Category: 2nd PUC

Students can Download 2nd PUC Sanskrit Previous Year Question Paper June 2016, Karnataka 2nd PUC Sanskrit Model Question Papers with Answers help you to revise complete Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Sanskrit Previous Year Question Paper June 2016

अङ्काः 100
समय : 3 घण्टा 15 निमेषाः।

I. एकवाक्येन संस्कृतभाषया उत्तरं लिखत। (10 × 1 = 10)

प्रश्न 1.
कः भरतः इति उच्यते?
उत्तरम्
मनुः भरतः इति उच्यते ।

प्रश्न 2.
विक्रमादित्यः कथं देशान्तरं निर्गतः?
उत्तरम्
विक्रमादित्यः योगिवेषेण देशान्तरं निर्गतः ।

प्रश्न 3.
परिव्राजकस्य अवस्थां दृष्ट्वा कः जहासः?
उत्तरम्
परिव्राजकस्य अवस्थां दृष्ट्वा सकलो जनः जहास ।।

प्रश्न 4.
“विधिना दर्शितं प्रभुत्वम्’ इति का वदति?
उत्तरम्
“विधिना दर्शितं प्रभुत्वम्’ इति शकुन्तला वदति ।

प्रश्न 5.
बालिकायाः हस्तात् रोटिकाम् आकृष्य कः अधावत् ?
उत्तरम्
बालिकायाः हस्तात् रोटिकाम् आकृष्य अरण्यमार्जालः अधावत् ।

प्रश्न 6.
महाश्वेता स्नातुं कुत्र आगच्छति?
उत्तरम्
महाश्वेता स्नातुं अच्छोदसरः आगच्छति ।

प्रश्न 7.
‘सा’ शान्तिः कुत्र कार्य करोति?
उत्तरम्
‘सा’ शान्तिः उद्योगार्थं कार्यं करोति।

प्रश्न 8.
सुन्दरः किमर्थं नगरम् आगच्छति?
उत्तरम्
सुन्दरः गुरोः आशीर्वादः नगरम् आगच्छति ।

प्रश्न 9.
के धनमिच्छन्ति? उत्तरम् अधमाः धनमिच्छन्ति ।

प्रश्न 10.
कृष्णशास्त्रिणां माता का?
उत्तरम्
कृष्णशास्त्रिणां माता शङ्करम्मा

II. द्वित्रः वाक्यैः संस्कृतभाषया कन्नडभाषया आङ्ग्लभाषावा वा उत्तरं लिखत। (पञ्चनामेव) (5 × 2 = 10)

प्रश्न 11.
पञ्चद्वीपानां नामानि लिखत ।

प्रश्न 12.
नगरवासिभिः जनैः राक्षसः किम् उक्तः ?

प्रश्न 13.
प्रव्राजकः निजान् अनुचरान् किम् उवाच ?

प्रश्न 14.
भारतसर्वकारः राणाप्रतापाय कथं गौरवम् असूचयत् ?

प्रश्न 15.
सुन्दरः किमर्थं संरक्ष्यः ?

प्रश्न 16.
शास्त्रिणाम् औदार्यम् ।

प्रश्न 17.
कोषः किमर्थं संरक्ष्यः ?

III. पाठनाम उल्लिख्य श्लोकानाम् अनुवाद कन्नड-आङ्ग्लभाषया वा कुरुत। (त्रयाणाम्) (3 × 3 = 9)

प्रश्न 18.
ज्ञेयं तद्भारतं वर्ष सर्वकर्मफलप्रदम् ।
अत्र कर्माणि कुर्वन्ति त्रिविधानि तु नारद ॥

प्रश्न 19.
प्रविष्टो जातु भिक्षार्थमेकस्य वणिजो गृहे ।
स ददर्श शुभां कन्यां भिक्षामादाय निर्गताम् ।।

प्रश्न 20.
कामं प्रत्यादिष्टां स्मरामि न परिग्रहं मुनेस्तनयाम् ।
बलवत्तु दूयमानं प्रत्याययतीव मे हृदयम् ।।

प्रश्न 21.
धन्योऽस्मि गमनात्पूर्वमीदृशी सम्पदागता ।
विना तु गुरुशुश्रूषां नान्यद्भाव्यं परं मम ॥

प्रश्न 22.
कामक्रोधौ तथा लोभं स्वादं शृङ्गारकौतुके ।
अतिनिद्रातिसेवे च विद्यार्थी ह्यष्ट वर्जयेत् ॥

IV. पाठनाम उल्लिख्य कः कं प्रति अवदत् ? इति संस्कृतभाषया लिखत । (चतुर्णामेव) (4 × 2 = 8)

प्रश्न 23.
यं कमेवं मा भक्षय ।

प्रश्न 24.
अथ लोकानुग्रहाय कुशली काश्यपः।

प्रश्न 25.
कुत्रापि गत्वा धनं संग्रहिष्यामि ।

प्रश्न 26.
किं माम् अन्यथा सम्भावयसि ।

प्रश्न 27.
दृढीक्रियतां चेतः ।

प्रश्न 28.
सः यत्रकुत्रापि भवतु कालेज आनेतव्यः एव ।

V. दशवाक्यैः संस्कृतभाषया कन्नडभाषया आङ्ग्लभाषया वा उत्तरं लिखत । (पञ्चानामेव) (5 × 5 = 25)

प्रश्न 29.
विक्रमादित्यस्य परोपकारगुणं वर्णयत ।

प्रश्न 30.
प्रव्राजकः

प्रश्न 31.
शारिव-दुष्यन्तयोः संवादः ।

प्रश्न 32.
भामाशाहस्य राजभक्तिः ।

प्रश्न 33.
पुण्डरीकस्य जन्मवृत्तान्तं लिखत ।

प्रश्न 34.
कर्णशापप्रसङ्गः ।

प्रश्न 35.
सा शान्तिः ।

प्रश्न 36.
शास्त्रिणाम् शिष्यवात्सल्यम् ।

VI. मञ्जूषातः सूक्तपदं चित्वा रिक्तं स्थानं पूरयत। (4 × 1 = 4)

प्रश्न 37.
उत्तरं यत् समुद्रस्य …………. दक्षिणं च यत् । (हिमवत्)

प्रश्न 38.
अङ्गुलीयकशून्या मे ………… । (अङ्गुलिः)

प्रश्न 39.
…………. गन्धम् अभ्यजिध्रम् । (कुसुमः)

प्रश्न 40.
विद्या …………. रक्ष्यते । (योगेन)

(अङ्गुलिः, कुसुमः, हिमवत्, योगेन)

VII. संयोजयत। (4 × 1 = 4)

प्रश्न 41.
क – ख
(a) जाह्नवी – (i) सरस्वत्याः
(b) बिल्लाः – (ii) ज्ञानमवाप्नोति
(c) विलासमिव – (iii) गङ्गा
(d) श्रुत्वा – (iv) योद्धारः
उत्तरम्
(iii) गङ्गा
(iv) योद्धारः
(i) सरस्वत्याः
(ii) ज्ञानमवाप्नोति

VIII. रेखाङ्कितानि पदानि आश्रित्य प्रश्ननिर्माणं कुरुत । (त्रयाणामेव) (3 × 1 =3)

प्रश्न 42.
पुत्थलिका भोजराजं प्रति अब्रवीत् ।
उत्तरम्
कं ।

प्रश्न 43.
अनुचराः प्रव्राजकस्य वचनम् अपालयन् ।
उत्तरम्
कस्य ।

प्रश्न 44.
मेवाड इति स्थलं राजस्थाने प्रसिद्धमासीत् ।
उत्तरम्
कुत्र ।

प्रश्न 45.
शान्तिः धनं न स्वीचकार ।
उत्तरम्
किं ।

प्रश्न 46.
विमर्शकेन कृतीनामध्ययनं सम्यक् कर्तव्यम् ।
उत्तरम्
केन ।

IX. यथानिर्देशं कुरुत । (चतुर्णामेव) (4 × 3 = 12)

प्रश्न 47.
सन्धिं विभजत /योजयत। (त्रयाणामेव)
भीषणाकृतिः, सर्वोऽपि
ततः + ततः, तथा + एव, हि + अकाले
उत्तरम्
भीषण + आकृतिः, सर्वो + अपि, ततस्ततः, तथैव, ह्यकाले

प्रश्न 48.
विग्रहवाक्यं / समस्तपदं लिखत । (त्रयाणामेव)
पुराणभारतम्, राजपुत्रः
अरण्यस्य मार्जालः, मनुजः पशु इव, नराः एव विग्रहाः
उत्तरम्
पुराणेषु वर्णितं भारतम्, राज्ञः पुत्रः, अरण्यमार्जालः, मनुजपशुः, नरविग्रहाः

प्रश्न 49.
लिङ्ग-विभक्ति-वचनानि लिखत । (द्वयोरेव)
वाचा, वचनात्, भगवति, पादयोः ।
उत्तरम्
वाचा – स्त्रीलिङ्गः, तृतीया विभक्तिः, एकवचनम्
वचनात् – नपुंसकलिङ्गः, पञ्चमी विभक्तिः, एकवचनम्
भगवति – पुंलिङ्गः, सप्तमी विभक्तिः, एकवचनम्
पादयोः – पुंलिङ्गः, षष्ठी विभक्तिः, द्विवचनम्

प्रश्न 50.
लकार-पुरुष-वचनानि लिखत । (द्वयोरेव)
लभते, निबोधत, अपनेष्यामि, ददर्श
उत्तरम्
लभते – लट्, प्रथम, एक
निबोधत – लोट्, मध्यम, बहुवचनम्
अपनेष्यामि – लुट्, उत्तम, बहुवचनम्
श्रुणु – लोट्, मध्यम, एकवचनम्
ददर्श – लिट्, प्रथम, एक

प्रश्न 51.
पदपरिचयं कुरुत । (त्रयाणामेन)
त्यक्त्वा, स्मृतम्, प्रवेषयितुम्, निरीक्ष्य, पतन्
उत्तरम्
त्यक्त्वा – क्त्वान्तः
स्मृतम् – क्त प्रत्ययान्तः भूतकृदन्तः
प्रवेषयितुम् – तुमुन्नान्तः
निरीक्ष्य – ल्यबन्तः
पतन् – शतृ प्रत्ययान्तः वर्तमानः कृदन्तः ।

प्रश्न 52.
प्रयोगं परिवर्तयत ।
वित्तेन धर्मः रक्ष्यते ।
उत्तरम्
वित्तं धर्मं रक्षति ।

अथवा

नरैः कर्माणि क्रियन्ते ।
उत्तरम्
नराः कर्माणि कुर्वन्ति ।

प्रश्न 53.
अलङ्कारं सलक्षणं निर्दिशत ।
मध्ये तपोधनानां किसलयमिव – पाण्डुपत्राणाम् ।
अथवा
अज्ञातहृदयेष्वेव वैरी भवति सौहृदम् ।

प्रश्न 54.
कन्नड भाषया-आङ्ग्लभाषया वा अनुवदत।
क्रीडा मानवस्य नितराम् आवश्यकी । क्रीडया शक्तिः वर्धते । उल्लासः उत्पद्यते । आरोग्यं जायते। मनः विकसति। जीवनोत्साहः एधते । एकताभावः सम्पद्यते । मैत्री विस्तारं गच्छति, अथ खलु ज्येष्ठाः एक घण्टापर्यन्तं तुल्यसीलवयोभिः क्रीडितव्यमिति उपदिशन्ति ।

प्रश्न 55.
संस्कृतभाषया अनुवदत।

Once the sage Vishwamitra came to the court of the king Dasharatha. He requested to send the boys Rama and Lakshmana to protect his sacrifice. Dasharatha felt sad due to affection towards his children. Vasishtha, the preceptor of the family, advised Dasharatha to fulfil Vishwamitra’s request. Dasharatha sent Rama and Lakshmana for the welfare of the world.

