# July 2020

## Telugu Grammer( తెలుగు వ్యాకరణం )

Telugu Grammer( తెలుగు వ్యాకరణం ): మన మనస్సు లోని భావాలను , అనుభూతులను పైకి చెప్పడానికి  మాతృభాష ఎంతో ఉపయోగం , అందుకే మాతృ భాష తల్లి లాంటిది అంటారు. మనం మన మాతృ భాషని గౌరవించాలి .

గ్రహణ సామర్థ్యం పెరగడానికి మాతృ భాష  లో విద్యా బోధన ఎంతగానో ఉపయోగ పడుతుంది.  మాతృ భాష లో బోధించడం వలన విద్యార్థుల్లో సృజనాత్మకత పెరుగుతుంది.

### భాషా భాగాలు :

భాషకు ప్రాణం భావ ప్రసరణ . ఈ భావ ప్రసరణ ఒకరి నుండి మరొకరికి చేరాలి . ఇలా చేరడానికి కొన్ని పదాలు వాక్యాలు అవసరం. ఇలా వాక్యం లోని ఉపయోగాన్ని బట్టి భాషకు ఐదు ప్రధాన భాగాలుగా విభజించారు.

అవి :− 1. నామవాచకం   2. సర్వనామం  3. విశేషణం  4. క్రియ  5. అవ్యయము

### 1.నామవాచకం :

నామము అనగా పేరు.ఒక వ్యక్తిని గాని, వస్తువుని గాని ,గుణమును గాని, జాతిని గాని తెలుపును.

ఉదా :- ధర్మరాజు , హైదరాబాద్ , బంతిపువ్వు , ఆవు మొ|| నవి .

#### 2. సర్వనామము :

నామ వాచకాలకు బదులుగా వాడే వాటిని “ సర్వనామాలు “ అంటారు. “సర్వ” అనగా సమస్తము.

ఉదా :- అది, ఇది, అతడు , ఆమె ,అన్ని ,కొన్ని మొ|| నవి.

#### 3.  విశేషణం  :

నామవాచకము మరియు సర్వనామముల యొక్క గుణమును తెలియజేయునది .

ఉదా :- మంచి , చెడు , లావు , పొట్టి , పొడుగు  , ఎత్తు  మొ|| నవి.

#### 4. క్రియ :

పనులను, స్తితిగతులను తెలియజేయునది .

ఉదా :- రాస్తున్నాడు , వెళ్తున్నాడు , పాడుతున్నాడు  మొ|| నవి.

#### 5. అవ్యయము :

వ్యయము అనగా నశించేది, అవయము అనగా నశించనిది .

లింగ, వచన, విభక్తుల ప్రసక్తిగాని, వచన ఆకాంక్ష లేని వాటిని అవ్యయములు అంటారు .

ఉదా :-  అక్కడ, ఇక్కడ, ఆహా , భళా  మొ|| నవి.

## సంధులు

వ్యాకరణ భాషలో రెండు  స్వరాల కలయికను  సంధి  అంటారు .

రెండు అచ్చుల మధ్య  జరిగే మార్పును  సంధి కార్యం  అంటారు.

సంధి జరిగే మొదటి పదo చివరి అక్షరం లోని అచ్చును ‘పూర్వ పదం’ అంటారు .

సంధి జరిగే రెండవ పదం మొదటి అక్షరం లోని అచ్చును ‘పర పదం ‘ అంటారు .

ఉదా :- రామ + అయ్యా:  ‘ రామ’  లోని     ‘మ’  లో  ‘అ’ పూర్వ పదం  ‘అయ్యా’  లోని  ‘అ ‘ పర పదం .

##### అత్వ సంధి(అకార సంధి ):

అత్తునకు సంధి  బహుళంగా వస్తుంది .

ఉదా :-1) మేఅల్లుడు = మేన + అల్లుడు                     3) లేకేమి  = లేక + ఏమి

2) రాకుంటే  = రాక + ఉంటే                          4) పోవుటెట్లు =  పోవుట  + ఎట్లు

##### ఇత్వ సంధి ( ఇకార సంధి):

ఏమ్యాదులకు ఇత్తునకు సంధి .

ఉదా :- 1) ఏమంటివి = ఏమి + అంటివి                       3) పైకెత్తినారు  = పైకి + ఎత్తినారు

2) వచ్చిరిపుడు  = వచ్చిరి + ఇపుడు                4) మనిషన్నవాడు  = మనిషి + అన్నవాడు

#### ఉత్వ సంధి ( ఉకార సంధి ):

ఉత్తునకు అచ్చు పరమైనపుడు సంధి నిత్యంగా వస్తుంది .

ఉదా :- 1) రాముడతడు = రాముడు + అతడు            3) మనమున్నాము = మనము + ఉన్నాము

2) అతడెక్కడ = అతడు + ఎక్కడ

4) మనసైన = మనసు + ఐన

#### యదగామ సంధి :

అంది లేని చోట అచ్చుల మద్య ‘య్’ వచ్చి  చేరడాన్ని “యడాగమం” అంటారు .

ఉదా :- 1) మాయమ్మ = మా + అమ్మ                             3) హరియతడు = హరి + అతడు

2) మాయిల్లు = మా + ఇల్లు

#### ఆమ్రేడిత సంధి :

అచ్చునకు ఆమ్రేడితం పరమైతే సంధి తరచుగా వస్తుంది .

ఉదా :- 1)  ఆహాహా = ఆహా + ఆహా                                  3)  ఔరౌర = ఔర  + ఔర

2) అరెరే  = అరె + అరె                                           4)  ఏమిటేమిటి = ఏమిటి + ఏమిటి

##### గసడదవాదేశ సంధి :

ప్రథమ మీది పరుషాలకు గ, స ,డ , ద ,వ   లు  బహుళంగా వస్తాయి .

ఉదా :-  1)  కొలువుసేసి = కొలువు + చేసి                   3) కూరగాయలు = కూర + కాయ

2) పాలువోయక = ఆలు + పోయక                4) తల్లిదండ్రులు  = తల్లి + తండ్రి

#### త్రిక సంధి :

త్రికము మీది అసంయుక్త హల్లునకు దిత్వం బహుళంగా వస్తుంది .

 ఆ , ఈ , ఏ  లు త్రికంఅనబడుతాయి

ద్విరుక్తమైన హల్లు పరమైనపుడు, అచ్చికమైన దీర్ఘానికి  హ్రస్వం వస్తుంది.

