Subject

गणित

Class

CBSEH Class 10

Pre Boards

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Sample Papers

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 Multiple Choice QuestionsLong Answer Type

11.

आकृति 6 में, बनी incrementABC, जिसके शीर्ष A के निर्देशांक (0, -1) हैं तथा भुजाओं AB तथा AC के मध्य-बिंदुओं D तथा E के निर्देशांक क्रमशः (1, 0) तथा (0, 1) हैं। यदि F भुजा BC का मध्य-बिंदु हैं तो त्रिभुजों DEF तथा ABC के क्षेत्रफल ज्ञात कीजिए। 
                  


म ा न ा space straight B space left parenthesis straight p comma space straight q right parenthesis space और space straight C space left parenthesis straight r comma space straight s right parenthesis
AB space क े space मध ् य space ब िं द ु space equals space straight D space क ा space समन ् वय
left parenthesis begin inline style fraction numerator 0 space plus space straight p over denominator 2 end fraction end style comma space begin inline style fraction numerator negative 1 space plus space straight q over denominator 2 end fraction end style right parenthesis space equals space left parenthesis 1 comma space 0 right parenthesis
rightwards double arrow space begin inline style straight p over 2 end style space equals space 1
rightwards double arrow space straight p space equals space 2
rightwards double arrow space begin inline style fraction numerator negative 1 space plus space straight q over denominator 2 end fraction end style space equals space 0
rightwards double arrow space minus 1 space plus space straight q space equals space 0
rightwards double arrow space straight q space equals space 1
therefore space straight B left parenthesis 2 comma space 1 right parenthesis
AC space क े space मध ् य space ब िं द ु space equals space straight E space क ा space समन ् वय
left parenthesis begin inline style fraction numerator 0 space plus space straight r over denominator 2 end fraction end style comma space begin inline style fraction numerator negative 1 space plus space straight s over denominator 2 end fraction end style right parenthesis space equals space left parenthesis 0 comma space 1 right parenthesis
rightwards double arrow space begin inline style straight r over 2 end style space equals 0
rightwards double arrow space straight r space equals space 0
rightwards double arrow space begin inline style fraction numerator negative 1 space plus space straight s space over denominator 2 end fraction end style space equals space 1
rightwards double arrow space minus 1 space plus straight s space equals 2 space
rightwards double arrow space straight s space equals 3
therefore space straight C left parenthesis 0 comma space 3 right parenthesis
straight F space क ा space समन ् वय space equals space BC space क े space मध ् य space ब िं द ु
space space space space space space space space space space space space space space space space space space left parenthesis straight x comma space straight y right parenthesis space equals space left parenthesis begin inline style fraction numerator 2 space plus space 0 over denominator 2 end fraction end style comma space begin inline style fraction numerator 1 space plus space 3 over denominator space 2 end fraction end style right parenthesis space equals space left parenthesis 1 comma space 2 right parenthesis
straight A space left parenthesis 0 comma space minus 1 right parenthesis comma space straight B space left parenthesis 2 comma space 1 right parenthesis comma space straight C space left parenthesis 0 comma space 3 right parenthesis
increment ABC space क ा space क ् ष े त ् रफल comma
equals space begin inline style 1 half end style left square bracket straight x subscript 1 left parenthesis straight y subscript 2 space minus space straight y subscript 3 right parenthesis space plus space straight x subscript 2 left parenthesis straight y subscript 3 space minus space straight y subscript 1 right parenthesis space plus space straight x subscript 3 left parenthesis straight y subscript 1 space minus space straight y subscript 2 right parenthesis right square bracket
equals space begin inline style 1 half end style left square bracket 0 left parenthesis 1 space minus space 3 right parenthesis space plus space 2 left parenthesis 3 space plus space 1 right parenthesis space plus space 0 left parenthesis negative 1 space minus 1 right parenthesis right square bracket
equals begin inline style 1 half end style left parenthesis 8 right parenthesis space equals space 4 space वर ् ग space य ू न ि ट
straight D left parenthesis 1 comma space 0 right parenthesis comma space straight E left parenthesis 0 comma space 1 right parenthesis space straight F left parenthesis 1 comma space 2 right parenthesis
increment DEF space क ा space क ् ष े त ् रफल comma
equals space begin inline style 1 half end style left square bracket 1 left parenthesis 1 space minus 2 right parenthesis space plus space 0 left parenthesis 2 space minus 0 right parenthesis space plus space 1 left parenthesis 0 space minus space 1 right parenthesis right square bracket space
equals space begin inline style 1 half end style space left parenthesis negative 1 space minus 1 right parenthesis space equals space minus 1
equals space 1 space वर ् ग space य ू न ि ट space space space space space space space space space space space space space space space space space space space space therefore space space space space space क ् ष े त ् रफल space हम े श ा space धन ा त ् मक left parenthesis plus ve right parenthesis space ह ो त ा space ह ै। space
72 Views