प्रश्न 56.
इमं परिच्छेदं पठित्वा प्रश्नानाम् उत्तराणि लिखत । (5 × 1 = 5)
ध्रुवः उत्तानपादस्य पुत्रः। तस्य मातुः नाम सुनीति इति । राज्ञः प्रियपत्नी सुरूचिः। एकदा ध्रुवः पितुः उत्सङ्गे उपवेष्टुम् पितुः समीपं गतवान् । तदा सुरूचिः कुपिता अभवत् । दुःखेन ध्रुवः स्वमातरं अवदत् – “नारायणं पूजयित्वा ध्रुवं स्थानं सम्पादयिष्यामि” इति। नारदमहर्षेः उपदेशं अनुसृत्य तपश्चरति। तस्य तपसा तुष्टः नारायणः तस्मै ध्रुवपदं अनुगृहीतवान् ।
प्रश्नाः
1. ध्रुवः कस्य पुत्रः ?
उत्तरम्
उत्तानपादस्य

2. उत्तानपादस्य प्रिय पत्नी का?
उत्तरम्
सुरुचिः

3. ध्रुवः स्वमातरं किं अवदत् ?
उत्तरम्
‘नारायणं पूजयित्वा ध्रुवं स्थानं सम्पादयिष्यामि ।

4. ध्रुवः कस्य उपदेशं अनुसृत्य तपश्चरति ।
उत्तरम्
नारदमहर्षेः ।

5. तुष्टः नारायणः तस्मै किं अनुगृहीतवान् ?
उत्तरम्
ध्रुवपदं ।
अथवा धनं प्रेषयितुं पितरं पत्रं लिखत ।

Students can Download 2nd PUC Sanskrit Previous Year Question Paper March 2016, Karnataka 2nd PUC Sanskrit Model Question Papers with Answers help you to revise complete Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Sanskrit Previous Year Question Paper March 2016

अङ्काः 100
समय : 3 घण्टा 15 निमेषाः।

I. एकवाक्येन संस्कृतभाषया उत्तरं लिखत। (10 × 1 = 10)

प्रश्न 1. अद्यापि देवाः कुत्र जन्म इच्छन्ति?
उत्तरम्
भारतभूतले।

प्रश्न 2.
परदुःखेन के अत्यन्तदुःखिनः भवन्ति?
उत्तरम्
साधवः।

प्रश्न 3.
प्रव्राजकः वणिजः गृहं किमर्थमग्छत्?
उत्तरम्
भिक्षार्थं।

प्रश्न 4.
शारद्वतः कः?
उत्तरम्
कण्वशिष्यः।

प्रश्न 5.
राणाप्रतापः कस्मिन् ग्रामे अजायत?
उत्तरम् कुम्बल।

प्रश्न 6.
महाश्वेता के प्रति स्ववृत्तान्तं कथयति?
उत्तरम्
पुण्डरीकं।

प्रश्न 7.
‘सा’ शान्तिः कुत्र कार्य करोति?
उत्तरम्
वस्त्रापणे।

प्रश्न 8.
परशुरामः कर्णं किमस्त्रं उपदिशति?
उत्तरम्
भार्गवास्त्रं।

प्रश्न 9.
कामधेनुगुणा का?
उत्तरम्
विद्या।

प्रश्न 10.
कृष्णशास्त्रिणां अध्यापकेषु एकस्य नाम लिखत।
उत्तरम्
बि.एम.श्री।

II. द्वित्रः वाक्यैः संस्कृतभाषया कन्नडभाषया आङ्ग्लभाषया वा उत्तरं लिखत। (पञ्चनामेव) (5 × 2 = 10)

प्रश्न 11.
भारतवर्ष इति किमर्थं कथ्यते?

प्रश्न 12.
विस्मितः राक्षसः विक्रमादित्यं किं पृच्छति?

प्रश्न 13.
राजपुत्रः मञ्जूषां प्राप्य किं करोति?

प्रश्न 14.
राणाप्रतापस्य प्रतिज्ञा का?

प्रश्न 15.
पुण्डरीकस्य अनुरागं महाश्वेता कथं जानाति?

प्रश्न 16.
कर्णं प्रति परशुरामोपदेशं लिखत।

प्रश्न 17.
शान्त्यः सौन्दर्यं वर्णयत।

III. पाठनाम उल्लिख्य श्लोकानाम् अनुवाद कन्नड-आङ्ग्लभाषया वा कुरुत। (त्रयाणाम्) (3 × 3 = 9)

प्रश्न 18.
अयं तु नवमस्तेषां द्वीपस्सागरसंवृत्तः ।
योजनानां सहस्रं तु द्वीपोऽयं दक्षिणोत्तरम् ॥

प्रश्न 19.
तदेतां वीक्ष्य दुःखं मे जातो भक्तो हि मे भवान् ।
तेनैवमुक्तवानस्मि त्यक्त्वा मौनं भवत्कृते ॥

प्रश्न 20.
कुतो धर्मक्रियाविघ्नः सतां रक्षितरि त्वयि ।
तमस्तपति धर्माशौ कथमाविर्भविष्यति ॥

प्रश्न 21.
अस्त्रप्रयोगवेलायां यदा ते सूतभावना ।
तदा सर्वाणि चास्त्राणि भवेयुनिष्फलानि. हि ॥

प्रश्न 22.
धनहीनो न हीनश्च धनिकः स सुनिश्चयः ।
विद्यारलेन यो हीनः स हीनः सर्ववस्तुषु ॥

IV. पाठनाम उल्लिख्य कः कं प्रति अवदत् इति संस्कृतभाषया लिखत । (चतुर्णामेव) (4 × 2 = 8)

प्रश्न 23.
अद्यप्रभृति मनुष्यभक्षणं परित्यज ।

प्रश्न 24.
भवतु, अनिर्वर्णनीयं परकलत्रम् ।

प्रश्न 25.
नैतद्धनं स्वीकर्तुं अहमिच्छामि ।

प्रश्न 26.
पुण्डरीक इति नाम चक्रे ।

प्रश्न 27.
अहो गुरुकुलवैभवम् ।

प्रश्न 28.
परीक्षार्थं इतोऽपि समयावकाशः विद्यते ।

V. दशवाक्यैः संस्कृतभाषया कन्नडभाषया आङ्ग्लभाषया वा उत्तरं लिखत । (पञ्चानामेव) (5 × 5 = 25)

प्रश्न 29.
विक्रमादित्यस्य परोपकारगुणं वर्णयत ।

प्रश्न 30.
प्रव्राजकः ।

प्रश्न 31.
शार्ङ्गरवदुष्यन्तयोः संवादः ।

प्रश्न 32.
कीदृशं मुनिकुमारकं अपश्यम् इति महाश्वेता वर्णयति?

प्रश्न 33.
कर्णशापप्रसङ्गः ।

प्रश्न 34.
भामाशाहस्य राजभक्तिः ।

प्रश्न 35.
शान्तिदर्शनात् तरुणस्य चिन्तासन्ततिः ।

प्रश्न 36.
विद्यार्थिनां कृते शास्त्रिणां सन्देशः ।

VI. मञ्जूषातः सूक्तपदं चित्वा रिक्तं स्थानं पूरयत। (4 × 1 = 4)

प्रश्न 37.
उत्तरं यत् समुद्रस्य …………. दक्षिणं च यत् । (हिमवत्)

प्रश्न 38.
वैरिणां …………. समुद्रोपमं आसीत् । (सैन्यं)

प्रश्न 39.
भगवतः अशेषत्रिभुवनसुन्दरं ………. आसीत् । (रूपं)

प्रश्न 40.
अङ्गुलीयकशून्या मे …………. । (अङ्गुलिः)

(रूपं, अङ्गुलिः, सैन्यं, हिमवत्)

VII. संयोजयत। (1 × 4 = 4)

प्रश्न 41.
क – ख
(a) जाह्नवी – (i) ऋषिकुमारः
(b) पौरवः – (ii) पापम्
(c) पुण्डरीकः – (iii) गङ्गा
(d) किल्बिषम् – (iv) दुष्यन्तः
उत्तरम्
(iii) गङ्गा
(iv) दुष्यन्तः
(i) ऋषिकुमारः
(ii) पापम्

VIII. रेखाङ्कितानि पदानि आश्रित्य प्रश्ननिर्माणं कुरुत। (त्रयाणामेव) (3 × 1 = 3)

प्रश्न 42.
पुत्थलिका भोजराजं प्रति अब्रवीत्।
उत्तरम्
पुत्थलिका कं प्रति अब्रवीत् ?

प्रश्न 43.
अनुचराः प्रव्राजकस्य वचनं अपालयन् ।
उत्तरम्
अनुचराः कस्य वचनं अपालयन् ?

प्रश्न 44.
मेवाड इति स्थलं राजस्थाने प्रसिद्धमासीत् ।
उत्तरम्
मेवाड इति स्थलं कुत्र प्रसिद्धमासीत् ?

प्रश्न 45.
शान्तिः धनं न स्वीचकार ।
उत्तरम्
शान्तिः किं न स्वीचकार ?

प्रश्न 46.
अम्बया सह स्नातुं अहं अभ्यागमम् ।
उत्तरम्
कया सह स्नातुं अहं अभ्यागमम् ?

IX. यथानिर्देशं कुरुत । (चतुर्णामेव) (4 × 3 = 12)

प्रश्न 47.
सन्धिं विभजत /योजयत। (त्रयाणामेव)
तथेति, प्रागेव, तावच्च
ततः + ततः, अस्त्रः + अभ्यासः
उत्तरम्
तथा + इति, प्राक् + एव, तावत् + च
ततस्ततः, अस्त्राभ्यासः।

प्रश्न 48.
विग्रहवाक्यं / समस्तपदं लिखत । (त्रयाणामेव)
कामक्रोधौ, राजपुत्रः, योगिन् वेषम्, स्वस्य जन्मस्थानम्, कमनीयं रूपं यस्याः सा।
उत्तरम्
कामक्रोधौ – कामः च क्रोधः च
राजपुत्रः – राज्ञः पुत्रः
योगिन् वेषम् – योगिवेषम्
स्वस्य जन्मस्थानम् – स्वजन्मस्थानम्
कमनीयं रूपं यस्याः सा – कमनीयरूपा

प्रश्न 49.
लिङ्ग-विभक्ति-वचनानि लिखत । (द्वयोरेव)
पौरवैः, रोटिकाम्, सलिलम्, पादयोः ।
उत्तरम्
पौरवैः – पुल्लिङ्गः, तृतीया विभक्तिः , बहुवचनम्
रोटिकाम् – स्त्रीलिङ्गः, द्वितीया विभक्तिः, एकवचनम्
सलिलम् – नपुंसकलिङ्गः, प्रथमा विभक्तिः, एकवचनम्
पादयोः – पुल्लिङ्गः, सप्तमी विभक्तिः, द्विवचनम्

