where the length PQ = 8 cm.
Area of rectangle PQRS = PQ x RS = 8 x 6 = 48 cm²

Question 5.
AB and CD are two intersecting lines. Find the position of a point which is at a distance of 2 cm from AB and 1.6 cm from CD.
Solution:
(i) AB and CD are the intersecting lines which intersect each other at O.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q5.1
(ii) Draw a line EF parallel to AB and GH parallel to CD intersecting each other at P
P is the required point.

Question 6.
Two straight lines PQ and PK cross each other at P at an angle of 75°. S is a stone on the road PQ, 800 m from P towards Q. By drawing a figure to scale 1 cm = 100 m, locate the position of a flagstaff X, which is equidistant from P and S, and is also equidistant from the road.
Solution:
1 cm = 100 cm
800 m = 8 cm.
Steps of Construction :
(i) Draw the lines PQ and PK intersecting each other
at P making an angle of 75°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q6.1
(ii) Take a point S on PQ such that PS = 8 cm.
(iii) Draw the perpendicular bisector of PS.
(iv) Draw the angle bisector of ∠KPS intersecting
the perpendicular bisector at X.
X is the required point which is equidistant from P and S
and also from PQ and PK.

Question 7.
Construct a rhombus PQRS whose diagonals PR, QS are 8 cm and 6 cm respectively. Find by construction a point X equidistant from PQ, PS and equidistant from R, S. Measure XR.
Solution:
Steps of Construction :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q7.1
(i) Take PR = 8 cm and draw the perpendicular bisector
of PR intersecting it at O.
(ii) From O, out. off OS = OQ = 3 cm
(iii) Join PQ, QR, RS and SP.
PQRS is a rhombus. Whose diagonal are PR and QS.
(iv) PR is the bisector of ∠SPQ.
(v) Draw the perpendicular bisector of SR intersecting PR at X
∴ X is equidistant from PQ and PS and also from S and R.
On measuring length of XR = 3.2 cm (approx)

Question 8.
Without using set square or protractor, construct the parallelogram ABCD in which AB = 5.1 cm. the diagonal AC = 5.6 cm and the diagonal BD = 7 cm. Locate the point P on DC, which is equidistant from AB and BC.
Solution:
Steps of Construction :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q8.1
(i) Take AB = 5.1 cm
(ii) At A, with readius \(\\ \frac { 5.6 }{ 2 } \) = 2.8 cm
and at B with radius \(\\ \frac { 7.0 }{ 2 } \) = 3.5 cm,
draw two arcs intersecting each other at O.
(iii) Join AO and produce it to C such that
OC = AD = 2.8 cm and
join BO and produce it to D such that
BO = OD = 3.5 cm
(iv) Join BC, CD, DA
ABCD is a parallelogram.
(v) Draw the angle bisector of ∠ABC intersecting CD at P.
P is the required point which is equidistant from AB and BC.

Question 9.
By using ruler and compass only, construct a quadrilateral ABCD in which AB = 6.5 cm, AD = 4cm and ∠DAB = 75°. C is equidistant from the sides AB and AD, also C is equidistant from the points A and B.
Solution:
Steps of Construction :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test Q9.1
(i) Draw a line segment AB = 6.5 cm.
(ii) At A, draw a ray making an angle of 75° and cut off AD = 4 cm.
(iii) Draw the bisector of ∠DAB.
(iv) Draw perpendicular bisector of AB intersecting the angle bisector at C.
(v) Join CB and CD.
ABCD is the required quadrilateral.

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3

More Exercise

Question 1.
Find the surface area of a sphere of radius :
(i) 14 cm
(ii) 10.5 cm
Solution:
(i) Radius (r) = 14 cm
Surface area = \(4\pi { r }^{ 2 }=4\times \frac { 22 }{ 7 } \times 14\times 14 \) cm2
= 2964 cm2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q1.1

Question 2.
Find the volume of a sphere of radius :
(i) 0.63 m
(ii) 11.2 cm
Solution:
(i) Radius (r) = 0.63 m
Volume = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q2.1

Question 3.
Find the surface area of a sphere of diameter: (i) 21 cm (ii) 3.5 cm
Solution:
(i) Diameter = 21 cm
Radius (r) = \(\\ \frac { 21 }{ 2 } \) cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q3.1

Question 4.
A shot-put is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm3, find the mass of the shot-put.
Solution:
Radius of the metallic shot-put = 4.9 cm
Volume = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q4.1

Question 5.
Find the diameter of a sphere whose surface area is 154 cm2.
Solution:
Surface area of a sphere = 154 cm2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q5.1

Question 6.
Find:
(i) the curved surface area.
(ii) the total surface area of a hemisphere of radius 21 cm.
Solution:
Radius of a hemisphere = 21 cm
(i) Curved surface area = 2πr2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q6.1

Question 7.
A hemispherical brass bowl has inner- diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm2.
Solution:
The inner diameter of hemispherical bowl = 10.5 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q7.1

Question 8.
The radius of a spherical balloon increases from 7 cm to 14 cm as air is jumped into it. Find the ratio of the surface areas of the balloon in two cases.
Solution:
Original radius of balloon = 7 cm
Radius after filling the air in it = 14 cm
The surface area of balloon, the original position
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q8.1

Question 9.
A sphere and a cube have the same surface. Show that the ratio of the volume of the sphere to that of the cube is √6 : √π
Solution:
Let the edge of a cube = a
Surface area = 6a2
and surface area of sphere = 6a2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q9.1

Question 10.
(a) If the ratio of the radii of two sphere is 3 : 7, find :
(i) the ratio of their volumes.
(ii) the ratio of their surface areas.
(b) If the ratio of the volumes of the two sphere is 125 : 64, find the ratio of their surface areas.
Solution:
(a) Ratio in radii of two spheres = 3 : 7
Let radius of the first sphere = 3x
and radius of the second sphere = 7x
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q10.2

Question 11.
A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.
Solution:
Side of a cube = 4 cm
Volume (side)³ = 4 × 4 × 4 = 64 cm³
Diameter of sphere contained by this cube is d = 4 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q11.1

Question 12.
Find the volume of a sphere whose surface area is 154 cm².
Solution:
Given that
Surface area of a sphere = 154 cm²
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q12.1

Question 13.
If the volume of a sphere is \(179 \frac { 2 }{ 3 } \) cm³, find its radius and the surface area.
Solution:
Given that
Volume of a sphere = \(179 \frac { 2 }{ 3 } \) cm³
= \(\\ \frac { 539 }{ 3 } \) cm³
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q13.1

Question 14.
A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?
Solution:
Radius of a hemispherical bowl (r) = 3.5 cm
= \(\\ \frac { 7 }{ 2 } \) cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q14.1

Question 15.
The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Find the volume of water pumped into the tank.
Solution:
Internal diameter of a hemispherical tank (r) = 14 m
Radius of the tank = \(\\ \frac { 14 }{ 2 } \) = 7 m
Water stored in it = 50 kilolitres of water
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q15.1

Question 16.
The surface area of a solid sphere is 1256 cm². It is cut into two hemispheres. Find the total surface area and the volume of a hemisphere. Take π = 3.14.
Solution:
Surface area of a solid sphere = 1256 cm²
By cutting it into two hemisphere,
Curved surface area of each hemisphere
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q16.1

Question 17.
Write whether the following statements are true or false. Justify your answer :
(i) The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.
(ii) The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals the volume of a hemisphere of radius r.
(iii) A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3.
Solution:
(i) The volume of a sphere is equal to the two third of the volume of a cylinder
whose height and diameter are equal to the diameter of the sphere.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q17.1
(ii) The volume of the longest right circular cone that can be filled in a cube
whose edge is 2r equal to the volume of a hemisphere of radius r
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q17.2
(iii) A cone, a hemisphere and a cylinder stand on equal bases and have the same height.
The ratio of their volumes is 1 : 2 : 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3 Q17.3

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

More Exercises

Question 1.
A point moves such that its distance from a fixed line AB is always the same. What is the relation between AB and the path traveled by P ?
Solution:
Let point P moves in such a way that
it is at a fixed distance from the fixed line AB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q1.1
∴ It is a set of two lines l and m parallel to AB
drawn on either side of it at equal distance from it.

Question 2.
A point P moves so that its perpendicular distance from two given lines AB and CD are equal. State the locus of the point P.
Solution:
(i) When two lines AB and CD are parallel,
then the locus of the point P which is equidistant
from AB and CD is a line (l)
in the midway of AB and CD and parallel to them
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q2.1
(ii) If AB and CD are intersecting lines,
then the locus of the point P will be a
pair of the straight lines l and m which bisect
the angles between the given lines AB and CD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q2.2

Question 3.
P is a fixed point and a point Q moves such that the distance PQ is constant, what is the locus of the path traced out by the point Q ?
Solution:
∴ P is a fixed point and Q is a moving point
such that it is always at an equidistant from P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q3.1
∴ P is the centre of the path of Q which is a circle.
The distance between P and Q is the radius of the circle.
Hence locus of point Q is a circle with P as centre.