ఉదా :-  1)  ఇక్కాలము  = ఈ + కాలము                       3)  అక్కోమరుండు =  ఆ + కొమరుండు

2) ఎవ్వాడు = ఏ + వాడు                                    4) అచ్చోట = ఆ + చోట

###### రుగాగమ సంధి :

పేదాది శబ్దాలకు  ‘ఆల‘ శబ్దo పరమైతే కర్మదారాయం లో రుగాగం వస్తుంది .

ఉదా :- 1) మనుమరాలు = మనుమా + ఆలు                3) ధీరురాలు  = దీరు + ఆలు

2) పేదరాలు = పేద + ఆలు                                     4)  బాలెంతరాలు = బాలెంత + ఆలు

5 ) ముద్దరాలు = ముద్ద +ఆలు                               6) జవరాలు = జావా + ఆలు

###### సవర్ణ దీర్ఘ  సంధి:

అ, ఇ , ఉ,  ఋ లకు అవే అచ్చులు పరమైతే వాని దీర్గాలు ఎకాదేశంగా వస్తాయి .

ఉదా :-    1)  రామానుజుడు = రామ + అనుజుడు    2 )  రామాలయం = రామ + ఆలయం

3)  భానూఉదయం  = భాను + ఉదయం                  4)  కవీంద్రుడు = కవి + ఇంద్రుడు

5 ) పితౄణం = పితృ +ఋణం        6) వదూపేతుడు  = వధు + ఉపేతుడు

#### గుణసంధి :

ఇ , ఉ , ఋ  పరమైతే ఏ, ఓ , ఆర్  లు క్రమంగా ఎకాదేసంగా వస్తాయి .

ఉదా :-    1)  రాజేంద్రుడు  = రాజ  + ఇంద్రుడు          2 )  పరోపకారం  = పర  + ఉపకారం

3)  రాజర్షి   = రాజ  + ఋషి               4)   మహోన్నతి = మహా  + ఉన్నతి

#### యణాదేశ సంధి:

ఇ , ఉ, ఋ లకు అసవర్ణ అచ్చులు పరమైతే య , వ ,ర  లు వస్తాయి .

ఉదా :-    1)  అత్యవసరం  = అతి  + అవసరం    2 )  ప్రత్యేకం  = ప్రతి  + ఏకం

3)  అణ్వస్త్రం   = అణు  + అస్త్రం      4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

5 ) పితౄణం = పితృ +ఋణం

#### వృద్ధి సంధి :

అకారినికి ఏ , ఐ  లు పరమైతే ‘ఐ ‘ కారము , ఓ , ఔ లు పరమైతే  ‘ఔ’ కారము వస్తాయి.

ఉదా :-    1)  వసుధైక  = వసుధ   + ఏక           2 )  సమైక్యం   = సమ   + ఐక్యం

3)  వనౌసది    = వన   + ఔసది         4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

##### అనునాసిక సంధి :

వర్గ ప్రతమాక్షరాలకు ‘న’ గాని , ‘మ’ గాని ప్రమైతే అనునాసికాలు.

ఉదా :-    1)  వాజ్మయం   = వాక్   + మయం           2 )  జగన్నాథుడు   = జగత్  + నాథుడు

3)  అణ్వస్త్రం   = అణు  + అస్త్రం               4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

5 ) తన్మయం  = తత్  + మయం

## సమాసాలు

సమాసం : వేరు వేరు అర్థాలు కల రెండు పదాలు కలిసి , ఏకంగా ఏర్పడితే దాన్ని ‘సమాసం’ అంటారు .

###### దంద్వ సమాసం:

రెండు కాని,  అంతకంటే ఎక్కువ కాని నామవాచకాల మద్య ఏర్పడే సమాసాన్ని ‘దంద్వ సమాసం ‘ అంటారు.

ఉదా : – 1) అన్నదమ్ములు  – అన్న, తమ్ముడు    2) తల్లిదడ్రులు – తల్లి, తండ్రి

3) మంచిచెడులు – మంచి, చెడు  4 ) కష్టసుఖాలు – కష్ట , సుఖము

###### ద్విగు సామాసం :

సమాసంలో మొదటి పదంలో సంఖ్య గల సమాసాన్ని ‘ద్విగు’ సమాసం అంటారు.

ఉదా : – 1) నవరసాలు   – నవ సంఖ్య  గల రసాలు    2) రెండుజడలు  – రెండు సంఖ్య  గల జడలు

3) నాలుగు వేదాలు – సంఖ్య గల వేదాలు

#### తత్పురుష సమాసం :

విభక్తి ప్రత్యాలు విగ్రహ వాక్యంలో ఉపయోగించే సమాసాలు ‘ తత్పురుష  సమాసాలు .

## విభక్తులు

• ఒక వాక్యం లోని వేరు వేరు పదాలకు అన్వయం కలిగించు పదాలను “ విభక్తులు” అంటారు
 విభక్తులు ప్రత్యయాలు ప్రథమా విభక్తి డు – ము – వు – లు ద్వితీయ విభక్తి నిన్ – నన్ – లన్ – కూర్చి – గురించి తృతీయ విభక్తి చేతన్ – చెన్ –  తోడన్ – తోన్ చతుర్థి విభక్తి కొరకున్  – కై పంచమ విభక్తి వలనన్ – కంటెన్ – పట్టి షష్ఠివిభక్తి యొక్క – లోన్ – లోపలన్ సప్తమి విభక్తి అందున్ – నన్ సంబోధన ప్రథమా  విభక్తి ఓరి – ఓయి – ఓసి
 సమాస పదం విగ్రహ వాక్యం సమాసం పేరు మద్యాహ్నము అన్నము యొక్క మద్య ప్రథమా తత్పురుష జటాధారి జడలను ధరించినవాడు ద్వితీయ  తత్పురుష రాజ పూజితుడు రాజు చే పూజితుడు తృతీయ  తత్పురుష వంట కట్టెలు వంట కొరకు కట్టెలు చతుర్థి  తత్పురుష అగ్నిభయం అగ్ని వల్ల భయం పంచమ  తత్పురుష భుజభలం భుజాల యొక్క భలం షష్ఠి తత్పురుష పుర జనులు పురమునందు  జనులు సప్తమి తత్పురుష

విశేషణ పూర్వపద కర్మధారయ సమాసం : విశేషణం పూర్వపదంగా (మొదటి) ఉండే సమాసం .