प्रश्न 50.
लकार-पुरुष-वचनानि लिखत । (द्वयोरेव)
भवन्तु, आसीत्, भवेत्, वदन्ति
उत्तरम्
भवन्तु – लोट्लकारः, प्रथम पुरुषः, बहुवचनम्
आसीत् – लङ् लकारः, प्रथम पुरुषः, एकवचनम्
भवेत् – विधिलिङ्, प्रथम पुरुषः, एकवचनम्
वदन्ति – लट् लकारः, प्रथम पुरुषः, बहुवचनम्

प्रश्न 51.
पदपरिचयं कुरुत । (त्रयाणामेव)
स्नातुम्, कृत्वा, निर्गतः, स्मरन्, आदाय ।
उत्तरम्
तुमुन्नान्ताव्ययम्, क्त्वान्ताव्ययम्, क्त प्रत्ययान्तः भूतकृदन्तः, शतृ प्रत्ययान्तः वर्तमान कृदन्तः, ल्यबन्त।

प्रश्न 52.
प्रयोगं परिवर्तयत ।
राणाप्रतापः प्रतिज्ञां स्वीकृतवान् ।
उत्तरम्
राणाप्रतापेन प्रतिज्ञा स्वीकृता ।

अथवा

स्त्रिया गृहं रक्ष्यते ।
उत्तरम्
स्त्री गृहं रक्षति ।

प्रश्न 53.
विद्यारत्नेन यो हीनः स हीनः सर्ववस्तुषु ।
वा
विलासमिव सरस्वत्याः मुनिकुमारकं अपश्यम् ।

प्रश्न 54.
कन्नड भाषया-आङ्ग्लभाषया वा अनुवदत।

पुस्तकानि सम्पर्कसाधनानि वर्तन्ते । बहून् विषयान् एकत्र संगृह्य सुचारुरूपेण निवेदयितुं पुस्तकमेव समर्थं माध्यमम् । पुस्तकामाध्यमेन शाश्वतरूपेण विचारविनिमयं कर्तुं शक्यते । यः पुस्तकानि अधिकं पठति सः अतीव बुद्धिमान् भवति। पूर्व, जनाः तालपत्रेषु भूर्जपत्रेषु च लिखन्ति स्म । मुद्रणयन्त्रस्य अन्वेषणानन्तरं पुस्तकप्रचारः अधिकः सञ्जातः।

प्रश्न 55.
संस्कृतभाषया अनुवदत।

Karthikeya the destroyer of demon Taraka, is the younger son of Lord Shiva. Ganesha the remover of obstacles is his elder son: Kubera the lord of wealth is his friend. Parvathi the mother of universe is his wife. Parvatharaja the king of mountains is his father-in-law. Nandi is his vehicle. Veerabhadra is his chief of army. In spite of all this, Lord Shiva begs. What a destiny!

प्रश्न 56.
इमं परिच्छेदं पठित्वा प्रश्नानाम् उत्तराणि लिखत । (5 × 1 = 5)
पुरा अयोध्यायां सगरो नाम राजा आसीत् । एकदा सः अश्वमेधयागं कर्तुं ऐच्छत्। तदर्थं सः यज्ञाश्वं अमुञ्चत्। इमं विषयं ज्ञात्वा, देवेन्द्रः यज्ञाश्वं अपहृतवान् । तं पातालं नीत्वा तत्र तपः कुर्वतः कपिलमुनेः समीपे स्थापितवान् । यज्ञाश्वं अन्विषन्तः सगरपुत्राः भूमिं खनन्तः पातालं अगच्छन् । तत्र यज्ञाश्वं दृष्ट्वा, कपिल एव चोर इति मत्वा, ते तं महामुनिः अवाच्यशब्दैः अनिन्दन् । तदा कुपितो मुनिः नेत्रे उन्मील्य तान् सर्वान् भस्मीकृतवान्। अतः ज्येष्ठाः वदन्ति ‘अविचार्य किमपि कर्म न करणीयम्’ इति।
प्रश्नाः
1. सगरो नाम राजा कुत्र आसीत् ?
उत्तरम्
अयोध्यायां।

2. राजा सगरः किं कर्तुं ऐच्छत्?
उत्तरम्
अश्वमेधयागं

3. सगरस्य यज्ञाश्वं कः अपहृतवान्?
उत्तरम्
देवेन्द्रः

4. कुपितो मुनिः किं अकरोत्?
उत्तरम्
नेत्रे उन्मील्य तान् सर्वान् भस्मीकृतवान्।

5. अतः ज्येष्ठाः किं वदन्ति?
उत्तरम्
‘अविचार्य किमपि कर्म न करणीयम्’।
अथवा
युष्माकं विद्यालयस्य वार्षिकोत्सवाथै आगन्तुं मित्राय पत्रमेकं लिखत।

You can Download Chapter 1 Accounting for Partnership : Basic Concepts Questions and Answers, Notes, 2nd PUC Accountancy Question Bank with Answers Karnataka State Board Solutions help you to revise complete Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Accountancy Question Bank Chapter 1 Accounting for Partnership : Basic Concepts

Students can Download Maths Chapter 13 Probability Ex 13.4 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Miscellaneous Exercise

Question 1.
A family has two children. Find the probability that both are boys, if it is known that
(i) At least one of them is boy
(ii) Elder child is a boy (CBSE – 2010)
Answer:
(i) Let A : At least one of them is a boy
{BB,BG,GB,GG}

Question 2.
A bag contains 4 balls
Two balls are drawn at random and are found to be White. What is the probability that all Balls are white.
Answer:
E₁: 2 balls are white, 2- are non white
E₂: 3 balls are white, 1 is non white
E₃: 4 balls are white
A : Two balls are white

Question 3.
Given three identical boxes I, II and III each containing two coins. In box I, both coins are gold, In box II both are silver and in box III there is one gold and one silver coin. A person choose a box and takes out a coin. If the coin is gold, what is the probability that the other coin is also gold  (CBSE – 2011)
Answer:
E₁: Both gold coin
E₂: Both silver coin
E₃: One gold and one silver

Question 4.
Two cards are drawn without replacement from a well shuffled pack of cards. Find the mean and variance of the number of Red cards.  (CBSE – 2012)
Answer:
Let X denote the no. of Red cards

Questions from Competitive Exams

Question 1.
A die is thrown. Let A be the event that the number obtained: is greater than 3. B be the event that the number obtained is less than 5. Find P(A∪B)
Answer:

Question 2.
It is given that, the events A and B are such that
\(\mathbf{P}(\mathbf{A})=\frac{1}{4}, \mathbf{P}(\mathbf{A} | \mathbf{B})=\frac{1}{2}, \mathbf{P}(\mathbf{B} | \mathbf{A})=\frac{2}{3}, \text { Find } P(B)\) [AIEEE – 2008]
Answer:

Question 3.
\(\mathbf{P}(\mathbf{A})=\frac{1}{4}, \mathbf{P}(\mathbf{B})=\frac{1}{2}, \mathbf{P}(\mathbf{A} \cap \mathbf{B})=\frac{1}{8}\),find P(not A, not B) (UPSEE -2008)
Answer:

Question 4.
[Latex]\mathbf{P}(\mathbf{A})=\frac{1}{2}, \mathbf{P}(\mathbf{B})=\frac{7}{12}, \mathbf{P}(\text { not } \mathbf{A} \text { or not } \mathbf{B})=\frac{1}{4}[/latex]
State whether A and B are independent. [AIEEE]
Answer:

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Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.5

2nd PUC Maths Probability NCERT Text Book Questions and Answers Ex 13.5

Ex 13.5 Class 12 Maths Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Solution:
There are 3 odd numbers on a die
∴ Probability of getting an odd number on a die = \(\frac { 3 }{ 6 }\) = \(\frac { 1 }{ 2 }\)

Ex 13.5 Class 12 Maths Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item ?
Solution:
Probability of getting one defective item = 5%
= \(\frac { 5 }{ 100 }\)
= \(\frac { 1 }{ 20 }\)
Probability of getting a good item = \(1-\frac { 1 }{ 20 }\) = \(\frac { 19 }{ 20 }\)
A sample of 10 item include not more than one defective item.
=> sample contains at most (me defective item Its probability = P (0) + P (1)

Ex 13.5 Class 12 Maths Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Solution:
n(S) = 36, A = {11,22,33,44,55,66}

Ex 13.5 Class 12 Maths Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Solution:
S = {52 cards}, n (S) = 52
Let A denotes the favourable events
A= {13 spade}, n(A)= 13

Ex 13.5 Class 12 Maths Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Solution:
Probability that a bulb gets fuse after 150 days of its use = 0.05
Probability that the bulb will not fuse after 150 days of its use = 1 – 0.05 = 0.95
(i) Probability that no bulb will fuse after 150

Ex 13.5 Class 12 Maths Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:
S = {0,1,2,3,4,5,6,7,8,9},n(S) = 10
Let A represents that the ball is marked with the digit 0.
A = {0}, n(A) = 1

Ex 13.5 Class 12 Maths Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true,’ if it falls tails, he answers “ false’. Find the probability that he answers at least 12 questions correctly.
Solution:
Probability that student answers a question true = \(\frac { 1 }{ 2 }\)
i.e., when a coin is thrown, probability that a head is obtained = \(\frac { 1 }{ 2 }\)
Probability that his answer is false = \(1-\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 2 }\)
Probability that his answer at least 12 questions correctly = P (12) + P (13) + P (14) +…….. P (20)

Ex 13.5 Class 12 Maths Question 8
Suppose X has a binomial distribution \(B\left( 6,\frac { 1 }{ 2 } \right) \). Show that X = 3 is the most likely outcome.
(Hint: P (X = 3) is the maximum among all P (Xi), xi. = 0,1,2,3,4,5,6)
Solution:
\({ \left( \frac { 1 }{ 2 } +\frac { 1 }{ 2 } \right) }^{ 6 } \)

Ex 13.5 Class 12 Maths Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Solution:
P = \(\frac { 1 }{ 3 }\). q = 1 – P = \(1-\frac { 1 }{ 3 }\) = \(\frac { 2 }{ 3 }\)

Ex 13.5 Class 12 Maths Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \(\frac { 1 }{ 100 }\) . What is the probability that he will win a prize?
(a) at least once,
(b) exactly once,
(c) at least twice?
Solution:
Probability that the person wins the prize = \(\frac { 1 }{ 100 }\)
Probability of losing = \(1-\frac { 1 }{ 100 }\) = \(\frac { 99 }{ 100 }\)
(a) Probability that he loses in all the loteries

Ex 13.5 Class 12 Maths Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Solution:
S = {1,2,3,4,5,6},n(S) = 6
A = {5} => n(A) = 1

Ex 13.5 Class 12 Maths Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:
When a die is thrown,
Probabiltiy of getting a six = \(\frac { 1 }{ 6 }\)
Probabiltiy of not getting a six = \(1-\frac { 1 }{ 6 }\) = \(\frac { 5 }{ 6 }\)
Probabiltiy of getting at most 2 sixes in 6 throws of a single die = P (0) + P (1) + P (2)

Ex 13.5 Class 12 Maths Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Solution:
p = \(\frac { 1 }{ 100 }\) = \(\frac { 1 }{ 10 }\)
q = 1 – p = \(1-\frac { 1 }{ 10 }\) = \(\frac { 9 }{ 10 }\)