Question 4.
(i) AB is a fixed line. State the locus of the point P so that ∠APB = 90°.
(ii) A, B are fixed points. State the locus of the point P so that ∠APB = 60°.
Solution:
(i) AB is a fixed line and P is a point
such that ∠APB = 90°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q4.1
The locus of P will be the circle whose diameter is AB.
We know that the angle in a semi-circle is always equal to 90°.
∠APB = 90°
(ii) AB is a fixed line and P is a point such that ∠APB = 60°.
The locus of P will be a major segment of a circle whose AB is a chord.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q4.2

Question 5.
Draw and describe the locus in each of the following cases :
(i) The locus of points at a distance 2.5 cm from a fixed line.
(ii) The locus of vertices of all isosceles triangles having a common base.
(iii) The locus of points inside a circle and equidistant from two fixed points on the circle.
(iv) The locus of centres of all circles passing through two fixed points.
(v) The locus of a point in rhombus ABCD which is equidistant from AB and AD. (1998)
(vi) The locus of a point in the rhombus ABCD which is equidistant from points A and C.
Solution:
1. Draw a given line AB.
2. Draw lines of l and m parallel to AB at a distance of 2.5 cm.
Lines l and m are the locus of point P which is at a distance of 2.5 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.1
(ii) ∆ABC is an isosceles triangle in which AB = AC.
From A, draw AD perpendicular to BC.
AD is the locus of the point A the vertices of ∆ABC.
In rt. ∆ABD and ∆ACD
Side AD = AD (Common)
Hyp. AB = AC (given)
∴ ∆ABD = ∆ACD (R.H.S. Axiom)
∴ BD = DC (c.p.c.t.)
Hence locus of vertices of isosceles triangles
having common base is the perpendicular bisector of BC.
(iii) (i) Draw a circle with centre O.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.2
(ii) Take points A and B on it and join them.
(iii) Draw a perpendicular bisector of AB
which passes from O and meets the circle at C.
CE the diameter, which is the locus of a point inside the circle
and equidistant from two points A and B at the circle.
(iv) Let C1, C2, C3 be the centres of the circle
which pass through the two fixed points A and B.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.3
Draw a line XY passing through these centres C1, C2, C3.
Hence locus of centres of circles passing through two points A and B
is the perpendicular bisector of the line segment joining the two fixed points.
(v) In rhombus ABCD, Join AC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.4
AC is the diagonal of rhombus ABCD
∴ AC bisect ∠A.
∴ Any point on AC, is the locus which is equidistant from AB and AD.
(vi) ABCD is a rhombus. Join BD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.5
BD is the locus of a point in the rhombus which is equidistant from A and C.
Diagonal BD bisects  ∠B and ∠D.
Any point on BD will be equidistant from A and C.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q5.6

Question 6.
Describe completely the locus of points in each of the following cases :
(i) mid-point of radii of a circle.
(ii) centre of a ball, rolling along a straight line on a level floor.
(iii) point in a plane equidistant from a given line.
(iv) point in a plane, at a constant distance of 5 cm from a fixed point (in the plane).
(v) centre of a circle of varying radius and touching two arms of ∠ADC.
(vi) centre of a circle of varying radius and touching a fixed circle, centre O, at a fixed point A on it.
(vii) centre of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with centre 0.
Solution:
(i) The locus of midpoints of the radii of a circle
is another concentric circle with radius is
half of the radius of the given circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.1
(ii) AB is the straight line on the ground and the ball is rolling on it
∴ locus of the centre of the ball is a line parallel A lo the given line AB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.2
(iii) AB is the given line and P is a point in the plane.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.3
From P, draw a line CD and another line EF from P’ parallel to AB.
Thus CD and EF are the lines which are the locus of the point equidistant from AB
(iv) Take a point O and another point P such that OP = 5 cm.
with centre O and radius equal to OP, draw a circle.
Thus this circle is the locus of point P
which is at a distance of 5 cm from O, the given point.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.4
(v) Draw the bisector BX of ∠ABC.
This bisector of an angle is the locus of the centre of a circle with different radii.
Any point on BX, is equidistant from the arms BA and BC of the ∠ABC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.5
(vi) A circle with centre O is given and one point A on it.
The locus of the centre of a circle which touches the circle
at the fixed point A on it, is a line joining the points O and A.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.6
(vii) (a) If the circle with 2 cm as radius touches the given circle
externally then the locus of the centre of the circle
will be a concentric circle with radius 3+2 = 5 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.7
If the circle with 2 cm as radius touches the given circle with 3 cm as radius internally,
then the locus of the centre of the circle will be a concentric circle with radius 3-2 = 1 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q6.8

Question 7.
Using ruler and compasses construct :
(i) a triangle ABC in which AB = 5.5 cm, BC = 3.4 cm and CA = 4.9 cm.
(ii) the locus of points equidistant from A and C.
Solution:
(i) Draw BC = 3.4 and mark the arcs of 5.5 add 4.9 cm from B and C.
Join A, B and C.
ABC is the required triangle.
(ii) Draw ⊥ bisector of AC.
(iii) Draw an angle of 90° at AB at A which intersects ⊥ bisector at O.
Draw circle taking 0 as centre and OA as the radius.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q7.1

Question 8.
Construct triangle ABC, with AB = 7 cm, BC = 8 cm and ∠ABC = 60°. Locate by construction the point P such that :
(i) P is equidistant from B and C and
(ii) P is equidistant from AB and BC
(iii) Measure and record the length of PB. (2000)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q8.1
(i) Take BC = 8 cm a long line segment. At B,
draw a ray BX making an angle of 60° with BC.
Cut off BA = 7 cm. and join AC.
(i) Draw the perpendicular bisector of BC.
(ii) Draw the angle bisector of ∠B which intersect
the perpendicular bisector of BC at P. P is the required point.
(iii) On measuring the length of BP = 4.6 cm (approx.)

Question 9.
A straight line AB is 8 cm long. Locate by construction the locus of a point which is :
(i) Equidistant from A and B.
(ii) Always 4 cm from the line AB.
(iii) Mark two points X and Y, which are 4 cm from AB and equidistant from A and B. Name the figure AXBY.(2008)
Solution:
Steps of construction:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q9.1
(i) Draw a line segment AB = 8 cm.
(ii) With the help of compasses and ruler,
draw the perpendicular bisector l of AB which intersects AB at O.
(iii) Then any point on l, is equidistant from A and B.
(iv) Cut off OX = OY = 4 cm. The X and Y are the required loci,
which is equidistant from AB and also from A and B.
(v) Join AX, XB, BY and YA.
The figures so formed AXBY is the shape of a square because
its diagonals are equal and bisect each other at right angles.

Question 10.
Use ruler and compasses only for this question.
(i) Construct ∆ABC, where AB = 3.5 cm, BC = 6 cm and ∠ABC = 60°.
(ii) Construct the locus of points inside the triangle which are equidistant from BA and BC.
(iii) Construct the locus of points inside the triangle which are equidistant from B and C.
(iv) Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure arid record the length of PB. (2010)
Solution:
In ∆ABC, AB = 3.5 cm, BC = 6 cm and ∠ABC = 60°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q10.1
Steps of construction :
(a) (i) Draw a line segment BC = 6 cm
(ii) At, B draw a ray BX making an angle of 60° and cut off BA = 3.5 cm.
(iii) Join AC.
The ∆ABC is the required triangle.
(b) Draw the bisector BY of ∠ABC.
(c) Draw the perpendicular bisector of BC which intersects BY at P.
P is the required point P which is equidistant from BC and BA
and also equidistant from B and C.
On measuring PB it is 3.4 cm (approx)

Question 11.
Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°. Hence:
(i) Construct the locus of points equidistant from BA and BC.
(ii) Construct the locus of points equidistant from B and C.
(iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.
Solution:
Steps of construction :
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q11.1
Construct the triangle ABC with AB = 5.5 cm
∠BAC = 105° and AC = 6 cm
(i) Points which are equidistant from BA and BC lies on the bisector of ∠ABC.
(ii) Points equidistant from B and C lies on the perpendicular bisector of BC.
Draw perpendicular bisector of BC.
The required point P is the point of intersection of the bisector of
∠ABC and the perpendicular bisector of BC.
(iii) Required length of PC = 4.8 cm.