ఉదా : – తెల్ల గుర్రం – తెల్లదైన  గుర్రం,  ఇస్టార్థములు – ఇష్టమైన అర్థములు

సంభావన  పూర్వపద కర్మధారయ సమాసం :  సమాసం లోని పూర్వపడం సంజ్ఞావాచాకంగా , ఉత్తరపదం జాతి వాచకంగా ఉంటుంది .

ఉదా :- కాశిక పట్టణం – కాశిక అను పేరు గల పట్టణం, తెలంగాణా రాష్ట్రము – తెలంగాణ అను పేరు గల రాష్ట్రం

నైతత్పురుష సమాసం :  వ్యతిరేఖ పదాన్ని ఇచ్చే పదం .

ఉదా :- అసత్యం – సత్యం కానిది , నిరాదారం – ఆదారం కానిది , అనుచితం – ఉచితం కానుది .

#### అలంకారాలు

అలంకారం : చెప్పదలచిన విషయాన్ని అందంగా మలిచేది.

అంత్యాను ప్రాస అలంకారం:  ఒకే అక్షరం లేదా రెండు , మూడు అక్షరాలు వాక్యం చివర మాటి మాటికి వస్తే దాన్ని  అంత్యాను ప్రాస అలంకారం అంటారు .

ఉదా : – భాగవతమున భక్తి – భారతమున యుక్తి – రామ కథయే రక్తి  ఓ కూనలమ్మ .

వృత్యానుప్రాస అలంకారం: ఒకటి గాని అంతకంటే ఎక్కువ గాని హల్లులు పలుమార్లు వచ్చునట్లు చెబితే  వృత్యానుప్రాస అవితుంది.

ఉదా : – వీరు పొమ్మను వారు  వారు పోగబెట్టు వారు

కాకి కోకికాదు దా

చేకానుప్రాస అలంకారం: అర్థ భేదం తో కూడిన హల్లుల జంట వెంట వెంటనే వస్తే చేక్కనుప్రాస అనబడుతుంది .

ఉదా : – అ నాథ నాథ  నంద నంద న నీకు వందనం

నీకు వంద వందనాలు.

లాటాను ప్రాస అలంకారం : అర్థంలో భేధం లేకపోయినా , తాత్పర్యంలో భేదం ఉండేటట్లు , ఒక పదం రెండు సార్లు ప్రయోగించబడితే లాతానుప్రాస అనబడుతుంది.

ఉదా : – కమలాక్షునకు అర్పించు కరములు కరములు

యమకం : అచ్చులలో హల్లులలో మార్పు లేనట్టి అక్షరాల సమూహం అర్థ భేదంతో మళ్ళీ  ప్రయోగిన్చినట్లయితే యమకం అనబడుతుంది.

ఉదా : –  పురము నందు నంతిపురము

ముక్త పద గ్రస్తo : విడిచి పెట్టబడ్డ పద భాగాలను వ్యవదానం లేకుండా వెంటనే ప్రయోగించి చెబితే ముక్త ప్రదగ్రస్తం .

ఉదా :- సుదతీ సూదన మదనా

మదనా గ తురంగ పూర్ణ మణి మాయ సదనా

సదనా మయ గజ రాదనా .

ఉపమాలంకారం : ఉపమేయానికి ఉపమానం తో చక్కని పోలిక వర్ణించబడిన యెడల ఉపమాన అలంకారం అనబడుతుంది .

ఉదా : – 1) చేనేత కార్మికులు ఎలుకల్ల మాడి పోతున్నారు    2) నీ కీర్తి హంష  లాగ ఆకాశ గంగలో మునుగుతుంది

రూపకాలoకారం : ఉపమానానినికి , ఉపమేయానినికి భేదం లేనట్లు వర్ణించి చెబితే రూపకాలoకారం అంటారు.

ఉదా : – 1) సంసార సాగరాన్ని తరించడం మిక్కిలి కష్టం   2 ) మౌనికి తేనె పలుకులు అందరికి ఇష్టమే

ఉత్ప్రేక్ష అలంకారం : ఉపమానానినికి ఉన్న ధర్మాలు  ఉపమేయంలో ఉండడం చేత , ఉపమేయాన్ని  ఉపమానo గా ఊహించి చెబితే  ఉత్ప్రేక్ష  అలంకారం అంటారు.

ఉదా : –  1) ఆ మేడలు ఆకాశాన్ని  ముద్దడుతున్నాయా అన్నట్లు ఉన్నాయి    2) ఈ వెన్నెల పాలవెళ్లి యో  అన్నట్లుంది .

అతి శయోక్తి అలంకారం : ఒక వస్తువు గురించి కాని సందర్భాన్ని గురించి కాని ఉన్నదాని కంటే ఎక్కువ చేసి చెబితే అతి శయోక్తి అంటారు.

ఉదా : – మా నగరం లోని మేడలు ఆకాశాన్ని అంటుతున్నాయి

శ్లేషాలo కారం : అనేకమైన అర్థాలు కల శబ్దాలను ఉపయోగించి చెబితే శ్లేష అనబడుతుంది

ఉదా : – రాజు కువలయానంద  కరుడు

రాజు = ప్రభువు , చంద్రుడు       కువలయం = భూమి , కలువ పూలు

స్వభావోక్తి అలంకారం :  జాతి గుణం క్రియాదు లలో ఉన్నది ఉన్నట్లు చెప్పడం

ఉదా : – చెట్ల ఆకులు గాలికి కదులుతున్నాయి

## ఛందస్సు

పద్య లక్షణాన్ని తెలిపే శాస్త్రాన్ని ఛందస్ శాస్త్రం అంటారు .

ఒక మాత్ర కాలం లో ఉచ్చరించబడేది లఘువు (I )

రెండు మాత్రల  కాలం లో ఉచ్చరించబడేది  గురువు ( U  )

య  గణం   IUU                                            జ  గణం    IUI

మ  గణం  UUU                                           భ  గణం    UII

త   గణం   UUI                                             న  గణం III

ర   గణం    UIU                                              స  గణం  IIU

 పద్యం పేరు గణాలు యతి స్తానం అక్షరాల సంఖ్య ఉత్పల మాల భ, ర, న, భ, భ, ర, వ 10 20 చంపక మాల న, జ, భ, జ, జ, జ, ర 11 21 శార్దూలం మ, స, జ, స, త, త, గ 13 19 మత్తేభం స, భ, ర, న, మ, య, వ 14 20

## వాఖ్య నిర్మాణము – రకాలు

వాక్యాలు మూడు రకాలు : 1 ) సామాన్య వాక్యము 2) సంశ్లిష్ట వాక్యము       3) సంయుక్త వాక్యము .