Ex 13.5 Class 12 Maths Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) \({ 10 }^{ -1 }\)
(b) \({ \left( \frac { 1 }{ 2 } \right) }^{ 5 }\)
(c) \({ \left( \frac { 9 }{ 10 } \right) }^{ 5 }\)
(d) \(\frac { 9 }{ 10 }\)
Solution:
p = \(\frac { 1 }{ 10 }\)
q = \(\frac { 9 }{ 10 }\) n = 5, r = 0, P(X=0) = \({ \left( \frac { 9 }{ 10 } \right) }^{ 5 }\)
Option (c) is correct

Ex 13.5 Class 12 Maths Question 15.
The probability that a student is not a swimmer is \(\frac { 1 }{ 5 }\). Then the probability that out of five students, four are swimmers is:

Solution:
p = \(\frac { 4 }{ 5 }\) , q = \(\frac { 1 }{ 5 }\) , n = 5,r = 4

Option (a) is true

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Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.4

2nd PUC Maths Probability NCERT Text Book Questions and Answers Ex 13.4

Question 1.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
(i)

x	0	1	2
P(X)	0.4	0.4	0.2

Answer:

X	0	1	2	T
P(X)	4	4	2	1

(ii)

X	0	1	2	3	4
P(X)	0.1	0.5	0.2	-0.1	0.3

Answer:

X	0	1	2	3	T
P(X)	.1	.5	.2	-.1	1

It is not P.d as P(X) < 0 when x = 3

(iii)

Y	-1	0	1
P(Y)	0.6	0.1	0.2

Answer:

Y	-1	0	1	T
P(Y)	.6	.1	.2	.9

It is not P.d as ΣP(X) = .9 ≠1

(iv)

Z	3	3	1	0	-1
P(Z)	0.3	0.2	0.4	0.1	0.05

Answer:

Z	3	3	1	0	T
P(Z)	.3	.2	.4	.1	1.05

It is not a P.d as ΣP(X) = 1.05 > 1

Question 2.
An urn contains 5 red and 2 black balls.Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X ? Is X a random variable ?
Answer:
X can take the values 0, 1, 2 It is a Random variable because the experiment is a random experiment.

Question 3.
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Answer:
When m = 6, n = 0  ⇒ |m-n| = 6
m = 5,n = 1                  |m-n| = 4
m = 4, n = 2                 |m-nl=2
m = 3,n = 3                  |m-n| = 0
m = 2, n = 4                 |m-n|=2
m = 1,n = 5                  |m-n| = 4
m = 0, n = 6                 |m-n|=6
∴ hence X can take values 0, 2, 4, 6.

Question 4.
Find the probability distribution of
(i) number of heads in two tosses of a coin.
(ii) number of tails in the simultaneous tosses of three coins.
(iii) number of heads in four tosses of a coin.
Answer:
(i) When a coin is tossed twice, the number of heads may be 0, 1,2
Sample space {HH, HT, TH, TT}

(ii) When a coin is tossed thrice, the number of heads are 0, 1, 2, 3
Sample space {HHH, HHT, HTH, HTT, ITT, TTH, THT, THH

(iii) When coin is tossed four times, the number of heads be 0, 1, 2, 3,4 Sample space {HHHH, HHHT, HHTH, HHTT, HTHT, HTHH, HTTH, TTTT, TTTH, HTTT, TTHT, TTHH, THTH, THTT, THHT, THHH]

Question 5.
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
Ans:
(i) Let the random variable be X no of successes then X can take the value 0,1,2.

(ii) Let X be random variable, then X can take values
O (no sixes) & 1 (at least 1 six)
P (x = 0) = P (no six)
= [1-P (getting a six)] x [1 – P (getting a six on single throw]

Question 6.
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with’ replacement. Find the probability distribution of the number of defective bulbs.
Answer:
Let X denote the no. of defective bulbs P denote the probability of obtaining defective bulb

Question 7.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
Answer:
Let X denote the number of tails in two tosses of a coin
∴ X can assain the values 0, 1,2
Let P be the probability getting head, q the tail

Question 8.
A random variable X has the following probability distribution:

X	0	1	2	3	4	5	6	7
P(X)	0	k	2k	2k	3k	k2	2k2	7k2+k

Determination
(i) K
(ii) P(X<3)
(iii) P(X>6)
(iv) P(0<X<3)
Answer:

Question 9.
The random variable X has a probability distribution P(X) of the following form, where k is some number:
\(P(X)=\left\{\begin{array}{ll}{k,} & {\text { if } \quad x=0} \\{2 k,} & {\text { if } x=1} \\{3 k,} & {\text { if } x=2} \\{0,} & {\text { otherwise }}\end{array}\right.\)
(a) Determine the value of k
(b) Find P (X < 2), P (X≤2), P (X≥2)
Answer:
(a) In P.d. Σ XP(X)= 1
6k = 1 ⇒ \(k=\frac{1}{6}\)

Question 10.
Find the mean number of heads in three tosses of a fair coin.
Answer:
If three coins are tossed, Let X denote the number of heads.
X can take the values 0, 1, 2, 3

Question 11.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Answer:
Let X denote the number of sires. Then X can take values 0,1,2

Question 12.
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
Ans:
X can take the values 2, 3, 4, 5, 6

Question 13.
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Ans:
Let X denote the sum of the numbers obtained
∴  X can take the values 2, 3,4, 5,6,7, 8,9, 10, 11, 12

Question 14.
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Ans:
Let X denote the age of the students.
Then X can take the values 14,15,16,17, 18,19,20, 21

Question 15.
In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X).
Answer:

Choose the correct answer in each of the following:

Question 16.
The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
(A) 1
(B) 2
(C) 5
(D) 1
Answer:

Question 17.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained . Then the value of E(X) is
(A) \(\frac{37}{221}\)
(B) \(\frac{5}{13}\)
(C) \(\frac{1}{13}\)
(D) \(\frac{2}{13}\)
Answer:

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Karnataka 2nd PUC Basic Maths Previous Year Question Paper June 2017

Time: 3.15 hours
Max Marks: 100

Instructions:

• The question paper has 5 parts A, B, C, D and E. Answer all the pats.
• Part-A carries 10 marks), Parts – B carries 20 marks, Part – C carries 3 marks, Part – D carries 30 marks and Part – E carries 1 marks.
• Write the question number properly as indicated in the question paper.

Part – A

Answer all the ten questions: (10 × 1 = 10)

Question 1.
Evaluate \(\left| \begin{matrix} 3200 & 3201 \\ 3202 & 3203 \end{matrix} \right| \)
Answer:

Question 2.
In how many ways can 10 people seated around a table ?
Answer:
(10 – 1)! = 9!

Question 3.
Write the verbal form of the compound proposition pvq where
P:x is an integer; Q: 5 is odd number.
Answer:
PVQ = x is an integer or 5 is odd number.

Question 4.
Find the triplicate ration of 3 : 5.
Answer:
3³ : 5³ = 27:125

Question 5.
A bill was drawn for 3 months was legally due on 18-08-2012. Find the date of drawing of the bill.
Answer:
Drawn date = L.D.D – B.P – 3 days = 15 – 5 – 12

Question 6.
Find the value of cos³ 20° – 3cos 20°.
Answer:
Cos 3(20) = cos60 = \(\frac { 1 }{ 2 }\)

Question 7.
If the length of the latus rectum of the parabola x² = 4k y is 8. Find the value of k.
Answer:
4k = 8 ⇒ k = 2

Question 8.
Evaluate line \(\lim _{x \rightarrow 3}\left(\frac{x^{2}-x}{x-2}\right)\)
Answer:
\(=\frac{3^{2}-3}{3-2}=\frac{9-3}{1}=6\)

Question 9.
Differentiate : 5e^x – log_e x – 3√x w.r.t.x.
Answer:
\(\frac{d y}{d x}=5 e^{x}-\frac{1}{x}-\frac{3}{2 \sqrt{x}}\)

Question 10.
Evaluate ∫sec² (x – 5)dx.
Answer:
Tan (x – 5)

Part – B

Answer any ten questions. (10 × 2 = 20)

Question 11.
If A = \(\left[ \begin{matrix} 2 \\ -1 \\ 3 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 1 & 4 & 2 \end{matrix} \right] \) find AB and BA.
Answer:

Question 12.
In a party each person shakes hands with everyone else. If there are 25 members in the party, calculate the number the number of hand shakes.
Answer:
\(25 C_{2}=\frac{25 \times 24}{2 \times 1}=25 \times 12=300\)

Question 13.
A bag contains 6 white beads and 4 red beads. A bead is drawn at random, what is the probability that the bead drawn is white.
Answer:
P(beaddrawniswhite) = \(\frac{10 C_{1}}{6 C_{1}} \frac{10}{6}=\frac{5}{3}\)

Question 14.
If p,q, r are the propositions with truth values F, T and F repectively. Then find the truth value of the ; compound proposition p → (q → r).
Answer:
P → (q → r)
F → (T → F)
F → F = T

Question 15.
If a : b = 4 : 5. Find \(\frac{3 a+2 b}{3 a-2 b}\)
Answer:

Question 16.
Banker’s gain on a bill due after 6 months at 4% p.a. is 24. Find true discount and Banker’s discount,
Answer:
Given B.G = 24. T = 6 months, = \(\frac { 1 }{ 2 }\) yr, r = 4% = 0.0
BG = TD + r
24 = TD × \(\frac { 1 }{ 2 }\) × 0.04

TD = 1200
BG = BD – TD ⇒ BD = BG + TD = 24 + 1200 = 1224

Question 17.
Prove that \(\frac{\sin 3 A}{1+2 \cos 2 A}\) = sin A.
Answer:

= Sin A = R.H.S

Question 18.
If tan A = \(\frac { 5 }{ 6 }\) tan B = \(\frac { 1 }{ 11 }\) Show that A + B = \(\frac{\pi}{4}\)
Answer:
Consider tan (A+B) = \(\frac{\tan A+\tan B}{1+\tan A \cdot \tan B}\)

Tan (A + B) = 1 ⇒ A + B = \(\frac{\pi}{4}\)

Question 19.
Find the equation of the circle whose centre is at (4,-2) and passing through the origin (0,0)
Answer:
C = (4,-2) & r = \(\sqrt{4^{2}+(-2)^{2}}=\sqrt{16+4}=\sqrt{20}\)
Equation of the circle (x + 4)² + (y + 2)² = (√ 20)
⇒ x² + 16 – 8x + y² + y + 4y – 20 = 0
⇒ x² + y² – 8x + 4y = 0

Question 20.

Answer:
\(\lim _{x \rightarrow 0}\) f(x) = \(\lim _{x \rightarrow 0}\) (1 + 3x)^1/3 = e³ = f(0)
∴ f(x) is continuous at x = 0

Question 21.
If siny = x sin(a+y), prove that \(\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}\)
Answer:

Question 22.
If s = at³ + bt, find a and b given that when t = 3, velocity is 0 and the acceleration is 14 units.
Answer:
If s = at³ + bt velocity = \(\frac{\mathrm{ds}}{\mathrm{dt}}\) = 3at² + b = 0
Acceleration = \(\frac{\mathrm{dv}}{\mathrm{dt}}\) = 6at = 14
At t = 3sec, 6a × 3 = 14 ⇒ a = \(\frac{14}{18}=\frac{7}{9}\)

Question 23.
Evaluate \(\int \frac{3^{x} \log 3}{\left(3^{x}+1\right)} d x\)
Answer:
= log(3^x + 1) + C ∴ \(\int \frac{f^{1}(x)}{f(x)} d x\) = log(3^x +1)

Question 24.
Evaluate \(\int_{1}^{2} \frac{1}{(2 x+3)} d x\)
Answer:

Part – C

Answer any ten questions: (10 × 3 = 30)

Question 25.
If 2A + B = \(\left[ \begin{matrix} 3 & -1 \\ -2 & 5 \end{matrix} \right] \) and A – 2B = \(\left[ \begin{matrix} 4 & 2 \\ -1 & 5 \end{matrix} \right] \) then find A and B.