Question 12.
In the given diagram, A, B and C are fixed collinear points; D is a fixed point outside the line: Locate
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q12.1
(i) the point P on AB such that CP = DP.
(ii) the points Q such that CQ = DQ = 3 cm. How many such points are possible?
(iii) the points R on AB such that DR = 4 cm. How many such points are possible?
(iv) the points S such that CS = DS and S is 4 cm away from the line CD. How many such points are possible?
(v) Are the points P, Q, R collinear?
(vi) Are the points P, Q, S collinear?
Solution:
Points A, B and C are collinear and D is any point outside AB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q12.2
(i) Join CD.
(ii) Draw the perpendicular bisector of CD which meets AB in P.
(iii) P is the required point such that CP = DP
(iv) With centres C and D, draw two arcs with 3 cm radius
which intersect each other at Q and Q’.
Hence there are two points Q and Q’ which are equidistant from C and D.
(v) With centre D, and radius 4 cm draw an arc which intersects AB at R and R’
∴ R and R’ are the two point on AB.
(vi) With centre C and D, draw arcs with a radius equal to
4 cm which intersects each other in S and S’.
∴ There can be two such points which are equidistant from C and D.
(vii) No P, Q, R are not collinear.
(viii) Yes, P, Q, S are collinear.

Question 13.
Points A, B and C represent position of three towers such that AB = 60 m, BC = 73m and CA = 52 m. Taking a scale of 10 m to 1 cm, make an accurate drawing of ∆ABC. Find by drawing, the location of a point which is equidistant from A, B and C, and its actual distance from any of the towers.
Solution:
AB = 60 mm = 6.0 cm, BC = 73 mm = 7.3 cm
and CA = 52 mm = 5.2 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q13.1
(i) Draw a line segment BC = 7.3cm
(ii) With Centre B and radius 6cm and with centre C
and radius 5.2 cm, draw two arcs intersecting each other at A
(iii) Joining AB and AC.
(iv) Draw the perpendicular bisector of AB, BC and CA respectively,
which intersect each other at point P. Join PB.
P is equidistant from A, B and C on measuring PB = 3.7 cm.
Actual distance = 37 m.

Question 14.
Draw two intersecting lines to include an angle of 30°. Use ruler and compasses to locate points which are equidistant from these lines and also 2 cm away from their point of intersection. How many such points exist ? (1990)
Solution:
(i) Two lines AB and CD intersect each other at O.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q14.1
(ii) Draw the bisector of ∠BOD and ∠AOD
(iii) With centre O and radius equal to 2 cm.
marks points on the bisector of angles at P, Q, R and S respectively.
Hence there are four points which are equidistant
from AB and CD and 2 cm from 0, the point of intersection of AB and CD.

Question 15.
Without using set square or protractor, construct the quadrilateral ABCD in which ∠BAD = 45°, AD = AB = 6 cm, BC = 3.6 cm and CD = 5 cm.
(i) Measure ∠BCD.
(ii) Locate the point P on BD which is equidistant from BC and CD. (1992)
Solution:
(i) Take AB = 6 cm long
(ii) AT A, draw the angle of 45° and cut off AD = 6 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q15.1
(iii) With centre D and radius 5 cm and with centre B,
and radius 3.5 cm draw two arcs intersecting each other at C.
(iv) Join CD and CB and join BD
ABCD is the required quadrilateral.
(v) On measuring ∠BCD = 65°.
(vi) Draw the bisector of ∠BCD which intersects BD at P.
P is the required point which is equidistant from CD and CB.

Question 16.
Without using set square or protractor, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ∠ABC. Find the point R on AD such that RB = RC. Measure the length of AR. (1990)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q16.1
Solution:
(i) Take AB = 4 cm
(ii) With centre A, draw an arc of 5 cm radius
and with B draw another arc of radius 4 cm intersecting each other at C.
(iii) Join AC and BD.
(iv) Again with centre A and C,
draw two arcs of radius 4 cm intersecting each other on D.
(v) Join AD and CD.
ABCD is the required rhombus and on measuring the ∠ABC, it is 78°.
(vi) Draw perpendicular bisector of BC intersecting AD at R.
On measuring the length of AR, it is equal to 1.2 cm.

Question 17.
Without using set-squares or protractor construct :
(i) Triangle ABC, in which AB = 5.5 cm, BC = 3.2 cm and CA = 4.8 cm.
(ii) Draw the locus of a point which moves so that it is always 2.5 cm from B.
(iii) Draw the locus of a point which moves so that it is equidistant from the sides BC and CA.
(iv) Mark the point of intersection of the loci with the letter P and measure PC. (1994)
Solution:
Steps of Construction :
(i) Draw BC = 3.2 cm long.
(ii) With centre B and radius 5.5 cm
and with centre C and radius 4.8 cm
draw arcs intersecting each other at A.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q17.1
(iiii) Join AB and AC.
(iv) Draw the bisector of ∠BCA.
(v) With centre B and radius 2.5 cm,
draw an arc intersecting the angle bisector of ∠BCA at P and P’.
P and P’ are two loci which satisfy the given condition.
On measuring CP and CP’
CP = 3.6 cm and CP’ =1.1 cm.

Question 18.
By using ruler and compasses only, construct an isosceles triangle ABC in which BC = 5 cm, AB = AC and ∠BAC = 90°. Locate the point P such that :
(i) P is equidistant from the sides BC and AC.
(ii) P is equidistant from the points B and C.
Solution:
Steps of Construction :
(i) Take BC = 5.0 cm and bisect it at D.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q18.1
(ii) Taking BC as diameter, draw a semicircle.
(iii) At D, draw a perpendicular intersecting the circle at A
(iv) Join AB and AC.
(v) Draw the angle bisector of C intersecting
the perpendicular at P. P is the required point.

Question 19.
Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, ∠B = 60°, AD = 5 cm and D is equidistant from AB and BC. Measure CD. (1983)
Solution:
Steps of Construction :
(i) Draw AB = 6 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q19.1
(ii) At B, draw angle of 60° and cut off BC = 5 cm.
(iii) Draw the angle bisector of ∠B.
(iv) With centre A and radius 5 cm. draw an arc
which intersects the angle bisector of ∠B at D
(v) Join AD and DC.
ABCD in the required quadrilateral.
On measuring CD, it is 5.3 cm (approx).

Question 20.
Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB (2001)
Solution:
Steps of Construction :
(i) Draw a line AB = 6 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q20.1
(ii) With centre A and B and radius 4 cm,
draw two arcs intersecting each other at C.
(iii) Join CA and CB
(iv) Draw the bisector of ∠C and cut off CP = 5 cm
(v) Draw a line XY parallel to AB at a distance of 5 cm.
(vi) From P, draw arcs of radius 5 cm each intersecting
the line XY at Q and R. Hence Q and R are the required points.

Question 21.
Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length 6 cm and 5 cm respectively.
(i) Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction.
(ii) Construct the locus of points, inside the circle, that are equidistant from AB and AC, (1995)
Solution:
Steps of Construction :
(i) With centre 0 and radius 4 cm draw a circle.
(ii) Take point A on this circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q21.1
(iii) With centre A and radius 6 cm draw an arc cutting the circle at B.
(iv) Again with radius 5 cm, draw another arc cutting the circle at C.
(v) Join AB and AC.
(vi) Draw the perpendicular bisector of AC.
Any point on it, will be equidistant from A and C.
(vii) Draw the angle bisector of ∠A intersecting
the perpendicular bisector of AC at P. P is the required locus.

Question 22.
Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct a triangle ABC, in which BC = 6 cm, AB = 9 cm. and ∠ABC = 60°.
(ii) Construct the locus of all points, inside ∆ABC, which are equidistant from B and C.
(iii) Construct the locus of the vertices of the triangles with BC as base, which are equal in area to ∆ABC.
(iv) Mark the point Q, in your construction, which would make ∆QBC equal in area to ∆ABC, and isosceles.
(v) Measure and record the length of CQ. (1998)
Solution:
Steps of Construction :
(i) Draw AB = 9 cm
(ii) At B draw an angle of 60° and cut off BC = 6 cm.
(iii) Join AC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 Q22.1
(iv) Draw perpendicular bisector of BC.
All points on it will be equidistant from B and C.
(v) From A, draw a line XY parallel to BC.
(vi) Produce the perpendicular bisector of BC to meet XY in Q.
(vii) Join QC and QB.
∆QBC will be the triangle equal in area to ∆ABC
because these are on the same base BC and between the same parallel lines.
On measuring, the length of CQ is 8.2 cm (approx.).

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14 are helpful to complete your math homework.

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2

More Exercise

Take π = \(\\ \frac { 22 }{ 7 } \) unless stated otherwise.