12.

x के लिए हल कीजिए: 
begin inline style fraction numerator 2 straight x over denominator straight x minus 3 end fraction plus fraction numerator 1 over denominator 2 straight x minus 3 end fraction plus fraction numerator 3 straight x plus 9 over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis 2 straight x plus 3 right parenthesis end fraction equals 0 comma space straight x not equal to 3 comma negative 3 divided by 2 end style


begin inline style fraction numerator 2 straight x over denominator straight x minus 3 end fraction plus fraction numerator 1 over denominator 2 straight x minus 3 end fraction plus fraction numerator 3 straight x plus 9 over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis 2 straight x plus 3 right parenthesis end fraction equals 0
rightwards double arrow space begin display style fraction numerator 2 straight x left parenthesis 2 straight x plus 3 right parenthesis space plus space left parenthesis straight x minus 3 right parenthesis space plus space 3 straight x space plus space 9 over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis 2 straight x plus 3 right parenthesis end fraction end style space equals space 0
rightwards double arrow space 4 straight x squared space plus space 6 straight x space plus space straight x space minus space 3 space plus space 3 straight x space plus space 9 space equals space 0
rightwards double arrow space 4 straight x squared space plus space 10 straight x space plus space 6 space equals space 0 space space space space space space space space space space space space space space space space therefore space द ो न ों space ओर space द ो space स े space भ ा ग space द े न े space पर
rightwards double arrow space 2 straight x squared space plus space 5 straight x space plus space 3 space equals space 0
rightwards double arrow space 2 straight x squared space plus space 3 straight x space plus space 2 straight x space plus space 3 space equals space 0
rightwards double arrow space straight x left parenthesis 2 straight x plus 3 right parenthesis space plus space 1 left parenthesis 2 straight x plus 3 right parenthesis space equals space 0
rightwards double arrow space left parenthesis 2 straight x plus 3 right parenthesis left parenthesis straight x plus 1 right parenthesis space equals space 0
rightwards double arrow space 2 straight x space plus space 3 space equals 0 space space space space space space space space और space space space space space space straight x space plus space 1 space equals space 0
rightwards double arrow space straight x space equals space fraction numerator negative 3 over denominator 2 end fraction space space space space space space space space space space space space space और space space space space space space straight x space equals space minus 1
space ल े क ि न comma space straight x space not equal to space fraction numerator negative 3 over denominator 2 end fraction
therefore space straight x space equals space minus 1 space ह ी space सम ा ध ा न space ह ै। space

end style

114 Views

13.

आकृति 3 में, O केंद्र वाले वृत्त की AP तथा BP ऐसी दो स्पर्श रेखाएँ हैं कि AP = 5 सेमी तथा  ∠APB =600 है। जीवा AB की लम्बाई ज्ञात कीजिए। 


PA space equals space PB
angle PAB space equals angle PBA space... left parenthesis straight I right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space..... left square bracket space इस space क ो ण space क े space सम ी प space बर ा बर space करन े space पर space
increment PAB comma space angle PAB space plus space angle PBA space plus space angle APB space equals space 180 degree
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space....... left square bracket increment space क ा space य ो ग space ग ु ण space प ् रम े य space
rightwards double arrow space angle PAB space plus angle PAB space plus space 60 degree space equals space 180 degree
rightwards double arrow space 2 angle PAB space equals space 180 degree space minus space 60 degree space equals space 120 degree
rightwards double arrow space angle PAB space equals space 60 degree
rightwards double arrow space angle PAB space equals space angle PBA space equals space angle APB space equals space 60 degree
space space space space space space space space space space AB space equals space AP space equals space 5 cm
सिध्द हुआ,
98 Views

14.