Question 26.
Solve \(\left| \begin{matrix} 2+x & 3 & -4 \\ 2 & 3+x & -4 \\ 2 & 3 & -4+x \end{matrix} \right| =0\)
Answer:

Question 27.
Find the number of permutations of the letters of the word ‘ENGINEERING’ How many of these
i. Begin with E and end with E
ii. Have all the 3E’s together?
Answer:
Total-11, N – 3, G – 2, 1 – 2, E = 3
Total permutations \(\frac{11 !}{3 ! \times 3 ! \times 2 ! \times 2 !}\)=
I. Number of permutations which begin with E & end with E is = \(\frac{9 !}{3 ! \times 2 ! \times 2 !}\)
II. Number of Permutations in Which all 3E’s are together (Total 11 – 3 + 1 = 9′) = \(\frac{9 !}{3 ! \times 2 ! \times 2 !}\)

Question 28.
A die is thrown, If E is the event “the number appearing is a multiple of 3” and F be the event “the number appearing is even”. Then find whether E are Fare independent.
Answer:
S = {1,2,3,4,5,6}
E = {3,6), F = {2,4,6}
N(S) = 6, n(E) = 2 n(F) = 3
P(E) = \(\frac{2}{5}=\frac{1}{3}\) P(E) = \(\frac{3}{6}=\frac{1}{3}\)
They are not independent

Question 29.
Two tap can separately fill a tank in 12 minutes and 15 minutes respectively. The tank when full can be emptied by a drain pipe in 20 minutes. When the tank was empty all the three were opened simultaneously in what time will the tank be filled up?
Answer:
A fills tank in 12 mins → In 1 min A will fill 1/12 of tank.
B fills tank in 15 mins → In 1 min B will fill 1/15 of tank.
Lets say C can alone empty the tank in x mins → In 1 min C will empty 1/x of tank.
According to question, if all taps are opened tank will fill in 20 mins.
In each min 1/20 tank is filled which is due to A, B and C. Now A and B fill the tank will C empty the tank.

1/12 + 1/15 – 1/x = 1/20

1/12 + 1/15 – 1/20 = 1/x

(5 + 4 – 3)/60 = 1/x

6/60 = 1/x

x = 10

Question 30.
Bharath bought a shirt for ₹ 336 including 12% sales tax and a neck tie for 110 including 10% sales tax. Find the printed price of shirt and neck tie together.
Answer:
Given S.P = ₹ 336, S.P (neck tie) = 110/
Let C.P of shirt = x, & C.P of neck tie = y
Here, setting price of shirt = C.P + 12% of C.P
336 = x + \(\frac{12 x}{100}\)
336 = \(=\frac{112 x}{100}\) ⇒ x = \(\frac{33600}{112}\)
Selling price of neck tie = C.P + 10% of C.P
110 = y + \(\frac{10 y}{100}\)
110 = \(\frac{10 y}{100}\) ⇒ \(\frac{1100}{11}\) = 100
∴ Printed price of shirt & neck tie = 300 + 100 = ₹ 400

Question 31.
A Banker pays ₹ 4,250 on a bill of ₹ 5,000. 146 days before the legally due date. Find the rate of discount charged by the Banker.
Answer:
Given F = ₹ 5000, Discounted Value = ₹ 4,520
t = 146 days = \(\frac{146}{365}\) year,r = ?
BD = F – Discounted value = 5000 – 4520 = ₹ 480
BD = F/r ⇒ 480 = \(\frac{5000 \times 146 \times r}{365}\)
∴ r = \(\frac{480 \times 365}{5000 \times 146}\) = 0.24 × 100=24

Question 32.
Rakshith decides to invest in TCS shares which are selling at 2020 per share. How many money is required to purchase 10 shares if the brokerage is 0.5%?
Answer:
Selling price of 10 shares at 2020 per share = 20,200
0.5 Brokerage = \(\frac{0.5}{100}\) × 20,200 = ₹ 101
100 Amount required to purchase = 20,200 + 101 = 20,301

Question 33.
Find the equation of the parabola if its focus is (4,0) and directrix is x = -4. Also find the equation of the tangent at the vertex.
Answer:
Given Directrix As x = -4 & Focus = (4,0) ⇒ a = 4
The standard equation is y² = 4ax ⇒ y² = 4 × 4 x X = 16x
∴ Equation of the parabola = 16x

Question 34.
If x = a cos⁴θ, y = a sin⁴6 show that \(\frac{d y}{d x}-\tan ^{2} \theta\)
Answer:
x = a cost⁴θ & y = a sin⁴
Differentiate both w.r.t θ we get
\(\frac{d y}{d \theta}\) = a.4cos³θ.sin θ, \(\frac{d y}{d \theta}\) = 2.4 sin³θ. cosθ

Question 35.
The side of an equilateral triangle is increasing at the rate of √3 cm/s. Find the rate at which its area is including when its side is 200 cms.
Answer:
Given \(\frac{\mathrm{dx}}{\mathrm{dt}}\) = \(\sqrt{3}\) cm/sec., x = 2 meters, \(\frac{\mathrm{dA}}{\mathrm{dt}}\) = ?
Area of equilateral Δ^le = A = \(\frac{\sqrt{3}}{4}\) x²
\(\frac{d A}{d t}=\frac{\sqrt{3}}{4} \cdot 2 x \frac{d x}{d t}\)
= \(\frac{\sqrt{3}}{4}\) . 2. 200. \(\sqrt{3}\) = 300cm² / sec.

Question 36.
Find two positive numbers whose sum is 14 and the sum of whose square is minimum.
Answer:
Let the two numbers be x and y
Given x + y = 14 & S = x² + y² where y = 14 – x
∴ S = x² + (14 – x)² = x² + 14² + x² – 28x = 2x² – 28x + 14²
\(\frac{d s}{d x}\) = 4x – 28 → (1) \(\frac{d s}{d x}\) = 0 ⇒ 4x – 28 = 0 ⇒ x = 7
\(\frac{\mathrm{d}^{2} \mathrm{s}}{\mathrm{d} \mathrm{x}^{2}}\) = 4 > 0, sum is minimum.
∴ y = 14 – x = 14 – 7 = 7
∴ the two positive number are 7 & 7.

Question 37.
Evaluate \(\int \frac{1+\cos 2 x}{1-\cos 2 x} d x\)
Answer:
= \(=\int \frac{2 \cos ^{2} x}{2 \cos } d x\) = ∫cos² dx = ∫(cosec²x – 1)dx = – cos x – x + c

Question 38.
Evaluate \(\int_{0}^{1}(6 x+1) \sqrt{3 x^{2}+x+5 d x}\)
Answer:
Put 3x² + x + 5 = t.
(6x + 1) dx = dt
When x = 0 then t = 5
When x = 1 then t = 9
\(\left.\int_{0}^{1}(6 x+1)(\sqrt{3 x^{2}+x+5}) d x=\int_{0}^{1} \sqrt{t} d t=\frac{2}{3} t^{\frac{3}{2}}\right]_{5}^{7}\)
\(=\frac{2}{3}\left(9^{\frac{3}{2}}-5^{\frac{3}{2}}\right)=\frac{2}{3}(27-5 \sqrt{5})\)

Part-D

Answer any six questions: (6 × 5 = 30)

Question 39.
Simplify (√2 + 1)⁶ – (√2 – 1)⁶ using Binomial theorem.
Answer:

Question 40.
Resolve \(\frac{2 x^{2}-4 x+1}{(x-2)(x-3)^{2}}\) into partial fractions.
Answer:
2x² – 4x + 1 = A[x – 3)² + B(x – 2) (x-3) + C (x – 2)
Put x = 2, 8-8 + 1 = A(-1)² = A = 1
Put x = 3, 18 – 12 + 1 = A(0) + B(0) + C(3 – 2) = 7 = C
Comparing coefficient of x² = 2 = A + B = B = 2-A = 2 -1 =1
∴ \(\frac{2 x^{2}-4 x+1}{(x-2)(x-3)^{2}}=\frac{1}{x-2}+\frac{1}{x-3}+\frac{7}{(x-3)^{2}}\)

Question 41.
Prove that (p ∨q)∧(~P∧q~) is a contradiction.
Answer:

Question 42.
Divide ₹ 1,647 into three parts such that \(\frac { 3 }{ 7 }\) th of the first, \(\frac { 2 }{ 3 }\) rd of second and \(\frac { 4 }{ 5 }\) th of the third are equal..
Answer:
Let the 3 parts be A,B & C
Given \(\frac{3}{7} A=\frac{2}{3} B \Rightarrow \frac{A}{B}=\frac{2}{7} \times \frac{7}{3}=\frac{14}{9}\)
∴ A : B = 14 : 9
Also given \(\frac{2}{3} B=\frac{4}{5} \Rightarrow \frac{B}{C}=\frac{4}{5} \times \frac{3}{2}=\frac{12}{10}=\frac{6}{5}\)
∴ B : C = 6 : 5
Multiple (1) by 2 and (2) by 3 we get A : B = 28 : 18 and B : C = 18 : 5
∴ A : B : C = 28 : 18 : 15
∴ 28x + 18x + 15x = 1647
⇒ 61x = 1647 ⇒ x = \(\frac{1647}{61}\) = 27
∴ A receives = 28 × 27 = ₹ 3 756
B receives = 18 × 27 = ₹ 486
Creceives = 15 × 27 = ₹ 405

Question 43.
A company requires 100 hours to produce the first 10 units at 15/hour. The learning effect is 80%. Find the total labour cost to produce a total of 160 units.
Answer:
Given
a = 100 hrs. 10 units = 1 lot
∴ 160 units = 16 lots

y = ax^b
log y = log a + b log x
= log 100 – 0.3219 log 16
= 2 – 0.3219 1.2041 = 2 – 0.3875 = 1.625
y = A(1.6125) = 40.98 hours/unit
∴ Total time required for 160 units or 16 lots = 40.98 16 = 655.68 hrs.
∴ Total cost to produce 16 units at ₹ 15/hr. = 655.68 × 15 = ₹ 9835.20
Table method

Total time for 16 lots = 655.36 hrs.
∴ Total cost to produce 16 lots at ₹ 15/hr. = 655.36 × 15 = ₹ 19830.4.

Question 44.
Solve the following L.P.P. graphically maximize z = 60x + 15y subject to the constraints x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0.
Answer:
Consider line x + y = 50
Put x = 0, y = 50, (0,50)
Put y = 0, x = 50 (50,0)
Consider the line 3x + y =- 90
Put x = 0, y = 90 (0,90)
Put y = 0, x = 30 (30,0)
Plot the above lines on the graph we get OABC is the solution region

≥ is maximum at c (30,0)
z_max = 1800

Question 45.
Prove that sin 20° sin 40° sin 60° sin 80°= \(\frac { 3 }{ 16 }\)
Answer:
L.H.S=sin 60° (sin 40° sin 20° )sin 80°

Question 46.
Find the equation of the circle passing through the points (5,1) and (3,4) and has its centre on the x – axis.
Answer:
Let the required equation of the circle is x² + y² + 2gx + 2fy + c = 0
It passes through (5, 1) & (3, 4)
(5, 1) (5)² + (1)² + 2g(5) + 2f (1) +C = 0
10g + 2f + C + 26 = 0 …..(1)
(3,4) 3² + 4² + 2g (3) + 2f (4) +c= 0
68 + 8f + C + 25 = 0 …..(2)
& the centre (- g, – f) lies on x-axis ⇒ f = 0 ….(3)
Solving 1 & 2 we get

⇒ 2x² + 2y² – x – 47 = 0.