• సామాన్య వాక్యము :- క్రియ ఉన్నా  లేకున్నా ఒకే ఒక్క భావాన్ని ప్రకటించే వాక్యాలను సామాన్య వాక్యాలు అంటారు .

ఉదా : (i ) సీత బజారుకు వెళ్ళింది .    (ii) పాము కాటేసింది   (iii ) మురళి మంచి బాలుడు

• సంశ్లిష్ట వాక్యము:- ఒక సమాపక క్రియ , ఒకటి గాని అంతకన్నా ఎక్కువ గాని అసామాపక క్రియలు ఉంటే  ఆ వాక్యాన్ని    సంశ్లిష్ట వాక్యము అంటారు .

ఉదా :- (i )  రాము అన్నము తిని , పడుకున్నాడు         (ii )  సీత బజారుకు వెళ్లి , సరుకులు కొన్నది

• సంయుక్త వాక్యము:- సమ ప్రాధాన్యం కల వాక్యాలను కలపడం వల్ల ఏర్పడే వాక్యాo ను సంయుక్త వాక్యము అంటారు.

ఉదా :-  (i ) అతడు నటుడు, రచయిత    (ii ) రాము మరియు సిత హైదరాబాద్ వెళ్లారు    (iii ) సీత చదువుతుంది ,

## కర్తరి – కర్మణి వాఖ్యాలు

• కర్తరి వాక్యము :- ఒక వాక్యంలో కర్తకు ప్రాధాన్యం ఇచ్చి , కర్మకు ద్వితీయ విభక్తి (నిన్ , నున్ , లన్ , కూర్చి , గురించి ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.
• కర్మణి వాక్యము :-  ఒక వాక్యంలో క్రియకు ధాతువు చేరి  , కర్మకు తృతీయ  విభక్తి  ( చేతస్ , చేన్ , తోన్ , తోడన్ ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.

ఉదా :-

 కర్తరి వాక్యము కర్మణి  వాక్యము 1 ప్రజలు శాంతిని కోరుతున్నారు ప్రజలచే శాంతి కోరబడుతుంది 2 మేం పెద్దలను గౌరవిస్తాము మాచే పెద్దలు గౌరవించ బడతారు 3 రాజు రైలును నడిపాడు రైలు రాజు చే నడపబడింది 4 భీముడు కొండలను పిండి చేసాడు కొండలు భీముని చే పిండి చేయబడెను 5 నా మీద రాళ్ళు విసురుతారు నా మీద రాళ్ళు విసరబడతాయి

ప్రత్యక్ష – పరోక్ష కథనాలు
ప్రతక్ష కథనం : ఒకరు చెప్పిన విషయాన్ని ఉన్నది ఉన్నట్లుగా చెప్పడం . ఒకరు చెప్పిన విషయం  “     “  చిహ్నాల మద్య ఉందును.

ఉదా:-   i )  “ నేను రస జీవిని “  అని చాసో అన్నాడు    (ii ) అంబేత్కర్  “  నేను ఎవరిని యాచిన్చను  “ అని అన్నాడు

• పరోక్ష కథనం :  ఒకరు చెప్పిన విషయాన్ని మన మాటల్లో చెప్పడం .    ఇందులో   “    “  చిహ్నాలు ఉండవు .

ఉదా:-  (i )  తాను రస  జీవినని చాసో అన్నాడు    (ii ) అబ్మేత్కర్ తాను ఎవరినీ  యాచిన్చనని  అన్నాడు

గమనిక : ప్రత్యక్ష కథనం  నుండి పరోక్షం లోకి మార్చునప్పుడు  జరుగు మార్పులు :

 ప్రత్యక్షం పరోక్షం నేను తాను ఆయన అతను, వాడు అది ఇది నాకు తనకు నా తన నన్ను తనను మేము తాము మాకు తమకు ఇది అది

## 3.SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS

Square:  Square number is the number raised to the power 2. The number obtained by the number multiplied by itself.

Ex: – 1) square of 5 = 52 = 5 × 5 = 25, 2) square of 3 = 32 = 3× 3 = 9

∗If a natural number p can be expressed as q2, where q is also natural, then p is called a square number.

Ex: – 1,4,9, …etc.

Test for a number to be a perfect square:

If a number is expressed as the product of pairs of equal factors, then it is called a perfect square.

Ex: – 36

Prime factors of 36 = 2× 2× 3× 3

36 can be expressed as the product of pairs of equal factors.

∴ 36 is a perfect square.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make pairs of similar factors.
3. The product of prime factors, choosing one out of every pair gives the square root of the given number.

Ex: – To find the square root of 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

1. Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend.
4. Now, the new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root.

Ex: – To find the square root of 225

Properties of a perfect square:

1. The square of an even number is always an even number.

Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number.

2. The square of an odd number is always an odd number.

Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number.

3. The square of a proper fraction is a proper fraction less than the given fraction.

Ex: –

4. The square of decimal fraction less than 1 is smaller than the given decimal.

Ex: – (0.3)2 = 0.09 < 0.03.

5. A number ending with 2, 3, 7, or 8 is never a perfect square.

Ex: – 72, 58, 23 are not perfect squares.

6. A number ending with an odd no. of zeros is never a perfect square

Ex: – 20, 120,1000 and so on.

The square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root as soon as the integral part is exhausted.

Ex: – To find the square root of 79.21

The square root of a decimal number which is not perfect square:

if the square root is required to correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required to correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find the square root of 0.8 up to two decimal places

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is a perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27, and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted by

Methods of finding the cube root of the given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make triplets of similar factors.
3. The product of prime factors, choosing one out of every triplet gives the cube root of the given number.

Ex: – 27

Prim factors of 27 = 3×3×3

= 3

∴ cube root of 27 = 3

Test for a number to be perfect cube:

A given number is a perfect cube if it can be expressed as the product of triplets of equal factors.

Ex: – 2744

Prime factors of 2744 = 2×2×2 × 7×7×7

∴ 2744 is a perfect cube.