4 मी व्यास का एक कुआँ, 21 मी की गहराई तक खोदा जाता है। इससे निकली हुई मिट्टी को कुएँ के चारों ओर 3 मी चौड़ी एक वृत्ताकार वलय (ring) बनाते हुए समान रूप से फैला कर एक बॉँध बनाया जाता है। बाँध की ऊँचाई ज्ञात कीजिए।  


क ु ऍ ं space क ी space त ् र ि ज ् य ा comma space straight r space equals space begin inline style 4 over 2 end style space equals space 2 space straight m
  
परिधि की त्रिज्या, R = 2 + 3 = 5 m
कुऍं की ऊँचाई, h = 21 m
अप े क ् ष ि त space ऊ ं च ा ई space equals space fraction numerator πr squared straight h over denominator straight pi left parenthesis straight R squared minus straight r squared right parenthesis end fraction
fraction numerator 2 space straight x space 2 space straight x space 21 over denominator left parenthesis 5 squared space minus space 2 squared right parenthesis end fraction space equals space fraction numerator 84 over denominator 25 space minus space 4 end fraction space equals space 84 over 21 space equals space 4 space straight m

सिध्द हुआ, कि परिधि की ऊँचाई 4m है।

 

186 Views

15.

आकृति 4 में, ABCD एक वर्ग है जिसकी भुजा 14 सेमी है। प्रत्येक भुजा को ब्यास मान कर अर्धवृत्त बनाए हैं। छायांकित भाग का क्षेत्रफल ज्ञात कीजिए।space left parenthesis straight pi space equals space 22 over 7 space ल ी ज ि ए right parenthesis
                                      


                               


वृत्त का त्रिज्या, r = straight D over 2 space equals space 14 over 2 space equals space 7 space c m
चूंकि सभी अर्धवृत्त का त्रिज्या एक जैसी है-

चारों अर्धवृत्त के क्षेत्रफल,
           AOB + DOC + AOD + BOC
equals space 4 left square bracket 1 half πr squared right square bracket
equals space 4 left square bracket 1 half space straight x space 22 over 7 space straight x space 7 space straight x space 7 right square bracket space equals space 308 space cm squared
space वर ् ग space क ा space क ् ष े त ् रफल comma space ABCD space equals space left parenthesis straight D right parenthesis squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis 14 right parenthesis squared space equals space 196 space cm squared
space छ ा य ां क ि त space क ् ष े त ् र space क ा space क ् ष े त ् रफल
equals space 2 left parenthesis वर ् ग space क ा space क ् ष े त ् रफल right parenthesis space minus space 4 left parenthesis space अर ् धव ृ त ् त space क ा space क ् ष े त ् रफल right parenthesis
equals space 2 space straight x space 196 space minus space 308
equals space 392 space minus space 308 space equals space 84 space cm squared 

423 Views

16.

बिंदु P का x-निर्देशांक उसके y-निर्देशांक का दुगुना है। यदि बिंदु Q(2, -5) तथा  R(-3, 6) से समदूरस्थ है, तो P के निर्देशांक ज्ञात कीजिए।  


माना कि बिंदु P (2K, K), Q(2, -5), R(-3, 6)
          
PQ = PR    ........ [दिया है
PQ2 = PR2                                     ...  दोनों तरफ वर्ग करने पर 
(2K - 2)2 + (K + 5)2 = (2K + 3)2 + (K-6)2
rightwards double arrow space 4 straight k squared space plus space 4 space minus space 8 straight k space plus space straight k squared space plus space 10 straight k space plus space 25 space equals space 4 straight k squared space plus space 9 space plus space 12 straight k space plus space straight k squared space minus space 12 straight k space plus space 36 space
rightwards double arrow space 2 straight k space plus space 29 space equals space 45
rightwards double arrow space 2 straight k space equals space 16
rightwards double arrow space straight k space equals space 8
space इसल ि ए space ब िं द ु space straight P space क ा space समन ् वय space ह ै space left parenthesis 16 comma space 8 right parenthesis space

267 Views

17.