Question 47.
If y = \((x+\sqrt{x^{2}+1})\) show that(x² – 1}y₂ + xy₁ – y = 0.
Answer:
y = x + \(\sqrt{x^{2} – 1}\) differentiate w.r.t x

∴ (x² – 1) y₁² = y² Differentiate agin w.r.t x
(x² – 1)2y₁y₂ + y₁² (2x) = 2y .y₁ (÷ 2y₁ we get)
(x² – 1)y₂ + xy₁ – y = 0 Hence proved

Question 48.
Find the area of the region between the parabolas y² = 4x and x² = 4y.
Answer:

Part – E

Answer any one question: (1 × 10 = 10)

Question 49.
(a) A school wants to award its students for the values of punctuality, good behavior and hardwork with a total cash award ₹ 6,000. Three times the award money for hardwork added to that given for punctuality amounts to ₹ 11,000. The award money for punctuality and hardwork together is double the one given for good behavior.

Represent the above situation algebraically and also find the award money for each value using matrix method.

(b) The total revenue function is given by R = 400x – 2x² and the total cost function given by C = 2x² + 40x + 4000 find
i. The marginal revenue and marginal cost function.
ii. The average revenue and average cost.
Answer:
(a) Let the values of punctuality, Good Behaviour and Hard work be denoted by x,y & z receptively
Given x + y +z = 6000 ———1
X + 3z = 11000 ———-2
X + z = 2y ⇒ x – 2y + z = 0 ——- 3
Solve these equations by matrix method

Thus x = ₹ 500, y = ₹ 2000 & z = ₹ 3500

(b) (i) MR = \(\frac{d}{d x}\) (TR) = \(\frac{d}{d x}\) (400x – 2x²) = 400 – 4x
MC = \(\frac{d}{d x}\) (TC) = \(\frac{d}{d x}\)(2x² + 40x + 400 )= 4x – 40

(ii) \(\mathrm{AR}=\frac{\mathrm{TR}}{\mathrm{x}}=\frac{400 \mathrm{x}-2 \mathrm{x}^{2}}{\mathrm{x}}=400-2 \mathrm{x}\)
\(A C=\frac{T R}{x}=\frac{2 x^{2}+40 x+4000}{x}=2 x+40+4000\)

Question 50.
(a) Prove that if \(\lim _{x \rightarrow a}\left(\frac{x^{n}-a^{n}}{x-a}\right)\) = na^n-1 for all values of’n’.
Answer:

Case 1: Let n be a positive integer.
xⁿ – aⁿ = (x-a)(x^n-1 + x^n-2. a + x^n-3. a² + ———- + 2n-1)

Case 2: Let n be a negative integer
put n = -m, m > 0

Case 3: Let n = p/q where p and q are integers and q ≠ 0

Hence it proved for al the values of n

(b) A person is at the top of a tower 75 feet high. From there he observes a vertical pole and finds the angles of depressions of top and bottom of the pole which are 30° and 60° respectively. Find the height of the pole.
Answer:

width=”278″ height=”194″ />
Let AB = Tower, CD = Pole = h
From triangle ABC we get
tan 60° = \(\frac{A B}{A C} \Rightarrow \frac{75}{A C}=\sqrt{3}\)
⇒ AC = \(\frac{75}{\sqrt{3}} ….. (1)\) …. (1)

From triangle BDE

⇒ AC = \(\sqrt{3}(75-h)\)

From 1 and 2 we get
\(\frac{75}{\sqrt{3}}=\sqrt{3}(75-h) \Rightarrow 75=(\sqrt{3})^{2}(75-h)\)
75 = 225 – 3h 3h ⇒ 150 ⇒ h = 50
∴ Height of the pole -50ft

Students can Download 2nd PUC Basic Maths Previous Year Question Paper June 2016, Karnataka 2nd PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Basic Maths Previous Year Question Paper June 2016

Time: 3.15 hours
Max Marks: 100

Instructions:

• The question paper has 5 parts A, B, C, D and E. Answer al the parts
• Part-A carries 10 marks\, Parts – B carries 20 marks, Part – C carries 3 marks, Part – D carries 30 marks and Part – E carries 1 marks.
• Write the question number properly as indicated in the question paper.

Part – A

Answer All the ten questions: (10 × 1 = 10)

Question 1.
Find x such that \(\left[ \begin{matrix} 3 & x \\ 4 & 7 \end{matrix} \right] \) is symmetric
Answer:
X = 4

Question 2.
Find n if ⁿC₈ = ⁿC₇
Answer:
ⁿC₈ = ⁿC₇ ⇒ ⁿC₈ = ⁿC₇ ⇒ 8 = n – 7 ⇒ n = 15

Question 3.
Negate the proposition “4 is an even integer or 7 is a prime number”.
Answer:
4 is not an even integer and 7 is not a prime number.

Question 4.
Find the value of x if 32 : x = 75 : 50.
Answer:
\(x=\frac{32 \times 50}{75}=\frac{64}{3}\)

Question 5.
Find the income obtained by investing ₹ 3,600 in 5% stock at 90.
Answer:
Income = \(\frac { 5 }{ 100 }\) × 3600 = 180

Question 6.
Find the value of 3 sin 10° – 4 sin 10°.
Answer:
3 sin 10° – 4 sin 10° = sin 3.10° = sin 30° = \(\frac { 1 }{ 2 }\)

Question 7.
If x² + y² – 4x – 8y + k = 0 represents a point circle find k.
Answer:
R = \(\sqrt{g^{2}+f^{2}-c}\), g = -2, f = -4,c = k, r = 0.
∴ \(\sqrt{4+16-k}=0\)
20 – k = 0 ⇒ k = 20

Question 8.
Evaluate \(\lim _{x \rightarrow 0}(1+3 x)^{\frac{1}{x}}\)
Answer:
\(\lim _{x \rightarrow 0}(1+3 x)^{\frac{1}{x}}=e^{3}\)

Question 9.
If y = cos (x³) find \(\frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}\)
Answer:
Let y = Cos(x³)
∴ \(\frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}\) = sin(x³).3x²

Question 10.
Evaluate : ∫(x^e + e^x – log a)dx
Answer:
\(\frac{x^{e+1}}{e+1}+e^{x}-\log a \cdot x+C\)

Part – B

Answer any ten questions. (10 × 2 = 20)

Question 11.
Solve using Cramer’r rule:
5x + 7y = 3
7x + 5y = 9
Answer:
5x + 7y = 3
7x + 5y = 9
Δ = \(\left| \begin{matrix} 5 & 7 \\ 7 & 5 \end{matrix} \right| \) = 25 – 69 = -24 Δ_x = \(\left| \begin{matrix} 3 & 7 \\ 9 & 5 \end{matrix} \right| \) = 15 – 63 = -48 Δ_y = \(\left| \begin{matrix} 5 & 3 \\ 7 & 9 \end{matrix} \right| \) = 45 – 21 = 24
∴ x = \(\frac{\Delta_{x}}{\Delta}=\frac{-48}{-24}\) = 2 y = \(\frac{\Delta_{y}}{\Delta}=\frac{24}{-24}\) = -1
∴ x = 2, y = -1

Question 12.
Find the number of parallelograms that can be formed form a set of 6 parallel lines intersecting another set of 4 parallel lines.
Answer:

Question 13.
What is the probability that there will be 53 Mondays in a randomly selected leap year?
Answer:
Every year has 52 complete weeks 8 = 364 days (52 × 7) A leap year has 366 days, in the remaining two days we have to see what is the probability I getting Monday, the two days may occur in any one of the following (Sunday, Monday) (Mon,Tue), (Wed, Thur) (Thur,Fri) (Fri, Satur) (Satur, Sun) Out of the 7 Possibilities, the first two cases have Monday
∴ P(53Mondays in a leap) = \(\frac { 2 }{ 7 }\)

Question 14.
Write the converse and income of the statement “If Maths is easy then child is brave”.
Answer:
Converse = (q → P) = If the child is brave then maths is easy
Inverse = (~ P → ~q) = If maths is not easy then the child is not brave.

Question 15.
Divide 6,000 in the ratio 3 : 4 : 5.
Answer:

II part = \(\frac{4}{12}\) × 6000 = 2000

III part = \(\frac{5}{12}\) × 6000 = 2500

Question 16.
A banker pays 2,380 on a bill of 2,500,73 days before the legally due date. Find the rate of discount charged by the banker.
Answer:
F = ₹ 2,500, discounted value = 2380, t = 73 days, t = \(\frac{73}{365}=\frac{1}{5}\) yr
Discounted value = F (1 – tr}
2380 = 2500 \(\left(1-\frac{1}{5} \times r\right)\) ⇒ \(\frac{2380}{2500}\) = 1 – \(\frac{r}{5}\) ⇒ 1 – 0.952
0.2r = 0.048
r = \(\frac{0.048}{0.2}\) = 24 = 24%

Question 17.
If A + B + C = 180° and tan A = 1, tan B=2, show that tan C = 3,
Answer:
Given A + B + C = 180° ⇒ A + B = 180 – C ∴ tan(A + B) = tan (180 – C)

Question 18.
Find the focus and equation of the directrix of the parabol x² = – 16y
Answer:
Given x² = – 16y
Compare x² = -4ay ⇒ 4a = 16 ⇒ a = 4
The curve turns down words ∴ focus S = (0,-4)
Equation of directrix is y = 4 or y – 4 = 0

Question 19.