Question 1.
Write whether the following statements are true or false. Justify your answer.
(i) If the radius of a right circular cone is halved and its height is doubled, the volume will remain unchanged.
(ii) A cylinder and a right circular cone are having the same base radius and same height. The volume of the cylinder is three times the volume of the cone.
(iii) In a right circular cone, height, radius and slant height are always the sides of a right triangle.
Solution:
(i) If the radius of a right circular cone is halved and its height is doubled,
then the volume will remain unchanged
It is wrong as
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q1.1
(ii) A cylinder and a right circular cone are having the same base radius
and same height the volume of the cylinder is three times the volume of cone – It is true as
Volume of cylinder = \(\pi { r }^{ 2 }h=3\times \frac { 1 }{ 3 } \pi { r }^{ 2 }h\) = 3(volume of cone)
(iii) In a right circular cone, height, radius and slant height are always the sides of a right triangle
It is true as in a cone and in a right-angled triangle.
Hypotenuse (slant x height) = r2 + h2
and cone is formed by revolving the right triangle about the perpendicular.

Question 2.
Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.
Solution:
10 Slant height of a cone (l) = 10 cm
and radius of the base = 7 cm
Curved surface area = πrl
= \(\\ \frac { 22 }{ 7 } \) × 7 × 10 = 220 cm2

Question 3.
Diameter of the base of a cone is 10.5 cm and slant height is 10 cm. Find its curved surface area.
Solution:
The diameter of the base of a cone = 10.5cm
Its radius (r) = \(\\ \frac { 10.5 }{ 7 } \) = 5.25 cm
and slant height (l) = 10 cm
Curved surface area = πrl
= \(\\ \frac { 22 }{ 7 } \) × 5.25 × 10 cm2
= 165.0 cm2

Question 4.
Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find ,
(i)radius of the base
(ii)total surface area of the cone.
Solution:
Curved surface area of a cone = 308 cm2
Slant height = 14 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q4.1

Question 5.
Find the volume of the right circular cone with
(i) radius 6 cm and height 7 cm
(ii) radius 3.5 cm and height 12 cm.
Solution:
(i) Radius of cone (r) = 6 cm
and height (h) = 7 cm
Volume = \(\frac { 1 }{ 3 } \pi { r }^{ 2 }h \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q5.1

Question 6.
Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm
Solution:
(i) Radius = 7 cm
and slant height (l) = 25 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q6.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q6.2

Question 7.
A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kiloliters ?
Solution:
Diameter of top of conical pit = 3.5 m
Radius (r) = \(\\ \frac { 3.5 }{ 2 } \) = 1.75 m
and depth (h) = 12m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q7.1

Question 8.
If the volume of a right circular cone of height 9 cm is 48π cm3, find the diameter of its base.
Solution:
Volume of a right circular cone = 48π cm3
Height (h) = 9 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q8.1

Question 9.
The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (Use π = 3.14)
Solution:
Height of cone (h) = 15 cm
Volume = 1570 cm3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q9.1

Question 10.
The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white washing its curved surface area at the rate of Rs 210 per 100 m2.
Solution:
Slant height of conical tomb (l) = 25 m
and base diameter = 14 m
Radius(r) = \(\\ \frac { 14 }{ 2 } \) = 7m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q10.1

Question 11.
A conical tent is 10 m high and the radius of its base is 24 m. Find :
(i) slant height of the tent.
(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is Rs 70.
Solution:
Height of a conical tent (h) = 10 m
and radius (r) = 24 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q11.1

Question 12.
A Jocker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the cloth required to make 10 such caps.
Solution:
Base radius of a conical cap = 7 cm
and height (h) = 24 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q12.1
= 550 x 10
= 5500 cm2

Question 13.
(a) The ratio of the base radii of two right circular cones of the same height is 3 : 4. Find the ratio of their volumes.
(b) The ratio of the heights of two right circular cones is 5 : 2 and that of their base radii is 2 : 5. Find the ratio of their volumes.
(c) The height and the radius of the base of a right circular cone is half the corresponding height and radius of another bigger cone. Find:
(i) the ratio of their volumes.
(ii) the ratio of their lateral surface areas.
Solution:
(i) The ratio in base radii of two right circular cones of the same height = 3 : 4
Let h be the height and radius of first cone = 3x and
Radius of second cone = 4x
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q13.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q13.3

Question 14.
Find what length of canvas 2 m in width is required to make a conical tent 20 m in diameter and 42 m in slant height allowing 10% for folds and the stitching. Also find the cost of the canvas at the rate of Rs 80 per metre.
Solution:
Diameter of the base of the conical tent = 20 m
Radius (r) = \(\\ \frac { 20 }{ 2 } \) = 10 m
and slant height (h) = 42 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q14.1

Question 15.
The perimeter of the base of a cone is 44 cm and the slant height is 25 cm. Find the volume and the curved surface of the cone.
Solution:
Perimeter of the base of a cone = 44 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q15.1

Question 16.
The volume of a right circular cone is 9856 cm3 and the area of its base is 616 cm2. Find
(i) the slant height of the cone.
(ii) total surface area of the cone.
Solution:
Volume of a circular cone = 9856 cm3
Area of the base = 616 cm2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q16.2

Question 17.
A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the volume and the curved surface of the cone so formed. (Take π = 3.14)
Solution:
Sides of a right triangle are 6 cm and 8 cm
It is revolved around 8 cm side
Radius (r) = 6 cm
Height (h) = 8 cm
Slant height (l) = 10 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q17.1

Question 18.
The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume be \(\\ \frac { 1 }{ 27 } \) of the volume of the given cone, at what height above the base is the section cut?
Solution:
Height of a cone (H) = 30 cm
A small cone is cut off from the top of the cone given
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q18.2
h = 10 cm
∴ A line parallel to base at a distance of 30 – 10 = 20 cm is drawn.

Question 19.
A semi-circular lamina of radius 35 cm is folded so that the two bounding radii are joined together to form a cone. Find
(i) the radius of the cone.
(ii) the (lateral) surface area of the cone.
Solution:
Radius of a semi-circular lamina = 35 cm
By folding it a cone is formed whose slant height (l) = r = 35
and half circumference = circumference of the top of the cone
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 Q19.1

We hope the ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2 help you. If you have any query regarding ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.2, drop a comment below and we will get back to you at the earliest.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

More Exercise

Take π = \(\\ \frac { 22 }{ 7 } \) unless stated otherwise.

Question 1.
Find the total surface area of a solid cylinder of radius 5 cm and height 10 cm. Leave your answer in terms of π.
Solution:
Radius of the cylinder (r) = 5 cm
Height (h) = 10 cm
Total surface area = 2πr (h + r)
= 2π x 5(10 + 5) cm2
= 10 x 15π
= 150π cm2

Question 2.
An electric geyser is cylindrical in shape, having a diameter of 35 cm and height 1.2m. Neglecting the thickness of its walls, calculate
(i) its outer lateral surface area,
(ii) its capacity in litres.
Solution:
Diameter of cylindrical geyser = 35 cm
Radius (r) = \(\\ \frac { 35 }{ 2 } \) cm
Height = 1.2 m = 120 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q2.1

Question 3.
A school provides milk to the students daily in cylindrical glasses of diameter 7 cm. If the glass is filled with milk upto a height of 12 cm, find how many litres of milk is needed to serve 1600 students.
Solution:
Number of students = 1600
Diameter of cylindrical glasses = 7 cm
Radius (r) = \(\\ \frac { 7 }{ 2 } \) cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q3.1

Question 4.
In the given figure, a rectangular tin foil of size 22 cm by 16 cm is wrapped around to form a cylinder of height 16 cm. Find the volume of the cylinder.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q4.1
Solution:
Length of rectangular tin foil (l) = 22 cm
and breadth (b) = 16 cm
By folding lengthwise, the radius of the cylinder
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q4.2

Question 5.
(i) How many cubic metres of soil must be dug out to make a well 20 metres deep and 2 metres in diameter?
(ii) If the inner curved surface of the well in part (i) above is to be plastered at the rate of Rs 50 per m2, find the cost of plastering.
Solution:
(i) Depth of well (h) = 20 m
and diameter = 2 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q5.2

Question 6.
A roadroller (in the shape of a cylinder) has a diameter 0.7 m and its width is 1.2 m. Find the least number of revolutions that the roller must make in order to level a playground of size 120 m by 44 m.
Solution:
Diameter of a road roller = 0.7 m
Radius (r) = \(\\ \frac { 0.7 }{ 2 } \) = 0.35 m
and width (h) = 1.2 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q6.1

Question 7.
If the volume of a cylinder of height 7 cm is 448 π cm3, find its lateral surface area and total surface area.
Solution:
Volume of a cylinder = 448 π cm3
Height (h) = 7 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q7.1