## ICSE IX Class Maths Concept

ICSE IX Class Maths Concept: This note is designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 9th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the ICSE 9th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of ICSE 9th

Math problems are asked in the CBSE board and entrance examinations.

## 1. RATIONAL AND IRRATIONAL NUMBERS

Natural numbers: counting numbers 1, 2, 3… called Natural numbers. The set of natural numbers is denoted by N.

N = {1, 2, 3…}

Whole numbers: Natural numbers including 0 are called whole numbers. The set of whole numbers denoted by W.

W = {0, 1, 2, 3…}

Integers: All positive numbers and negative numbers including 0 are called integers. The set of integers is denoted by I or Z.

Z = {…-3, -2, -1, 0, 1, 2, 3…}

Rational number: The number, which is written in the form of, where p, q are integers and q ≠ o is called a rational number. It is denoted by Q.

∗ In a rational number, the numerator and the denominator both can be positive or negative, but our convenience can take a positive denominator.

Ex: – $\inline&space;\fn_jvn&space;-\frac{2}{3}$ can be written as $\inline&space;\fn_jvn&space;\frac{-2}{3}=\frac{2}{-3}$  but our convenience we can take $\inline&space;\fn_jvn&space;\frac{-2}{3}$

Equal rational numbers:

For any 4 integers a, b, c, and d (b, d ≠ 0), we have $\inline&space;\fn_jvn&space;\frac{a}{b}=\frac{c}{d}$ ⇒ ad = bc

The order of Rational numbers:

If  are two rational numbers such that b> 0 and d > 0 then $\inline&space;\fn_jvn&space;\frac{a}{b}>&space;\frac{c}{d}$ ⇒ ad > bc

The absolute value of rational numbers:

The absolute value of a rational number is always positive. The absolute value of  $\inline&space;\fn_jvn&space;\frac{a}{b}$ is denoted by $\inline&space;\fn_jvn&space;\left&space;|&space;\frac{a}{b}&space;\right&space;|$ .

Ex: – absolute value of $\inline&space;\fn_jvn&space;-\frac{2}{3}=\frac{2}{3}$

To find rational number between given numbers:

• Mean method: – A rational number between two numbers a and b is $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{a&space;+&space;b}{2}$

Ex: – insert two rational number between 1 and 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{1&space;+&space;2}{2}$ < 2   ⟹     1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$  < 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$ $\inline&space;\dpi{120}&space;\fn_jvn&space;<&space;\frac{\frac{3}2{+2}}{2}$< 2   ⟹   1 < $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}<&space;\frac{7}{4}$ $\dpi{120}&space;\fn_jvn&space;<$  2

To rational numbers in a single step: –

Ex:- insert two rational numbers between 1 and 2

To find two rational numbers, we 1 and 2 as rational numbers with the same denominator 3

(∵ 1 + 2 = 3)

1 =   $\fn_jvn&space;\frac{1\times&space;3}{3}$  and 2 = $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{2\times&space;3}{3}$

$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{3}\left&space;(&space;1&space;\right&space;)<&space;\frac{4}{3}<&space;\frac{5}{3}<&space;\frac{6}{3}\left&space;(&space;2&space;\right&space;)$

Note: – there are infinitely many rational numbers between two numbers.

The decimal form of rational numbers

∗ Every rational number can be expressed as a terminating decimal or a non-terminating repeating decimal.

Converting decimal form into $\dpi{120}&space;\fn_jvn&space;\frac{p}{q}$  the form:

1. Terminating decimals: –

1.2 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{12}{10}=\frac{6}{5}$

1.35 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{135}{100}=\frac{27}{20}$

2. Non-Terminating repeating decimals: –

Irrational numbers:

• The numbers which are not written in the form of  $\dpi{120}&space;\fn_cm&space;\frac{p}{q}$, where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S.
• Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:- $\dpi{120}&space;\fn_cm&space;\sqrt{2},\,&space;\sqrt{5},\pi$ and so on.

• Calculation of square roots:
• There is a reference of irrationals in the calculation of square roots in Sulba Sutra.
• Procedure to finding $\dpi{120}&space;\fn_cm&space;\sqrt{2}$ value:

## ICSE X Class Maths

ICSE X Class Maths Concept designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 10th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the ICSE 10th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of 10th class Mathematics problems asked in the ICSE board and entrance examinations.

1. Goods and Service Tax

Two types of taxes in the Indian Government:

1.Direct taxes: –

These are the taxes paid by an organisation or individual directly to the government. These include Income tax, Capital gain tax and Corporate tax.

2.Indirect taxes: –

These are the taxes on goods and services paid by the customer, collected by an individual or an organisation and deposited with the Government. Earlier there were several indirect taxes levied by the central and state Governments.

Goods and Service Tax (GST):

GST is a comprehensive indirect tax for the whole nation. It makes India one unified common market.

Registration under GST:

Any individual or organisation that has an annual turnover of more than ₹ 20 lakh is to be registered under GST.

Input and Output GST:

For any individual or organisation, the GST paid on purchases is called the ‘Input GST’ and the GST collection on sale of goods is called the ‘Output GST’. The input GST is set off against the output GST and the difference between the two is payable in the Government account.

One currency one tax:

There is a uniform GST rate on any particular goods or services across all states and Union Territories of India. This is called ‘One currency one tax’.

Note: Assam was the first state to implement GST and Jammu & Kashmir was the last.

GST rate slabs:

However, the tax on gold is kept at 3% and on rough precious and semi-precious is kept at 0.25%.

The multitier GST tax rate system in India has been developed keeping in mind that essential commodities should be taxed less than luxury goods.

• Simple tax system.

• Elimination of multiplicity of taxes.

• Development of a common market nation-wide.

• Lower taxes result in the reduction of costs making in the domestic market.

Benefits of GST for Consumers:

• Single and transparent System.

• Elimination of cascading effect has resulted in the reduction in the costs of goods and services.

• Increase in purchasing power and savings.

• Single tax system, simple and easy to administer.

• Higher revenue efficiency.

• Better control on leakage and tax evasion.

Types of GST in India

Central GST (CGST): For any intrastate supply half of the GST collected as the output GST is deposited with the Central Governments as CGST.

State GST or Union Territory GST (SGST/UGST): For any local supply (supply with in the same state or Union Territory) half of the GST is deposited with the respective state or Union Territory Government as the beneficiary. This is called SGST/UGST.

Integrated GST (IGST): The GST levied on the supply of goods or services in the case of interstate trade within India or in the case of exports/imports is known as IGST.