Answer:
Given the function is continuous \(\lim _{x \rightarrow 4}\) f(x) = f(4)
\(\lim _{x \rightarrow 4} \frac{x^{4}-4^{4}}{x-4}=K\) ∴ k = 4.4^+-1 = 4.4³ = 4⁴ = 256

Question 20.
If y = x^x find \(\frac{d y}{d x}\)
dx
Answer:
Let y = x^x
log y = x.log x
\(\frac{1}{y} \frac{d y}{d x}=x, \frac{1}{x}+\log x, 1 \Rightarrow \frac{d y}{d x}\) = y (1 + log x) = x^x(1 + logx)

Question 21.
When brakes are applied to a moving car, the car travels a distance ‘s’ feet in ‘t’ secs is given by s = 20t – 40t². When does the car stop?
Answer:
Given s = 20t – 40t²
v = \(\frac{d s}{d t}\) = 20 – 80t
The car stops when the velocity = 0 ⇒ 20 – 80t = 0 ⇒ t = \(\frac{1}{4}\) sec

Question 22.
The radius of a sphere is increasing at the rate of 0.5 mts / sec. Find the rate of Increase of its volume when r=1.5 mts.
Answer:
Given \(\frac{d r}{d t}\) = 0.5 mts / sec. \(\frac{d v}{d t}\) = ? when r =1.5mts v = \(\frac{4}{3}\)πr³

Question 23.
Evaluate : \(\int \frac{e^{x}}{e^{x}-1} d x\)
Answer:
\(\int \frac{e^{x}}{e^{x}-1} d x\) = log (e^x+ 1) + C ∴ \(\int \frac{f^{\prime}(x)}{f(x)}\) dx = log(f(x))

Question 24.
Evaluate \( \int_{1}^{2}\left(x+e^{x}\right) d x\)
Answer:

Part – C

Answer any ten questions: (10 × 3 = 30)

Question 25.
If A = \(\left[ \begin{matrix} 2 & -1 \\ 1 & 4 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -3 & 2 \\ -1 & 4 \end{matrix} \right] \) show that (AB)’ = B’ A’.
Answer:

From 1 and 2 we get (AB)’ = B’A’

Question 26.
Solve for \(\left[ \begin{matrix} x+2 & 3 & 4 \\ 2 & x+3 & 4 \\ 2 & 3 & x+4 \end{matrix} \right] =\quad 0\)
Answer:

Question 27.
Find the number of permutations of the letters of the word Engineering How many of these
(a) Begin with GRIN
(b) Have all 3 E’s together?
Answer:
The word has 11 letters, E = 3, N = 3, G = 2, I = 2
Total permutations \(\frac{11 !}{3 !, 3 !, 2 ! 2 !}\)
(a) Permutations word which begin with GRIN = \(\frac{7 !}{3 ! 2 !}\)
(b) All 3E’s are together Consider 3E as one unit remaining letters = 8 + 1 = 9
N = 3,G = 2, 1 – 2 No of permutation = \(\frac{9 !}{3 ! 2 ! 2 !}\)

Question 28.
One Card is randomly drawn from a pack of 52 cards, Find the probability that
(a) Card is either black or jack
(b) Card is red
(c) Card is diamond.
Answer:
Totally there are 52 cards and any one can be drawn
∴ Total no cases = n = 52

(a) There are 26 black cards and 2 jack totally = 28 cards ∴ no of favourable case = m = 28
∴ P(black or jack) = \(\frac{28}{52}\)

(b) There are 26 red cards
∴ m = 26,

(C) There are 13 diamonds cards
∴ m = 13,
P(getting diamond) = \(\frac{13}{52}=\frac{1}{4}\)

Question 29.
Walking 4 kmph a student reaches his college 5 minutes late and if he walks at 5 kmph, he reaches \(2 \frac{1}{2}\) minutes early. What is the distance from his house to the college?
Answer:
Let the distance from his house to the college = x km
Time taken to Cover x kns at 4km ph = \(\frac{x}{4}\) hr⁵
Time to cover 5 km ph = \(\frac{x}{5}\) hrs

Question 30.
A bill for ₹ 12,900 was drawn on 3 Feb, 2014 at 6 months and discounted on 13 March 2014 at 8% p.a. For what sum was the bill discounted ?
Answer:
F = 12,900, bill period = 6 months, Drawn date = 03 – 2 – 14
∴ LDD = DD + BP + Grace period

Date of discount 13-3-2014. This means, the no of days from 13-2-2014 to 06-08-14 becomes the unexpired period

∴ t = 145 days = \(\frac{146}{365}\) year, r = 80% = 0.08, DV = ?
BD = Ftr = 12,900 × \(\frac{146}{365}\) × 0.08 = 412.8
∴ Discounted value = 12,900 – 412.8 = 12,487.20

Question 31.
Prathik sells out ₹ 6,000 of 7.5% stock at 108 and reinvests the proceeds in 9% stock. If Prathik’s income increase by ₹ 270, at what price did Prathik buy 9% stock?
Answer:
To calculate income

Income = \(\frac{6000 \times 7.5}{10}\) = 459.0 = ₹ 450

For 9% stock given income increase by ₹ 270
∴ New income = 450 + 270 = ₹ 270.

Question 32.
A furniture dealer sold furniture for ₹ 21,000 and added 5% sales tax to the quoted price. The customers agree to buy it for ₹ 21,000 including sales tax. Find the discount he received.
Answer:
Discount = 5% 21,00 = \(\frac{5}{10}\) × 21,000 = 1050

Question 33.
Prove that \(\frac{\cos 75^{\circ}+\cos 15^{\circ}}{\sin 75^{\circ}-\sin 15^{\circ}}=\sqrt{3}\)
Answer:

Question 34.
Find equation of the parabola given that its vertex is (0,0) axis is Y-axis and passed through (-1,-3).
Answer:
Given v = (0,0) and axis is y-axis
The parabola is of the form x² = 4ay or x² = -4ay
But this equation passes through (-1,-0) which is in III qhadrant.
∴ The curve turn down words and is of the form x² = – 4ay, But it passes through (-1,-3)
∴ (-1)² = -4a(-3) ⇒ 1 = 12a ⇒ a = \(\frac{1}{12}\)
∴ The required equation is x² = -4 . \(\frac{1}{12}\)y ⇒ x² = \(\frac{-1}{3}\) or 3x² + y = 0

Question 35.
If x = \(\frac{1-t^{2}}{1+t^{2}}\) y = \(\frac{2 t}{1+t^{2}}\) find \(\frac{\mathrm{dy}}{\mathrm{dt}}\)
Answer:
Differentiate both w.r.tt


If the total cost function is given by C(x) = 350 + 12x + \(\frac{x^{2}}{4}\) and revenue function is given by R(x) = 75x – 2x², find the level of output at which profit is maximum.
Profit function = TR – TC = 75x – 2x – 350 – 12x – \(\frac{x^{2}}{4}\) p(x) = -2x² – x² + 63x – 350
For maximum profit differentiate w.r.t x \(\frac{d}{d x}\) P(x) = 0

126 = 9x ⇒ x = \(\frac{126}{9}\) = 14 ∴ x = 14

Question 37.
Evaluate ∫x² log xdx
Answer:

Question 38.
Evaluate \(\int \frac{1+\cos x}{1-\cos x} d x\)
Answer:

Part – D

Answer any six questions: (6 × 5 = 30)

Question 39.
Find the term independent of x in \(\left(x^{3}-\frac{3}{x^{2}}\right)^{15}\)
Answer:
T_r+1 = ⁿC_r x^n-r a^r
Here x → x³a → \(\frac{-3}{x^{2}}\) n → 15
T_r+1 = ¹⁵C_r (x³)^{15 – r} \(\left(\frac{-3}{x^{2}}\right)\) = ^{15C_{r .}} x^{45 – 3r} (-3)^r . x^-2r
T_r+1 = ¹⁵C_r(-3)^r . x^{45 – 5r}
To find the term independent of x equate the power of x = 0
⇒ 45 – 5r = 0 ⇒ 45 = 5r = 0 ⇒ r = 9
T₉₊₁ = ¹⁵C₉(-3)⁹ . x⁰ = ¹⁵C₉(-3)⁹
T₁₀ = 3⁹, ¹⁵C₉ is the term independent of x

Question 40.
Resolve into partial fractions \(\frac{3 x+2}{(x-2)(x+3)^{2}}\)
Answer:

3x + 2 = A(x +3 )² + B(x – 2)(x – 3) + C(x – 2)
Put x = 2,6 + 2 = A (2 + 3))² ⇒ A = \(\frac { 8 }{ 25 }\)
Put x = -3 -9 + 2 = 0 + 0 + C(-5)
-7 = -5C ⇒ C = 715
Put x = 0 we get B = \(-\frac { 8 }{ 25 }\)

Question 41.
Verify whether the proposition ~(p ∨ q) → (~P ∧ ~q) is a tautology contradiction or neither.
Answer:

Question 42.
Two taps can separately fill a tank in 12 minutes and 15 minutes respectively. The tank when full can be emptied by a drain tap in 20 minutes. When the tank was empty all the three taps were opened simultaneously. In what time will the tank be filled up?
Answer:
1^st tap can fill \(\frac{1}{12}^{\text {th }}\) tank in 1 min 2nd tap can fill \(\frac{1}{15}^{\text {th }}\) tank in 1 min
Drain pipe drain out = \(\frac{1}{20}^{\text {th }}\) tank in 1 min
∴ In 1 min = \(\frac{1}{12}+\frac{1}{15}-\frac{1}{20}=\frac{15+12-9}{180}=\frac{18}{180}=\frac{1}{10}\) of tank will get filled
∴ The tank will get filled in 10 minutes.

Question 43.
ABC company required 1000 hours to produce first 30 engines. If the learning effect is 90%, find the total labour cost at ₹ 20 per hour to produce total of 120 engines.
Answer:
Assume 1 lot = 30 engines
∴ 120 engines = 4 lots

The total no of hrs to produce 4 lots = 3240
The lab cost at 20 per hour = 3240 × 20 = ₹ 64,800

Question 44.
Solve the L.P.P graphically
Maximize Z = 2x + 3y
Subject to constraints x + y ≤ 400, 2x + y ≤ 400, x,y ≥ 0.
Answer:
Consider
X + y = 400 2x + y = 600

Point of intersection of x + y = 400 & 2x + y = 600

Function is maximum at the point A (0.400) and maximum value = 1200

Question 45.
The angles of elevation of the top of a tower the base and the top of a building are 60° and 30°. 3x + 5y ≤ 15, 5x + 2y ≤ 10 and x ≥ 0, y ≥ 0.
Answer:
Let AB = Building & CD is the tower. From the rt angled ∆ ACD
We have tan 60 = \(\frac { h }{ x }\)
√3 = \(\frac { h }{ x }\) ⇒ x = √3(h -20)
From 1 and 2 wet √3(h – 20) = \(\frac{\mathrm{h}}{\sqrt{3}}\) ⇒ 3(h – 20) = h ⇒ 3h – 60 = h ⇒ 3h – 60 = h ⇒ 2h = 60 ⇒ h = 30 mts
∴ Height of the tower = 30mts

Question 46.
Find the equation of circle passing through points (5,3) (1,5) (3,-1)
Answer:
Let the required equation of the circle is
x² + y² + 2gx + 2fg + c = 0
But this eqn passes through the points
(5,3) 10g + 6f + C + 34 = 0
(1,5) 2g + 10 f + C + 26 = 0
(3,-1) 6g – 2f + C + 10 = 0
Solving 2,3 and 4 we get
2 – 3 ⇒ 2g – f + 2 = 0
3 – 4 ⇒ g – 3f – 4 = 0
Again solving we get y = -2, f = -2, c = -2
This the required egn of the circle is x² + y² – 4x – 4y -2 = 0

Question 47.
If y = \((x+\sqrt{1+x^{2}})^{m}\) prove that (1 + x²)y₂ + xy₁ – m²y=0
Answer:
Given y = \((x+\sqrt{1+x^{2}})^{m}\)
Diff. w.r.tx

\(\sqrt{1+x^{2}} y_{1}\) = my S.BS
(1 + x²)y²₁ = m²y²
Again differentiate w.r.t.x (1 + x²).2y₁y₂ + y²₁ (2x) = m².2yy₁ ÷ 2y₁ ∴ (1 + x²).y₂ + xy₁ – m²y = 0

Question 48.
Find the area enclosed between parabola y² = x and the line x + y = 2.
Answer:
Given y² = x and x + y = 2
⇒ (2 – x)² = x ∴ y = 2 – x
⇒ 4 + x² – 4x – x = 0 ⇒ x² – 5x – 4 = 0 ⇒ x = 4 or

Part – E

Answer any one questions: (1 × 10 = 10)

Question 49.
(a) A school wants to award to award its students for the values of leadership, good behavior and hard work with a total cash award of ₹ 6,000. Three times the award money for hard work added to that given for leadership amounts to 11,000. The award money for leadership and hard work together is double the one given for good behavior. Calculate the award money given for each value, using matrix method.
Answer:
Let the values of punctuality, good behavior and hard work be denoted by x,y & z respectively
x + y + z = 6000
x + oy + 3z = 11,000
x + z = 2y or x – 2y + z = 0
The given equations in matrix form is \(\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] \left[ \begin{matrix} 6000 \\ 11,000 \\ 0 \end{matrix} \right] \)
A x = B
⇒ X = A^-1B
Where A = \(\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{matrix} \right] \)

X = ₹ 500, y = ₹ 2000, z = ₹ 3500.