Question 8.
A wooden pole is 7 m high and 20 cm in diameter. Find its weight if the wood weighs 225 kg per m3.
Solution:
Height of a wooden pole (h) = 7 m
Diameter = 20 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q8.1

Question 9.
The area of the curved surface of a cylinder is 4400 cm2, and the circumference of its base is 110 cm. Find
(i) the height of the cylinder.
(ii) the volume of the cylinder.
Solution:
Area of the curved surface of a cylinder = 4400 cm2
Circumference of base = 110 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q9.1

Question 10.
A cylinder has a diameter of 20 cm. The area of curved surface is 1000 cm2. Find
(i) the height of the cylinder correct to one decimal place.
(ii) the volume of the cylinder correct to one decimal place. (Take π = 3.14)
Solution:
Diameter of a cylinder = 20 cm
Radius (r) = \(\\ \frac { 20 }{ 2 } \) = 10 cm
Curved surface area = 1000 cm2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q10.1

Question 11.
The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up when writing 310 words on an average. How many words would use up a bottle of ink containing one-fifth of a litre?
Answer correct to the nearest. 100 words.
Solution:
Height of cylindrical barrel of a pen (h) = 7 cm
Diameter = 5 mm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q11.1

Question 12.
Find the ratio between the total surface area of a cylinder to its curved surface area given that its height and radius are 7.5 cm and 3.5 cm.
Solution:
Radius of a cylinder (r) = 3.5 cm
and height (h) = 7.5 cm
Total surface area = 2πr(r + h)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q12.1

Question 13.
The radius of the base of a right circular cylinder is halved and the height is doubled. What is the ratio of the volume of the new cylinder to that of the original cylinder?
Solution:
Let the radius of the base of a right circular cylinder = r
and height (h) = h
Volume = πr2h
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q13.1

Question 14.
(i) The sum of the radius and the height of a cylinder is 37 cm and the total surface area of the cylinder is 1628 cm2. Find the height and the volume of the cylinder.
(ii) The total surface area of a cylinder is 352 cm2. If its height is 10 cm, then find the diameter of the base.
Solution:
Sum of radius and height of a cylinder = 37 cm
Total surface area = 1628 cm2
Let r be radius and h be height, then r × h = 37
and 2πr(r + h) = 1628
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q14.2

Question 15.
The ratio between the curved surface and the total surface of a cylinder is 1 : 2. Find the volume of the cylinder, given that its total surface area is 616 cm2.
Solution:
Ratio between curved surface area and total surface area = 1 : 2
Total surface area = 616 cm2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q15.1

Question 16.
Two cylindrical jars contain the same amount of milk. If their diameters are in the ratio 3 : 4, find the ratio of their heights.
Solution:
Volume of two cylinders is the same
Diameter of both cylinder are in the ratio = 3 : 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q16.1

Question 17.
A rectangular sheet of tin foil of size 30 cm x 18 cm can be rolled to form a cylinder in two ways along length and along breadth. Find the ratio of volumes of the two cylinders thus formed.
Solution:
Size of the sheet = 30 cm × 18 cm
(i) By rolling lengthwise,
The circumference of the cylinder = 2πr = 30
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q17.2

Question 18.
A cylindrical tube open at both ends is made of metal. The internal diameter of the tube is 11.2 cm and its length is 21 cm. The metal thickness is 0.4 cm. Calculate the volume of the metal.
Solution:
Internal diameter of a metal tube = 11.2 cm
and radius (r) = \(\\ \frac { 11.2 }{ 2 } \) = 5.6 cm
Length (h) = 21 cm
Thickness of metal = 0.4 cm
External radius (R) = 5.6 + 0.4 = 6.0 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q18.1

Question 19.
The given figure shows a metal pipe 77 cm long. The inner diameter of a cross-section is 4 cm and the outer one is 4.4 cm. Find its
(i) inner curved surface area
(ii) outer curved surface area
(iii) total surface area.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q19.1
Solution:
In the given figure,
Length of metal pipe (h) = 77 cm
Inner diameter = 4 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q19.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q19.3

Question 20.
A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.
Solution:
Diameter of the pencil = 7 mm
Radius (R) = \(\\ \frac { 7 }{ 2 } \) mm = \(\\ \frac { 7 }{ 20 } \) cm
Diameter of graphite (lead) = 1 mm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q20.1

Question 21.
A soft drink is available in two packs
(i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and
(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?
Solution:
(i) Base of the tin of rectangular base = 5 cm × 4 cm
Height = 15 cm
Volume = lbh = 5 × 4 × 15 = 300 cm³
(ii) Base diameter of cylindrical plastic cylinder = 7 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q21.1

Question 22.
A cylindrical roller made of iron is 2 m long. Its inner diameter is 35 cm and the thickness is 7 cm all round. Find the weight of the roller in kg, if 1 cm³ of iron weighs 8 g.
Solution:
Length of cylindrical roller (h) = 2 m = 200 cm
Diameter = 35 cm
Inner radius = \(\\ \frac { 35 }{ 2 } \) cm
Thickness = 7 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 Q22.1

We hope the ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1 help you. If you have any query regarding ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1, drop a comment below and we will get back to you at the earliest.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

More Exercises

Question 1.
Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 cm from the centre.
Solution:
In a circle with centre O and radius 3 cm
and P is at a distance of 5 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q1.1

Question 2.
A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.
Solution:
CT is the radius
CP = 13 cm and tangent PT = 12 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q2.1

Question 3.
The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q3.1
Solution:
Radius of the circle = 6 cm
and length of tangent = 8 cm
Let OP be the distance
i.e. OA = 6 cm, AP = 8 cm .
OA is the radius
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q3.2

Question 4.
Two concentric circles are of the radii 13 cm and 5 cm. Find the length of the chord of the outer circle which touches the inner circle.
Solution:
Two concentric circles with centre O
OP and OB are the radii of the circles respectively, then
OP = 5 cm, OB = 13 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q4.2

Question 5.
Two circles of radii 5 cm and 2-8 cm touch each other. Find the distance between their centres if they touch :
(i) externally
(ii) internally.
Solution:
Radii of the circles are 5 cm and 2.8 cm.
i.e. OP = 5 cm and CP = 2.8 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q5.1
(i) When the circles touch externally,
then the distance between their centres = OC = 5 + 2.8 = 7.8 cm.
(ii) When the circles touch internally,
then the distance between their centres = OC = 5.0 – 2.8 = 2.2 cm

Question 6.
(a) In figure (i) given below, triangle ABC is circumscribed, find x.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed, find x.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q6.1
Solution:
(a) From A, AP and AQ are the tangents to the circle
∴ AQ = AP = 4cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q6.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q6.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q6.4

Question 7.
(a) In figure (i) given below, quadrilateral ABCD is circumscribed; find the perimeter of quadrilateral ABCD.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed and AD ⊥ DC ; find x if radius of incircle is 10 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q7.1
Solution:
(a) From A, AP and AS are the tangents to the circle
∴AS = AP = 6
From B, BP and BQ are the tangents
∴BQ = BP = 5
From C, CQ and CR are the tangents
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q7.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q7.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q7.4

Question 8.
(a) In the figure (i) given below, O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle.
(b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q8.1
Solution:
(i) Join OB
∠OBA = 90°
(Radius through the point of contact is
perpendicular to the tangent)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q8.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q8.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q8.4

Question 9.
(a) If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by
\(r= \frac { a+b-c }{ 2 } \)
(b) In the given figure, PB is a tangent to a circle with centre O at B. AB is a chord of length 24 cm at a distance of 5 cm from the centre. If the length of the tangent is 20 cm, find the length of OP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q9.1
Solution:
(a) Let the circle touch the sides BC, CA and AB
of the right triangle ABC at points D, E and F respectively,
where BC = a, CA = b
and AB = c (as showing in the given figure).
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q9.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q9.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q9.4

Question 10.
Three circles of radii 2 cm, 3 cm and 4 cm touch each other externally. Find the perimeter of the triangle obtained on joining the centres of these circles.
Solution:
Three circles with centres A, B and C touch each other externally
at P, Q and R respectively and the radii of these circles are
2 cm, 3 cm and 4 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q10.1

Question 11.
(a) In the figure (i) given below, the sides of the quadrilateral touch the circle. Prove that AB + CD = BC + DA.
(b) In the figure (ii) given below, ABC is triangle with AB = 10cm, BC = 8cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q11.1
Solution:
(a) Given: Sides of quadrilateral ABCD touch the circle at
P, Q, R and S respectively.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q11.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q11.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q11.4

Question 12.
(a) ln the figure (i) PQ = 24 cm, QR = 7 cm and ∠PQR = 90°. Find the radius of the inscribed circle ∆PQR
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.1
(b) In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12cm, find BP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.2
Solution:
(a) In the figure, a circle is inscribed in the triangle PQR
which touches the sides. O is centre of the circle.
PQ = 24cm, QR = 7 cm ∠PQR = 90°
OM is joined.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q12.6

Question 13.
(a) In the figure (i) given below, AB = 8 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all three semi-circles as shown, find its radius.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q13.1
(b) In the figure (ii) given below, equal circles with centres O and O’ touch each other at X. OO’ is produced to meet a circle O’ at A. AC is tangent to the circle whose centre is O. O’D is perpendicular to AC. Find the value of :
(i) \(\\ \frac { AO’ }{ AO } \)
(ii) \(\frac { area\quad of\quad \Delta ADO’ }{ area\quad of\quad \Delta ACO } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q13.2
Solution:
(a) Let x be the radius of the circle
with centre C and radii of each equal
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q13.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q13.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q13.5

Question 14.
The length of the direct common tangent to two circles of radii 12 cm and 4 cm is 15 cm. Calculate the distance between their centres.
Solution:
Let R and r be the radii of the circles
with centre A and B respectively
Let TT’ be their common tangent.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q14.1
Hence distance between their centres = 17 cm Ans.