आकृति 7 में, दो चाप PAQ दर्शाई गई हैं। चाप PAQ, O केन्द्र वाले वृत्त का भाग है, जिसकी त्रिज्या OP है तथा चाप PBQ, PQ को व्यास मान कर बनाया गया अर्धवृत्त है जिसका केंद्र M है। यदि OP = 10 सेमी तथा PQ = 10 सेमी तो दर्शाइए कि रेखांकित भाग का क्षेत्रफल 25 left parenthesis square root of 3 space minus space begin inline style straight pi over 6 end style right parenthesis स े म ी squared है। 
                                  


हम जानते हैं कि,
OP = OQ = 10 cm
PQ = OQ = PQ = 10 cm
therefore space increment POQ space समभ ु ज space ह ै comma
rightwards double arrow space angle POQ space equals space 60 degree
straight D space equals space 70 space cm comma space straight r space equals space 5 space cm comma space straight theta space equals space 60 degree comma space straight R space equals space 10 space cm
छ ा य ां क ि त space क ् ष े त ् र space क ा space क ् ष े त ् रफल space equals space increment OPQ space क ा space क ् ष े त ् रफल space plus space अर ् धव ृ त ् त space क ा space क ् ष े त ् रफल left parenthesis PBQM right parenthesis space minus space क ् ष े त ् र space क ा space क ् ष े त ् रफल left parenthesis OPAQ right parenthesis
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis straight D right parenthesis squared space plus space begin inline style 1 half end style πr squared space minus space begin inline style fraction numerator straight theta over denominator 360 degree end fraction end style πR squared
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis 10 right parenthesis squared space plus space begin inline style 1 half end style space straight x space straight pi left parenthesis 5 right parenthesis squared space minus begin inline style fraction numerator 60 degree over denominator 360 degree end fraction end style straight pi left parenthesis 10 right parenthesis squared
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis 100 right parenthesis space plus space begin inline style 25 over 2 end style straight pi space minus space begin inline style fraction numerator 100 straight pi over denominator 6 end fraction end style
equals space 25 square root of 3 space minus space begin inline style fraction numerator 25 straight pi over denominator 6 end fraction end style
equals space 25 left parenthesis square root of 3 space minus space begin inline style straight pi over 6 end style right parenthesis space cm squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space स ि ध ् द space ह ु आ । space

90 Views

18.

समान्तर श्रेढ़ी 18, 16, 14, ....... के कितने पद लिए जाएँ कि उनका योग शून्य हो?


प्रथम पद, a = 18
सार्व अंतर, d = 16 - 18 = -2
sn = 0                       ........ दिया है,
rightwards double arrow straight n over 2 left parenthesis 2 straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d right parenthesis space equals space 0
rightwards double arrow straight n over 2 left square bracket 36 space plus space left parenthesis straight n space minus space 1 right parenthesis left parenthesis negative 2 right parenthesis right square bracket space equals space 0
rightwards double arrow space 2 straight n over 2 space left parenthesis 18 space minus space straight n space plus space 1 right parenthesis space equals space 0
rightwards double arrow space straight n space left parenthesis 19 minus straight n right parenthesis space equals space 0
rightwards double arrow space straight n space equals space 0 space space space space space space space space space space space bold और bold space space space space space space space space space space space space space space space space space 19 space minus space straight n space equals space 0 space space space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight n space equals space 19 space space space space space space space space space space space space space space space space space space space space space space space space space space space space
परन्तु n शून्य नहीं हो सकता है। 
इसलिये, शब्दों की संख्या, n = 19



124 Views

19.

आकृति 5 में, सजावट के लिए बना एक ब्लाक दर्शाया गया है जो दो ठोसों-एक घन तथा एक अर्धगोले से बना  है। ब्लाक का आधार एक 6 सेमी भुजा का घन है तथा उसके ऊपर एक अर्धगोला है जिसका ब्यास 3.5 सेमी है। ब्लाक का कुल पृष्ठीय क्षेत्रफल ज्ञात कीजिए। left parenthesis straight pi space equals space 22 over 7 space ल ी ज ि ए right parenthesis
                       