Reverse charge Mechanism:

There are cases where the chargeability gets reversed, that is the receiver becomes liable to pay the tax and deposit it to the Government Account.

Composition shame:

The composition is meant for small dealers and service providers with an annual turnover less than ₹ 1.5 crores and also for Restaurant service providers. Under this scheme the rates of GST are:

Input Tax Credit (ITC)

When a dealer sells his goods, he charges the output GST from his customer which he has to deposit in the government account, but in running his business he had paid input GST on the goods he had availed. This input GST, he utilizes as Input Tax credit and deposits the exes amount of output GST with the Government.

Input Tax credit is a provision of reducing the GST already paid on inputs in order to avoid the cascading of taxes.

GST payable = Output GST – ITC

Claiming ITC: A dealer registered under GST can claim ITC only if:

• He possesses the tax invoice.
• He has received the said goods/services
• He has filed the returns.
• The tax paid by him has been paid to the government by his supplier.

Utilization of ITC:

The Amount of ITC available to any registered dealer shall be utilized to reduce the out put tax liability in the sequence shown in the table.

E – ledgers under GST:
An E – ledger is an electronic form of a pass book available to all GST registrants on the GST portal. These are of three types:

(i) Electric cash ledger (ii) Electric credit ledger and (iii) Electric Liability Register

(i) Electric cash ledger: It contains the amounts of GST deposited in each to the government.

(ii) Electric credit ledger: It contains the balance of ITC available to the dealer.

(iii) Electric credit ledger: It contains all the Tax liability of the dealer.

GST Returns:

These are the information provided from time to time by the dealer to the Government regarding the ITC, output Tax liability and the amounts of GST deposited.

A GST registered person has to submit the following returns:

E – Way bill:
E – Way bill is an electronic way bill that can be generated on the E – Way bill portal. A registered person can not transport goods whose value exceeds ₹ 50,000 in a vehicle without an e – way bill. When an E – way bill is generated, a unique e – way bill number (EBN) is allocated and is available to the supplier, the transporter and recipient. A dealer must generate an E – way bill if he has to transport them for returning to the supplier.

## 2.Banking

To encourage the habit of saving income groups, banks and post offices provide recurring deposit schemes.

Maturity period: An investor deposits a fixed amount every month for a fixed time period is called the maturity period,

Maturity value:  At the end of the maturity period, the investor gets the amount deposited with the interest. The total amount received by the investor is called Maturity value.

Interest =    $p&space;times&space;frac{n(n+1))}{2times&space;12}times&space;frac{r}{100}$

Where p is the principle

n is no. of months

r is the rate of interest

Maturity value = (p × n) + I

3.Shares and Dividend

Capital: The total amount of money needed to run the company is called Capital.

Nominal value (N.V): – The original value of a share is called the nominal value. It is also called as face value (F.V), printed value (P.V) or registered Value (R.V).

Market value: – The price of a share at a particular time is called market value (M.V). This value changes from time to time.

Shares: The whole capital is divided in to small units is called shares.

Share at par: – If the market value of a share is equal to face value of a share, then that share is called a share at par.

Share at a premium or Above par: – If the market value of a share is greater than the face value of the share then, the share is called share at a premium or above par.

Share at discount: – If the market value of a share is lesser than the face value of the share then, the share is called share at discount.

Dividend: – The profit distributed to the shareholders from a company at the end of the year is called a dividend.

The dividend is always calculated as the percentage of face value of the share.

Some formulae:

Note:

• The face value of a share always remains the same
• The market value of a share changes from time to time.
• Dividend is always paid on the face value of a share

4. Linear In equations

Linear inequations: A statement of inequality between two expressions involving a single variable x with highest power one is called linear inequation.

Ex: 3x – 3 < 3x + 5; 2x + 10 ≥ x – 2 etc.

General forms of Inequations: The general forms of the linear inequations are: (i) ax + b < c   (ii) ax + by ≤ c    (iii)  ax + by ≥ c    (iv) ax + by > c, where a, b and c are real numbers and a ≠ 0.

Domain of the variable or Replacement Set: The set form which the value of the variable x is replaced in an inequation is called the Domain of the variable.

Solution set: The set of all whole values of x from the replacement set which satisfy the given inequation is called the solution set.

Ex: Solution set of x < 6, x ∈ N is {1, 2, 3, 4, 5}

Solution set of x ≤ 6, x ∈ W is {0, 1, 2, 3, 4, 5, 6}

Inequations – Properties:

• Adding the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

3 + 2< 5 + 2

5 < 7 (no change in inequality)

• Subtracting the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

subtract 2 on both sides

3 – 2 < 5 – 2

1 < 3 (no change in inequality)

• Multiplying or Dividing the same positive number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × 2< 5 × 2

6 < 10 (no change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ 2< 8 ÷ 2

3< 4 (no change in inequality)

•Multiplying or Dividing the same negative number or expression to each side of an inequation can change(reverse) the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × –2< 5 × –2

–6 > –10 (change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ –2< 8 ÷ –2

–3 > –4 (change in inequality)

Note:

• a < b iff b > a
• a > b iff b < a

Ex: x < 4 ⇔ 4 > x

x > 3 ⇔ 3 < x

Method of solving Liner Inequations:

• Simplify both sides by removing group symbols and collecting like terms.
• Remove fractions by multiplying both sides by an appropriate factor.
• Collect all variable terms on one side and all constants on the other side of the inequality sign.
• Make the coefficient of the variable 1.
• Choose the solution set from the replacement set.

Ex: Solve the inequation 3x – 2 < 2 + x, x ∈ W

Sol: given in equation is

3x – 2 < 2 + x

3x – 2 + 2< 2 + x + 2

3x < 4 + x

3x – x < 4

2x < 4

Dividing both sides by 2

x < 2

∴ Solution set = { 0, 1}

Quadratic Equation: An equation of the form ax2 + bx + c = 0, where a, b, and c are real and a ≠ 0 is called a Quadratic equation in a variable ‘x’.

Ex: x 2 – 3x + 4 = 0 is a quadratic equation in a variable ‘x’

t2 + 5t = 6 is a quadratic equation in a variable ’t’

Roots of a quadratic equation: A number α is called a root of the quadratic equation ax2 + bx + c = 0, if aα2 + bα + c = 0.

Solution set:  The set of elements representing the roots of a quadratic equation is called solution set of the give quadratic equation.