Question 15.
Calculate the length of a direct common tangent to two circles of radii 3 cm and 8 cm with their centres 13 cm apart.
Solution:
Let A and B be the centres of the circles
whose radii are 8 cm and 3 cm and
let TT’ length of their common tangent and AB = 13 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q15.1

Question 16.
In the given figure, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively. Given that AB = 8 cm, calculate PQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q16.1
Solution:
AC is a transverse common tangent to the two circles
with centre P and Q and of radii 6 cm and 3 cm respectively
AB = 8 cm. Join AP and CQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q16.3

Question 17.
Two circles with centres A, B are of radii 6 cm and 3 cm respectively. If AB = 15 cm, find the length of a transverse common tangent to these circles.
Solution:
AB = 15 cm.
Radius of the circle with centre A = 6 cm
and radius of second circle with radius B = 3 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q17.2

Question 18.
(a) In the figure (i) given below, PA and PB are tangents at a points A and B respectively of a circle with centre O. Q and R are points on the circle. If ∠APB = 70°, find (i) ∠AOB (ii) ∠AQB (iii) ∠ARB
(b) In the figure (ii) given below, two circles touch internally at P from an external point Q on the common tangent at P, two tangents QA and QB are drawn to the two circles. Prove that QA = QB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q18.1
Solution:
(a) To find : (i) ∠AOB, (ii) ∠AQB, (iii) ∠ARB
Given: PA and PB are tangents at the points A and B respectively
of a circle with centre O and OA and OB are radii on it.
∠APB = 70°
Construction: Join AB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q18.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q18.3

Question 19.
In the given figure, AD is a diameter of a circle with centre O and AB is tangent at A. C is a point on the circle such that DC produced intersects the tangent of B. If ∠ABC = 50°, find ∠AOC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q19.1
Solution:
Given AB is tangent to the circle at A and OA is radius, OA ⊥ AB
In ∆ABD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q19.2

Question 20.
In the given figure, tangents PQ and PR are drawn from an external point P to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ, Find ∠RQS
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q20.1
Solution:
In the given figure,
PQ and PR are tangents to the circle with centre O drawn from P
∠RPQ = 30°
Chord RS || PQ is drawn
To find ∠RQS
∴ PQ = PR (tangents to the circle)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q20.2

Question 21.
(a) In the figure (i) given below, PQ is a tangent to the circle at A, DB is a diameter, ∠ADB = 30° and ∠CBD = 60°, calculate (i) ∠QAB (ii) ∠PAD (iii) ∠CDB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q21.1
(b) In the figure (ii) given below, ABCD is a cyclic quadrilateral. The tangent to the circle at B meets DC produced at F. If ∠EAB = 85° and ∠BFC = 50°, find ∠CAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q21.2
Solution:
(a) PQ is tangent and AD is chord
(i) ∴ ∠QAB = ∠BDA = 30°
(Angles in the alternate segment)
(ii) In ∆ADB,
∠DAB = 90° (Angle in a semi-circle)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q21.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q21.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q21.5
⇒ ∠CAB = 35°

Question 22.
(a) In the figure (i) given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the value of x, y and z. (2015)
(b) In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q22.1
Solution:
Consider the following figure:
TS ⊥ SP,
∠TSR = ∠OSP = 90°
In ∆TSR,
∠TSR + ∠TRS + ∠RTS = 180°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q22.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q22.3

Question 23.
In the given figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO (ii) ∠AOR (iii) ∠APB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q23.1
Solution:
(i) ∠BCO = ∠ACO = 30°
(∵ C is the intersecting point of tangent AC and BC)
(ii) ∠OAC = ∠OBC = 90°
∵∠AOC = ∠BOC = 180° – (90° + 30°) = 60°
(∵ sum of the three angles a ∆ is 180°)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q23.2

Question 24.
(a) In the figure (i) given below, O is the centre of the circle. The tangent at B and D meet at P. If AB is parallel to CD and ∠ ABC = 55°. find: (i)∠BOD (ii) ∠BPD
(b) In the figure (ii) given below. O is the centre of the circle. AB is a diameter, TPT’ is a tangent to the circle at P. If ∠BPT’ = 30°, calculate : (i)∠APT (ii) ∠B OP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q24.1
Solution:
(a) AB || CD
(i) ∠ABC = ∠BCD (Alternate angles)
⇒ ∠BCD = 55°
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q24.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q24.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q24.4

Question 25.
In the adjoining figure, ABCD is a cyclic quadrilateral.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q25.1
The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measure of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.
Solution:
ABCD is a cyclic quadrilateral.
PAQ is the tangent to the circle at A.
∠CAD : ∠CAP = 1 : 2.
AB and AD are the bisectors of ∠CAQ and ∠CAP respectively
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q25.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q25.3

Question 26.
In a triangle ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate (i) ∠QOR (ii) ∠QPR given that ∠A = 60°.
Solution:
OQ and OR are the radii and AC and AB are tangents.
OQ ⊥ AC and OR ⊥ AB
Now in the quad. AROQ
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q26.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q26.2

Question 27.
(a) In the figure (0 given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find
(i) ∠CBA (ii) ∠CQA (2006)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q27.1
(b) In the figure (ii) given below, AP and BP are tangents to the circle with centre O. Given ∠APB = 60°, calculate.
(i) ∠AOB (ii) ∠OAB (iii) ∠ACB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q27.2
Solution:
(a) AB is the diameter.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q27.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q27.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q27.5

Question 28.
(a) In the figure (i) given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY. (1994)
(b) In the figure (ii) given below, O is the centre of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate (i) ∠PRQ (ii) ∠POQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q28.1
Solution:
(a) Join OY, OX and OY are the radii of the circle
and XT and YT are the tangents to the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q28.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q28.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q28.4

Question 29.
Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.
Solution:
In a circle, two chords AB and CD intersect
each other at P internally.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q29.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q29.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q29.3

Question 30.
(a) In the figure (i) given below, PT is a tangent to the circle. Find TP if AT = 16 cm and AB = 12 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q30.1
(b) In the figure given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find (i) AB. (ii)the length of tangent PT.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q30.2
Solution:
(a) PT is the tangent to the circle and AT is a secant.
PT² = TA × TB
Now TA = 16 cm, AB = 12 cm
TB = AT – AB = 16 – 12 = 4 cm
∴ PT² = 16 + 4 = 64 = (8)²
⇒ PT = 8 cm or TP = 8 cm
(b) PT is tangent and PDC is secant out to the circle
∴ PT² = PC × PD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q30.3

Question 31.
PAB is secant and PT is tangent to a circle
(i) PT = 8 cm and PA = 5 cm, find the length of AB.
(ii) PA = 4.5 cm and AB = 13.5 cm, find the length of PT.
Solution:
∵ PT is the tangent and PAB is the secant of the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q30.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q30.5

Question 32.
In the adjoining figure, CBA is a secant and CD is tangent to the circle. If AB = 7 cm and BC = 9 cm, then
(i) Prove that ∆ACD ~ ∆DCB.
(ii) Find the length of CD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q32.1
Solution:
In ∆ACD and ∆DCB
∠C = ∠C (common)
∠CAD = ∠CDB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q32.2

Question 33.
(a) In the figure (i) given below, PAB is secant and PT is tangent to a circle. If PA : AB = 1:3 and PT = 6 cm, find the length of PB.
(b) In the figure (ii) given below, ABC is an isosceles triangle in which AB = AC and Q is mid-point of AC. If APB is a secant, and AC is tangent to the circle at Q, prove that AB = 4 AP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q33.1
Solution:
(a) In the figure (i),
PAB is secant and PT is the tangent to the circle.
PT² = PA × PB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q33.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q33.3