ब ् ल ॉ क space क ा space क ु ल space धर ा तल space क ् ष े त ् रफल space
equals space घनक ् ष े त ् र space क ा space क ु ल space धर ा तल space क ् ष े त ् रफल space plus space straight C. straight S. space ग ो ल ा र ् ध space क ा space क ् ष े त ् रफल space minus space व ृ त ् त space क ा space क ् ष े त ् रफल space
equals space 6 left parenthesis straight D right parenthesis squared space plus space 2 πr squared space
equals space 6 left parenthesis straight D right parenthesis to the power of 2 space plus space πr squared end exponent
equals space 6 left parenthesis 6 right parenthesis squared space plus space begin inline style 22 over 7 end style space straight x space begin inline style 7 over 4 end style space straight x space begin inline style 7 over 4 end style
space space space space space space space space space space space space space space space space space space space space space space space open curly brackets table attributes columnalign left end attributes row cell therefore space घन space क ी space द ि श ा space equals space 6 space cm end cell row cell ग ो ल ा र ् ध space क ी space त ् र ि ज ् य ा space equals begin inline style fraction numerator 3.5 over denominator 2 end fraction end style space equals space begin inline style 7 over 4 end style space cm end cell end table close
equals space 216 space plus space begin inline style 77 over 8 end style
equals space 216 space plus space 9.625 space
equals space 225.625 space cm squared space space space space space space space space space space space space space space space
91 Views

20.

यदि एक समान्तर श्रेढ़ी के प्रथम 7 पदों का योग 49 तथा प्रथम 17 पदों का योग 289 है, तो समान्तर श्रेढ़ी के प्रथम n पदों का योग ज्ञात कीजिए। 


म ा न ा space प ् रथम space पद space equals space straight a comma space स ा र ् व space अ ं तर space equals space straight d
द ि य ा space ह ै colon space straight S subscript 7 space equals 49 comma space straight S subscript 17 space equals space 289
straight S subscript straight n space end subscript equals space begin inline style straight n over 2 end style left square bracket 2 straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d right square bracket
straight S subscript 7 space equals space 7 over 2 left parenthesis 2 straight a space plus space 6 straight d right parenthesis space
49 space straight x space begin inline style 2 over 7 end style space equals space 2 straight a space plus space 6 straight d
2 straight a space plus space 6 straight d space equals space 14 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space................. left parenthesis straight i right parenthesis
straight S subscript 17 space equals space begin inline style 17 over 2 end style left parenthesis 2 straight a space plus space 16 straight d right parenthesis
289 space straight x space begin inline style 2 over 17 end style space equals space 2 straight a space plus space 16 straight d
2 straight a space plus space 16 straight d space equals space 34 space space space space space space space space space space space space space space space space space space space space space space space space space space................. left parenthesis ii right parenthesis
सम ी करण space left parenthesis straight i right parenthesis space straight व space left parenthesis ii right parenthesis space क ो space हल space करन े space पर comma

space space space space space 2 straight a space plus space space 6 straight d space equals space 14 space
minus 2 straight a space space plus-or-minus space 16 straight d space equals plus-or-minus 34
space space space space space space space space space space minus space 10 straight d space equals negative 20
minus negative negative negative negative negative negative negative negative negative negative
rightwards double arrow space space space space space space straight d space equals space 2
minus negative negative negative negative negative negative negative negative negative negative

समीकरण (i) में d का मान रखने पर,
2 straight a space plus space 6 left parenthesis 2 right parenthesis space equals space 14
2 straight a space equals space 14 space minus space 12 space
straight a space equals space begin inline style 2 over 2 end style space equals space 1
straight S subscript straight n space equals space begin inline style straight n over 2 end style left square bracket 2 straight a space plus space left parenthesis straight n space minus space 1 right parenthesis straight d right square bracket
space space space space space space equals space begin inline style straight n over 2 end style left square bracket 2 left parenthesis 1 right parenthesis space plus space left parenthesis straight n space minus space 1 right parenthesis 2 right square bracket space equals space begin inline style straight n over 2 end style left parenthesis 2 space plus space 2 straight n space minus space 2 right parenthesis
space space space space space space equals space begin inline style fraction numerator 2 straight n squared over denominator 2 end fraction end style space equals space straight n squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space स ि ध ् द space ह ु आ । space
221 Views