Solving Quadratic equation by using Factorization met

hod:

Step – 1: Make the given equation into the standard form of ax2 + bx + c = 0.

Step – 2: Factorise ax2 + bx + c into two linear factors.

Step – 3: Put each linear factor equal to zero.

Step – 4: Solve these linear equations and get two roots of the given quadratic equation.

Ex: Solve x2 – 3x – 4 = 0

x2 – 4x + x – 4 = 0

x (x – 4) + 1 (x – 4) = 0

(x – 4) (x + 1) = 0

x – 4 = 0 or x + 1 = 0

x = 4 or x =– 1

∴ Solution set = {– 1, 4}

Solving Quadratic equation by using Formula:

The roots of the quadratic equation ax2 + bx + c = 0 are:

Ex: Solve x2 – 3x – 4 = 0

Sol: Given equation is x2 – 3x – 4 = 0

Compare with ax2 + bx + c = 0

a = 1, b = – 3, c = – 4

x = 4 or x = – 1

∴ Solution set = {– 1, 4}

Nature of the roots:

Discriminant: – For a quadratic equation ax2 + bx + c = 0, b2 – 4ac is called discriminant.

(i) If b2 – 4ac > 0, then roots are real and un equal.

Case – 1: b2 – 4ac > 0 and it is a perfect square, then roots are rational and unequal.

Case – 2: b2 – 4ac > 0 and it is not a perfect square, then roots are irrational and unequal.

(ii) If b2 – 4ac = 0, then roots are equal and real.

(iii) b2 – 4ac < 0, then roots are imaginary and un equal.

To solve word problems and determine unknown values, by forming quadratic equations from the information given and solving them by using methods of solving Quadratic equation.

The problems may be based on numbers, ages, time and work, time and distances, mensuration etc.

Method of Solving word problems in Quadratic equation:

Step – 1: Read the given problem carefully and assume the unknown be x.

Step – 2: Translate the given statement and form a quadratic equation in x.

Step – 3: Solve for x.

## 7.Ratio and Proportion

Ratio: Comparing two quantities of same kind by using division is called a ratio.

The ratio between two quantities ‘a’ and ‘b’ is written as a : b and read as ‘a is to b’

In the ratio a : b, ‘a’  is called ‘first term’ or ‘antecedent’ and ‘b’ is called ‘second term’ or ‘consequent’.

Note:  The value of a ratio remains un changed if both of its terms are multiplied or divided by the same number, which is not a zero.

Lowest terms of a Ratio:

In the ratio a : b, if a, b have no common factor except 1, then we say that a : b is in lowest terms.

Ex: 4 : 12 = 1 : 3 ( lowest terms)

Comparison of Ratios:

• (a : b) > (c : d) ⇔ ad > bc
• (a : b) = (c : d) ⇔ ad = bc
• (a : b) < (c : d) ⇔ ad < bc

Proportion:

An equality of ratios is called a proportion.

a, b, c and d are said to be in proportion if a : b = c : d and we write as a : b : : c : d.

a and d are ‘extremes’, b and c are ‘means’

product of extremes = product of means

Continued proportion: If a, b, c, d, e and f are in continued proportion, then

Mean proportion:  If then b2 = ac or b =  , b is called mean proportion between a and b.

Third proportional: If a : b = b : c, then c is called third proportional to a and b.

Note:

Results on Ratio and Proportion:

8.Remainder Theorem and Factor Theorem

Polynomial: An expression of the form p(x) = a0 xn + a1 xn-1 + a2 xn-2 + …+ an-1 x + an, where a0, a1, …, an are real numbers and a0 ≠ 0. Is called a polynomial of degree n.

Value of a polynomial: The value of a polynomial p(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by p(a).

Ex: If p(x) = 2x + 3, then find the value of p (1), p (0).

Sol: given p(x) = 2x + 3

p (1) = 2 (1) + 3 = 2 + 3 = 5

p (0) = 2 (0) + 3 = 0 + 3 = 3

Division algorithm: On dividing a polynomial p(x) by a polynomial g(x), there exist quotient polynomial q(x) and remainder polynomial r(x) then

p(x) = g(x) q(x) + r(x)

p(x) is dividend; g(x) is divisor; q(x) is quotient; r(x) is remainder.

Remainder theorem:

If a polynomial p(x) is divided by (x – a), then the remainder is p(a).

Ex: If p(x) = 2x – 1 is divided by (x – 3), then find reminder.

Sol: Given p(x) = 2x – 1

Remainder = p (3)

= 2(3) – 1

= 6 – 1 = 5

∴ remainder is 5

Note:

• If p(x) is divided by (x + a), then the remainder is p (– a).
• If p(x) is divided by (ax + b), then remainder is .
• If p(x) is divided by (ax – b), then remainder is .

Factor theorem: Let p(x) be a polynomial and ‘a’ be given real number, then (x – a) is a factor of p(x) ⇔ p(a) = 0.

Note:

• If (x + a) is the factor of p(x), then p (– a) = 0.
• If (ax + b) is the factor of p(x), then  = 0.
•  If (ax – b) is the factor of p(x),    = 0

9. Matrices

Matrix: A rectangular arrangement of numbers in the form of horizontal and vertical lines and enclosed by the brackets [ ] or parenthesis ( ), is called a matrix.

The horizontal lines in a matrix are called its rows.

The vertical lines in a matrix are called its columns.

Oder of Matrix: A matrix having ‘m’ rows and ‘n’ columns is said to be of order m x n read as m by n.

Ex:

Elements of a matrix:

An element of a matrix appearing in the ith row and jth column is called the (i, j)th element of the matrix and it is denoted by aij.

A = [aij]m × n

A =

a11 means element in first row and first column

a12 means element in first row and second column

a22 means element in second row and second column

a32 means element in third row and second column

and so on.

Types of Matrices

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called column matrix.

Ex:

Rectangular Matrix: A matrix in which the no. of rows is not equal to no. of columns is called Rectangular matrix.

Ex:

Square Matrix: A matrix in which the no. of rows is equal to no. of columns is called square matrix.

Ex:

Diagonal Matrix: If each non-diagonal elements of a square matrix is ‘zero’ then the matrix is called diagonal matrix.

Ex:

Identity Matrix or Unit Matrix: If each of non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called unit matrix

Ex:

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called null matrix.

Ex:

Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.