Question 34.
Two chords AB, CD of a circle intersect externally at a point P. If PA = PC, Prove that AB = CD.
Solution:
Given: Two chords AB and CD intersect
each other at P outside the circle. PA = PC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q34.1

Question 35.
(a) In the figure (i) given below, AT is tangent to a circle at A. If ∠BAT = 45° and ∠BAC = 65°, find ∠ABC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q35.1
(b) In the figure (ii) given below, A, B and C are three points on a circle. The tangent at C meets BA produced at T. Given that ∠ATC = 36° and ∠ACT = 48°, calculate the angle subtended by AB at the centre of the circle. (2001)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q35.2
Solution:
(a) AT is the tangent to the circle at A
and AB is the chord of the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q35.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q35.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q35.5

Question 36.
In the adjoining figure ∆ABC is isosceles with AB = AC. Prove that the tangent at A to the circumcircle of ∆ABC is parallel to BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q36.1
Solution:
Given: ∆ABC is an isosceles triangle with AB = AC.
AT is the tangent to the circumcircle at A.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q36.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q36.3

Question 37.
If the sides of a rectangle touch a circle, prove that the rectangle is a square.
Solution:
Given: A circle touches the sides AB, BC, CD and DA
of a rectangle ABCD at P, Q, R and S respectively.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q37.1
To Prove : ABCD is a square.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q37.2

Question 38.
(a) In the figure (i) given below, two circles intersect at A, B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circle at C and D respectively. Prove that CD is parallel to the tangent at P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q38.1
(b) In the figure (ii) given below, two circles with centres C, C’ intersect at A, B and the point C lies on the circle with centre C’. PQ is a tangent to the circle with centre C’ at A. Prove that AC bisects ∠PAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q38.2
Solution:
Given: Two circles intersect each other at A and B.
From a point P on one circle, PAC and PBD are drawn.
From P, PT is a tangent drawn. CD is joined.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q38.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q38.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q38.5

Question 39.
(a) In the figure (i) given below, AB is a chord of the circle with centre O, BT is tangent to the circle. If ∠OAB = 32°, find the values of x and y.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.1
(b) In the figure (ii) given below, O and O’ are centres of two circles touching each other externally at the point P. The common tangent at P meets a direct common tangent AB at M. Prove that:
(i) M bisects AB (ii) ∠APB = 90°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.2
Solution:
AB is a chord of a circle with centre O.
BT is a tangent to the circle and ∠OAB = 32°.
∴ In ∆OAB,
OA = OB (radii of the same circle)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3 Q39.7

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

More Exercises

Question 1.
Draw a circle of radius 3 cm. Mark its centre as C and mark a point P such that CP = 7 cm. Using ruler and compasses only, Construct two tangents from P to the circle.
Solution:
Steps of Construction :

  1. Draw a circle with centre C and radius 3 cm.
  2. Mark a point P such that CP = 7 cm.
  3. With CP as diameter, draw a circle intersecting the given circle at T and S.
  4. Join PT and PS.
  5. Draw a tangent at Q to the circle given. Which intersects PT at D.
  6. Draw the angle bisector of ∠PDQ intersecting CP at E.
  7. With centre E and radius EQ, draw a circle.
    It will touch the tangent T and PS and the given circle at Q.
    This is the required circle.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q1.1

Question 2.
Draw a line AQ = 7 cm. Mark a point P on AQ such that AP = 4 cm. Using ruler and compasses only, construct :
(i) a circle with AP as diameter.
(ii) two tangents to the above circle from the point Q.
Solution:
Steps of construction :

  1. Draw a line segment AQ = 7 cm.
  2. From AQ,cut off AP = 4cm
  3. With AP as diameter draw a circle with centre O.
  4. Draw bisector of OQ which intersect OQ at M.
  5. With centre M and draw a circle with radius MQ
    which intersects the first circle at T and S.
  6. Join QT and QS.
    QT and QS are the tangents to the first circle.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q2.1

Question 3.
Using ruler and compasses only, construct a triangle ABC having given c = 6 cm, b = 1 cm and ∠A = 30°. Measure side a. Draw carefully the circumcircle of the triangle.
Solution:
Steps of Construction :

  1. Draw a line segment AC = 7 cm.
  2. At C, draw a ray CX making an angle of 30°
  3. With centre A and radius 6 cm draw an arc
    which intersects the ray CX at B.
  4. Join BA.
  5. Draw perpendicular bisectors of AB and AC intersecting each other at O.
  6. With centre O and radius OA or OB or OC,
    draw a circle which will pass through A, B and C.
    This is the required circumcircle of ∆ABC
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q3.1

Question 4.
Using ruler and compasses only, construct an equilateral triangle of height 4 cm and draw its circumcircle.
Solution:
Steps of Construction :

  1. Draw a line XY and take a point D on it.
  2. At D, draw perpendicular and cut off DA = 4 cm.
  3. From A, draw rays making an angle of 30°
    on each side of AD meeting the line XY at B and C.
  4. Now draw perpendicular bisector of AC intersecting AD at O.
  5. With centre O and radius OA or OB or OC
    draw a circle which will pass through A, B and C.
    This is the required circumcircle of ∆ABC.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q4.1

Question 5.
Using ruler and compasses only :
(i) Construct a triangle ABC with the following data: BC = 7 cm, AB = 5 cm and ∠ABC = 45°.
(ii) Draw the inscribed circle to ∆ABC drawn in part (i).
Solution:
Steps of construction :

  1. Draw a line segment BC = 7 cm.
  2. At B, draw a ray BX making an angle of 45° and cut off BA = 5 cm.
  3. Join AC.
  4. Draw the angle bisectors of ∠B and ∠C intersecting each other at I.
  5. From I, draw a perpendicular ID on BC.
  6. With centre, I and radius ID, draw a circle
    which touches the sides of ∆ABC at D, E and F respectively.
    This is the required inscribed circle.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q5.1

Question 6.
Draw a triangle ABC, given that BC = 4cm, ∠C = 75° and that radius of the circumcircle of ∆ABC is 3 cm.
Solution:
Steps of Construction:

  1. Draw a line segment BC = 4 cm
  2. Draw the perpendicular bisector of BC.
  3. From B draw an arc of 3 cm radius which intersects the perpendicular bisector at O.
  4. Draw a ray CX making art angle of 75°
  5. With centre O and radius 3 cm draw a circle which intersects the ray CX at A.
  6. Join AB.
    ∆ABC is the required triangle
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q6.1

Question 7.
Draw a regular hexagon of side 3.5 cm construct its circumcircle and measure its radius.
Solution:
Steps of construction:

  1. Draw a regular hexagon ABCDEF whose each side is 3.5 cm.
  2. Draw the perpendicular bisector of AB and BC
    which intersect each other at O.
  3. Join OA and OB.
  4. With centre O and radius OA or OB, draw a circle
    which passes through A, B, C, D, E and P.
    Then this is the required circumcircle.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q7.1

Question 8.
Construct a triangle ABC with the following data: AB = 5 cm, BC = 6 cm and ∠ABC = 90°.
(i) Find a point P which is equidistant from B and C and is 5 cm from A. How many such points are there ?
(ii) Construct a circle touching the sides AB and BC, and whose centre is equidistant from B and C.
Solution:
Steps of Construction :

  1. Draw a line segment BC = 6 cm.
  2. At B, draw a ray BX making an angle of 90° and cut off BA = 5 cm.
  3. Join AC.
  4. Draw the perpendicular bisector of BC.
  5. From A with 5 cm radius draw arc which intersects the perpendicular bisector of BC at P and P’.
    There are two points.
  6. Draw the angle bisectors of ∠B and ∠C intersecting at 0.
  7. From O, draw OD ⊥ BC.
  8. With centre O and radius OD, draw a circle which will touch the sides AB and BC.
    This is the required circle.
    ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test Q8.1

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5

More Exercise

Question 1.
The diameter of a metallic sphere is 6 cm. The sphere is melted and drawn into a wire of uniform cross-section. If the length of the wire is 36 m, find its radius.
Solution:
Diameter of metallic sphere = 6 cm
Radius(r) = \(\\ \frac { 6 }{ 2 } \) = 3cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q1.1

Question 2.
The radius of a sphere is 9 cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire in metres.
Solution:
Radius of sphere = 9 cm
Volume = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }=\frac { 4 }{ 3 } \pi \times { \left( 9 \right) }^{ 3 }{ cm }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q2.1
h = 97200 cm = 972 m
Length of wire = 972 m