(b). A company produces two products P and Q. Each P require 4 hours of grinding and 2 hours of polishing and each Q requires 2 hours of grinding and 5 hours of polishing. The total available hours for grinding is 20 and for polishing is 24. Profit per unit of P is 6 and that of Q is ₹ 8. Formulate the L.P.P
Answer:
Let x be the units of P manufactured and y be the units of Q manufactured


Maximize z = 6x + 8y
Subject to constraints

Question 50.
(a). Prove that \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\), where θ is in radius.
Answer:
Let ‘O’ the centre of unit circle, assume that is measured in positive radians.

From the figure, we have area of ∆AOB = area of sector AOB = area of ∆AOD

∴ Area of sector = \(\frac { 1 }{ 2 }\) (radius)² × angle in radians
From the triangle DOA, tan θ = \(\frac{\mathrm{DA}}{\mathrm{OA}}\) [∴ DA = r tanθ]
From the triangle BOC, sin θ = \(\frac{\mathrm{BC}}{\mathrm{OB}}\) [∴ BC = r sin θ]
(1) becomes
⇒ BC ≤ θ ≤ DA
⇒ sin θ ≤ θ ≤ tan θ

Dividing by sin θ

(b). Find the value of (1.1)⁵ using binomial theorem upto 4 decimal places.
Answer:
(1.1)⁵ = (1 + 0.1)⁵
= 1⁵ + ⁵C₁ (0.1) + ⁵C₂(0.1)² + ⁵C₃ (0.1)³ + ⁵C₄(0.1)⁴ + ⁵C₅(0.1)⁵
= 1 + 5(0.1) + 10(0.1) + 10(0.001) + 1(0.00001)
= 1 + 0.5 + 0.1 + 0.01 + 0.005 + 0.00001
(1.1)⁵ = 1.61051

Students can Download Maths Chapter 13 Probability Ex 13.3 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.3

2nd PUC Maths Probability NCERT Text Book Questions and Answers Ex 13.3

Question 1.
An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red ?
Answer:
In the first draw the ball may be red or black

(1) If red ball is drawn, 2 more red balls are put and then draw a red ball
(2) If black ball is drawn 2 more black ball is put and then again draw a red ball
∴ Required probability is
\(\frac{5}{10} \times \frac{7}{12}+\frac{5}{10} \times \frac{5}{12}=\frac{60}{120}=\frac{1}{2}\)

Question 2.
A bag contains 4 red and 4 black balls , another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first
Answer:
Let E₁ : The first bag is selected
E₂ : The second bag is selected
A : The ball is red

Question 3.
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year result report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier ?
Answer:
E₁ : Student is a hostler
E₂ : Student is a day scholar
A : student has an A grade

Question 4.
In answering a question on a multiple r. choice test, a student either knows the answer or guesses. Let \(\frac{3}{4}\) be the probability that he knows the answer and \(\frac{1}{4}\) be the probability that he guesses.Assuming that a student who guesses at the answer will be correct with probability \(\frac{1}{4}\) . What is the probability that the student knows the answer given that he answered it correctly ?
Answer:
E₁: student knows the answer
E₂ : student guesses the answer
A : Answer is correct

Question 5.
A laboratory blood test is 99% effective in detecting a certain disease when it is in
fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?
Answer:
E₁: person has the disease
E₂: person is healthy
A : Test is positive

Question 6.
There are three coins. One is a two headed coin (having head on both faces), another is a based coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
Answer:
E₁ : 1^st coin is chosen
E₂ 2^nd coin is chosen
E₂: 3^rd coin is chosen
Let ‘A’ be the even that shows head

Question 7.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Answer:
E₁: Insured person is scooter driver
E₂: Insured person is car driver
E₃: Insured person is truck driver
A : Meets with an accident

Question 8.
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
Answer:
E₁ : Item produce by machine A
E₂: Item produce by machine B
A : Item is defective

Question 9.
Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
Answer:
E₁ : 1^st group will win
E₂ : 2^nd group will win
A : New product is introduced

Question 10.
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
Answer:
E₁ : 5 or 6 shows on the die     HHH
E₂ : 1, 2, 3,4 shows on the die  HHT
A : Shows Head exactly once   HTH

Question 11.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Answer:
E₁ : item produce by A
E₂ : item produce by B
E₃ : item produce by C
A : item produced is defective

Question 12.
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are . drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Answer:
Let E₁ : Lost card is diamond.
E₂ : Lost card is not diamond
A : Two cards drawn are diamond

Question 13.
Probability that A speaks truth is \(\frac{4}{5}\) A coins is tossed. A reports that a head appears. The probability that actually there was head is
(A) \(\frac{4}{5}\)
(B) \(\frac{1}{2}\)
(C )\(\frac{1}{5}\)
(D) \(\frac{2}{5}\)
Answer:
Let E₁ : Head appers
E₂ : Tail appers
A : Reports that head appears

Question 14.
If A and B are two events such that A⊂B and P(B) ≠ 0,then which of the following is correct?
(A) \(\mathbf{P}(\mathbf{A} | \mathbf{B})=\frac{\mathbf{P}(\mathbf{B})}{\mathbf{P}(\mathbf{A})}\)
(B) P(A|B) < p(A)
(C) P(A|B) ≥ p(A)
(D) None of these
Answer:

Students can Download Maths Chapter 13 Probability Ex 13.2 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 13 Probability Ex 13.2

2nd PUC Maths Probability NCERT Text Book Questions and Answers Ex 13.2

Question 1.
If \(\mathbf{P}(\mathbf{A})=\frac{3}{5} \text { and } \mathbf{P}(\mathbf{B})=\frac{1}{5}, \text { find } \mathbf{P}(\mathbf{A} \cap \mathbf{B})\) ,find P(A∩B) if A and B are independent events.
Answer:
If two events A and B are independents P (A∩B) = P(A). P(B)
\((A \cap B)=\frac{3}{5} \times \frac{1}{5}=\frac{3}{25}\)

Question 2.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Answer:
There are 26 black cards out 52 cards. When cards are drawn without replacement,
\(P(\text { both black cards })=\frac{26 \mathrm{C}_{2}}{52 \mathrm{C}_{2}}=\frac{26 \times 25}{52 \times 51}=\frac{25}{102}\)

Question 3.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Answer:
Since selection is without replacement

Question 4.
A fair coin and an unbiased die are tossed.Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’.Check whether A and B are independent events or not.
Answer:

Question 5.
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is event, and B be the event, ‘the number is red’. Are A and B independent?
Answer:
A : number is even, B : The number is red

Question 6.
Let E and F be events with P(E) \({ P }({ E })=\frac { 3 }{ 5 } ,{ P }({ F })=\frac { 3 }{ 10 } { \quad and\quad }{ P }({ E }\cap { F })=\frac { 1 }{ 5 } \). Are E and F independent?
Answer:

hence they are not independent.

Question 7.
Given that the events A and B are such that \(P(A)=\frac { 1 }{ 2 } ,P(A\cup B)=\frac { 3 }{ 5 } \quad { and }\quad P(B)=p\) Find p if they are
(i) mutually exclusive
(ii) independent.
Answer:
(i) If the events are mutually exclusive

Question 8.
Let A and B be independent events with
P(A) = 0.3 and P (B) = 0.4. Find
(i) P(A∩B)
(ii)P(A∪B)
(iii) P (A|B)
(iv) P (B|A)
Answer:

Question 9.
If A and B are two events such that
\(P(A)=\frac { 1 }{ 4 } ,{ P }({ B })=\frac { 1 }{ 2 } { and }{ P }({ A }\cap { B })=\frac { 1 }{ 8 } \)
Answer:

Question 10.
Events A and B are such that \({ P }({ E })=\frac { 3 }{ 5 } ,{ P }({ F })(B)=\frac { 7 }{ 12 } \) P (not A or not B) = \(\frac{1}{4}\). State whether A and B are independent ?
Answer:

Question 11.
Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find
(i) P (A and B)
Answer:
i.e. P (A∩B) = P (A) x P (B) = .3 x .6 = .18

(ii) P (A and not B)
Answer:
P(A and not B) = \(P(A \cap \bar{B})\)
= P(A) x \(P(\bar{B})\) ⇒ P (A) x [1-P(B)]
= .3 x (1 – 0.6) = .3 x .4 = .12

(iii) P (A or B)
Answer:
P (A or B) = P (A∪B)
= P (A) + P (B) – P (A∩B)
= .3+ .6-.18 = .9 -.18 = .72

(iv) P (neither A nor B)
Answer:
P (neither A nor B) = \(P(\bar{A} \cap \bar{B})\)
= \(P(\overline{A \cup B})\)
= 1 -P(A∪B)= 1 -.72 = .28

Question 12.
A die is tossed thrice. Find the probability of getting an odd number at least once.
Answer:
E : an odd number at least once
\(\overline{\mathrm{E}}\) : not an odd number
\(\overline{\mathrm{E}}\) : an even number on all times
we get even all time 3 x 3 x 3 = 27
\(P(E)=1-P(\bar{E})=1-\frac{27}{216}=\frac{7}{8}\)

Question 13.
Two balls are drawn at random With replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red
(iii) one of them is black and other is red
Answer:

Question 14.
Probability of solving specific problem independently by A and B are \(\frac{1}{2} \text { and } \frac{1}{3}\) respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.
Answer:

Question 15.
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ?
(i) E : ‘the card drawn is a spade’
F : ‘the card drawn is a ace’
(ii) E : ‘the card drawn is black’
F : ‘the card drawn is a king’
(iii) E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’.
Answer:

Question 16.
In a hostel, 60% of the students read Hindi news paper, 40% read English news paper and 20% read both Hindi^and English news papers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English news papers.
(b) If she reads Hindi news paper, find the probability that she reads English news paper.
(c) If she reads English news paper, find the probability that she reads Hindi news paper.
Answer:
(a) 20% read neither Hindi nor english

Choose the correct answer in Exercises 17 and 18.

Question 17.
The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0
(B) \(\frac{1}{3}\)
(C) \(\frac{1}{12}\)
(D) \(\frac{1}{36}\)
Answer:
Even Prime no is (2,2)
P (even prime no) = \(\frac{1}{36}\)
Option ‘D’

Question 18.
Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P (A’B’) = [1 – P(A)] [1 – P(B)]
(C) P (A) = P (B)
(D) P (A) + P (B) = 1
Answer:
P(A’B’) = P(A’∩B’)
= \(P\bar { A\cup B } \)
= 1 – P (A∪B)
= 1 – P(A) – P(B) – P(A) – P(B)
= 1 – P (A) – P (B) (1 – P (A)]
= [1 – P (A)] [1 -P(B)
Option ‘B’