Question 3.
A solid metallic hemisphere of radius 8 cm is melted and recasted into right circular cone of base radius 6 cm. Determine the height of the cone.
Solution:
Radius of a solid hemisphere (r) = 8 cm
Volume = \(\frac { 2 }{ 3 } \pi { r }^{ 3 }=\frac { 2 }{ 3 } \pi \times { \left( 8 \right) }^{ 3 }{ cm }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q3.1

Question 4.
A rectangular water tank of base 11 m x 6 m contains water upto a height of 5 m. if the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank.
Solution:
Base of a water tank = 11 m × 6 m
Height of water level in it (h) = 5 m
Volume of water =11 × 6 × 5 = 330 m³
Volume of water in the cylindrical tank
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q4.1

Question 5.
The rain water from a roof of dimensions 22 m x 20 m drains into a cylindrical vessel having diameter of base 2 m and height; 3.5 m. If the rain water collected from the roof just fill the cylindrical vessel, then find the rainfall in cm.
Solution:
Dimensions of roof = 22 m × 20 m
Let rainfall = x m
.’. Volume of water = 22 × 20 × x m³
Volume of water in cylinder = 22 × 20 × x m³
Diameter of its base = 2 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q5.1

Question 6.
The volume of a cone is the same as that of the cylinder whose height is 9 cm and diameter 40 cm. Find the radius of the base of the cone if its height is 108 cm.
Solution:
Diameter of a cylinder = 40 cm
Radius (r) = \(\\ \frac { 40 }{ 2 } \) = 20 cm
Height(h) = 9 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q6.1

Question 7.
Eight metallic spheres, each of radius 2 cm, are melted and cast into a single sphere. Calculate the radius of the new (single) sphere.
Solution:
Radius of each metallic sphere (r) = 2 cm
Volume of one sphere = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q7.1

Question 8.
A metallic disc, in the shape of a right circular cylinder, is of height 2.5 mm and base radius 12 cm. Metallic disc is melted and made into a sphere. Calculate the radius of the sphere.
Solution:
Height of disc cylindrical shaped = 2.5 mm
and base radius = 12 cm
Volume of the disc = πr²h
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q8.1

Question 9.
Two spheres of the same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the big sphere.
Solution:
Weight of first sphere = 1 kg
and weight of second sphere = 7 kg
Radius of smaller sphere = 3 cm
Let r be the radius of a larger sphere
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q9.1
R = 3 x 2 = 6 cm
Diameter of big sphere = 2 x 6 = 12 cm

Question 10.
A hollow copper pipe of inner diameter 6 cm and outer diameter 10 cm is melted and changed into a solid circular cylinder of the same height as that of the pipe. Find the diameter of the solid cylinder.
Solution:
Inner diameter of a hollow pipe = 6 cm
and outer diameter = 10 cm
Inner radius (r) = \(\\ \frac { 6 }{ 2 } \) = 3cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q10.1

Question 11.
A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm and height is 72 cm, find the uniform thickness of the cylinder.
Solution:
Radius of a solid sphere (r) = 6 cm
Volume = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }=\frac { 4 }{ 3 } \pi \times { \left( 6 \right) }^{ 3 }{ cm }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q11.1

Question 12.
A hollow metallic cylindrical tube has an internal radius of 3 cm and height 21 cm. The thickness of the metal of the tube is \(\\ \frac { 1 }{ 2 } \) cm. The tube is melted and cast into a right circular cone of height 7 cm. Find the radius of the cone correct to one decimal place.
Solution:
Internal radius of a hollow metallic cylindrical tube (r) = 3 cm
and height (h) = 21 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q12.1

Question 13.
A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, is melted into a cone of base diameter 8 cm. Find the height of the cone. (2002)
Solution:
Internal diameter of a hollow sphere = 4 cm
and external diameter = 8 cm
Internal radius (r) = 2 cm
and external radius (R) = 4 cm
Volume of hollow sphere
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q13.1

Question 14.
A well with inner diameter 6 m is dug 22 m deep. Soil taken out of it has been spread evenly all round it to a width of 5 m to form an embankment. Find the height of the embankment.
Solution:
Inner diameter of a well = 6 m
Depth (h) = 22 m
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q14.2

Question 15.
A cylindrical can of internal diameter 21 cm contains water. A solid sphere whose diameter is 10.5 cm is lowered into the cylindrical can. The sphere is completely immersed in water. Calculate the rise in water level, assuming that no water overflows.
Solution:
Internal diameter of cylindrical can = 21 cm
Radius (R) = \(\\ \frac { 21 }{ 2 } \) cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q15.2

Question 16.
There is water to a height of 14 cm in a cylindrical glass jar of radius 8 cm. Inside the water there is a sphere of diameter 12 cm completely immersed. By what height will the water go down when the sphere is removed?
Solution:
Radius of the cylindrical jar (R) = 8 cm
Height of water level (h) = 14 cm
Volume of water = πR²h
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q16.1

Question 17.
A vessel in the form of an inverted cone is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped in it so that they are fully submerged. As a result, one-third of the water in the original cone overflows. What is the volume of each of the solid cone submerged? (2002)
Solution:
Height of conical vessel (h) = 20 cm
and diameter = 16.8 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q17.2

Question 18.
A solid metallic circular cylinder of radius 14 cm and height 12 cm is melted and recast into small cubes of edge 2 cm. How many such cubes can be made from the solid cylinder?
Solution:
Radius of a solid metallic cylindrical (r) = 14 cm
and height (h) = 12 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q18.1

Question 19.
How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm x 11 cm x 12 cm?
Solution:
Diameter of a shot = 3 cm
Radius (r) = \(\\ \frac { 3 }{ 2 } \) cm
Volume of one shot = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q19.1

Question 20.
How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm?
Solution:
Diameter of lead shot = 4 cm
Radius (r) = \(\\ \frac { 4 }{ 2 } \) = 2 cm and
volume =\(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q20.1

Question 21.
Find the number of metallic circular discs with 1.5 cm base diameter and height 0.2 cm to be melted to form a circular cylinder of height 10 cm and diameter 4.5 cm.
Solution:
Radius of the circular disc (r) = 0.75 cm
Height of circular disc (h) = 0.2 cm
Radius of cylinder (R) = 2.25 cm
Height of cylinder (H) = 10 cm
Now,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q21.1

Question 22.
A solid metal cylinder of radius 14 cm and height 21 cm is melted down and recast into spheres of radius 3.5 cm. Calculate the number of spheres that can be made.
Solution:
Radius of a solid metallic cylinder (r) = 14 cm
and height (h) = 21 cm
Volume of cylinder = πr²h
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q22.1

Question 23.
A metallic sphere of radius 10.5 cm is melted and then recast into small cenes, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained. (2005)
Solution:
Radius of a metallic sphere (r) = 10.5 cm
Volume = \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)
= \(\\ \frac { 4 }{ 3 } \) × π × 10.5 × 10.5 × 10.5 = 1543.5π cm³
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q23.1

Question 24.
A certain number of metallic cones each of radius 2 cm and height 3 cm are melted and recast in a solid sphere of radius 6 cm. Find the number of cones. (2016)
Solution:
Radius of each cone (r) = 2 cm
and height (h) = 3 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q24.1

Question 25.
A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top, which is open, is 2.5 cm. It is filled with water upto the rim. When some lead shots, each of which is a sphere of radius 0.25 cm, are dropped into the vessel, \(\\ \frac { 2 }{ 5 } \) of the water flows out. Find the number of lead shots dropped into the vessel. (2003)
Solution:
Radius of the top of the inverted conical vessel (R) = 2.5 cm
and height (h)= 11 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q25.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q25.2

Question 26.
The surface area of a solid metallic sphere is 616 cm². It is melted and recast into smaller spheres of diameter 3.5 cm. How many such spheres can be obtained? (2007)
Solution:
Surface area of a metallic sphere = 616 cm²
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q26.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q26.2

Question 27.
The surface area of a solid metallic sphere is 1256 cm². It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate
(i) the radius of the solid sphere.
(ii) the number of cones recast. (Use π = 3.14).
Solution:
Surface area of a solid metallic sphere = 1256 cm²
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q27.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q27.2

Question 28.
Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboid pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?
Solution:
Speed of water flow = 15 km/h
Diameter of pipe = 14 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q28.1

Question 29.
A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate :
(i) the total surface area of the can in contact with water when the sphere is in it.
(ii) the depth of the water in the can before the sphere was put into the can. Given your answer as proper fractions.
Solution:
Radius of a cylindrical can = 3.5 cm
Radius of the sphere = 3.5 cm
and height of water level in the can = 3.5 × 2 = 7 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q29.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5 Q29.2

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