Ex: If  ⟹ a=p; b = q; c = r; d = s

Comparing Matrices: Comparison of two matrices is possible, if they have same order.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by   AT.

Ex:

Addition of Matrices: If A and B are two matrices of the same order, then their sum A + B is the matrix obtained by adding the corresponding elements of A and B.

Ex:

Subtraction of Matrices: If A and B are two matrices of the same order, then their difference A + B is the matrix obtained by subtracting the elements of B from the corresponding elements of A.

Ex:

Product of Matrices:

Let A = [aik]mxn and B = [bkj]nxp be two matrices, then the matrix C = [cij]mxp   where

Note: Matrix multiplication of two matrices is possible when no. of columns of first matrix is equal to no. of rows of second matrix.

## 10. Arithmetic Progressions

Sequence:  The numbers which are arranged in a different order to some definite rule are said to form a sequence.

Ex: 1, 2, 3, ……

2, 4, 6, 8, ….

2, 4, 8, 16, …

Arithmetic Progression (A.P.):

A sequence in which each term differ from its preceding term by a constant is called an Arithmetic Progression (A.P.). The constant difference is called the common difference.

Terms: a, a + d, a + 2d…, a + (n – 1) d

First term: a

Common Difference: d = a2 – a1 = a3 – a2 = … = an – an -1

nth term: Tn = a + (n – 1) d

Sum of the n terms of A.P.:

Sum of the n terms of A.P. is

Where a is first term and l is last term.

To find the nth term from the end of an A.P.:

Let a be first term, d be the common difference and ‘l’ be the last term of a given A.P. then its nth term from the end is l – (n – 1) d .

## 11. Geometric Progressions

Terms: a, a r, a r2…, a rn – 1

First term: a

Common ratio:

nth term: Tn = a rn – 1

Sum of the n terms of G.P.:

Sum of the n terms of G.P. is

To find the nth term from the end of an G.P.:

Let a be first term, r be the common ratio and ‘l’ be the last term of a given G.P. then its nth term from the end is

## 12. Reflection

Coordinate Axes:

The position of the point in a plane is determined by two fixed mutually perpendicular lines XOX’ and YOY’ intersecting each other at ‘O’. These lines are called coordinate axes.

The horizontal line XOX’ is called X – axis.

The vertical line YOY’ is called Y – axis.

The point of intersection axes is called ‘origin’.

Coordinates of a point:

Let P be any point on the plane, the distance of P from X – axis is ‘x’ units and the distance of P from Y – axis is ‘y’ units, then we say that coordinates of P are (x, y).

x is called x coordinate or abscissa of P

y is called y coordinate or ordinate of P

The distance of any point on X – axis from X – axis is 0

∴ Any point on the X – axis is (x, 0)

The distance of any point on Y – axis from Y – axis is 0

∴ Any point on the Y – axis is (0, y).

The coordinates of the origin O are (0, 0).

The equation of X – axis is y = 0.

The equation of any line parallel to X – axis is y = k, where k is the distance from X – axis.

The equation of Y – axis is x = 0.

The equation of any line parallel to Y – axis is x = k, where k is the distance from Y – axis.

Reflection

Image of an object in a mirror: When an object is placed in front of a plane mirror, then its image is formed at the same distance behind the mirror as the distance of the object from the mirror.

Image of a point in a line:

Let P be a point and AB is a given line. Draw PM perpendicular to AB and produce PM

to Q such that PM = QM, then Q is called image of P with respect to the line AB.

Reflection of a point in a line:

Assume the given line as a mirror, the image of a given point is called the reflection of that point in the given line.

Reflection of P (x, y) in X – axis is P (x, –y) ⇒ Rx (x, y) = (x, –y)

Reflection of P (x, y) in Y – axis is P (–x, y) ⇒ Rx (x, y) = (–x, y)

Reflection of P (x, y) in the origin is P (–x, –y) ⇒ Rx (x, y) = (–x, –y)

Combination of Reflection:

• Rx. Ry = Ry. Rx = Ro
• Rx. Ro = Ro. Rx = Ry
• Ry. Ro = Ro. Ry = Rx

Invariant Points: A point P is said to be invariant in a given line if the image of P (x, y) in that line is P (x, y).

## 13. Section and Mid – Point Formula

Section formula: If P (x, y) divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m : n, then

P (x, y) =

Mid pint Formula:

The mid-point of the line segment joining the points A (x1, y1) and B (x2, y2) is

Centroid of the triangle:

The point of concurrence of medians of a triangle is called centroid of the triangle. It is denoted by G.

The centroid of the triangle formed by the vertices A (x1, x2), B (x2, y2) and C (x3, y3) is

G =

## 14. Equation of a Straight line

Inclination of a line: The angle of inclination of a line is the angle θ which is the part of the line above the X – axis makes with the positive direction of X – axis and measured in anticlockwise direction.

Horizontal line: A line which is parallel to X – axis is called horizontal line.

Vertical line: A line which is parallel to Y – axis is called vertical line.

Oblique line: A line which is neither parallel to X – axis nor parallel to Y – axis is called an oblique line.

Slope or Gradiant of a line:

A line makes an angle θ with the positive direction of x – axis then tan θ is called the slope of the line, it is denoted by ‘m’

m = tan θ

1. Slope of the x- axis is zero.
2. Slope of any line parallel to x- axis is zero.
3. Slope of y- axis is undefined.
4. Slope of any line parallel to y- axis is also undefined.
5. Slope of the line joining the points A (x1, y1) and B (x2, y2) is

1. Slope of the line ax + by + c = 0 is  =

Condition for collinearity: If three points A, B and C are lies on the same line then they are collinear points.

Condition for the collinearity is slope of AB = Slope of BC = slope of AC

Types of equation of a straight line:

• Equation of x- axis is y = 0.
• Equation of any line parallel to x – axis is y = k, where k is distance from above or below the x- axis.
• Equation of y- axis is x = 0.
• Equation of any line parallel to y – axis is x = k, where k is distance from left or right side of the y- axis.

Slope intercept form:

The equation of the line with slope m and y- intercept ‘c’ is y = mx + c.

Slope point form:

The equation of the line passing through the point (x1, y1) with slope ‘m’ is

y – y1 = m (x – x1)

Two points form:

The equation of the line passing through the points (x1, y1) and (x2, y2) is

Intercept form:

The equation of the line with x- intercept a, y – intercept b is

Note: –

1. If two lines are parallel then their slopes are equal

m1 = m2

1. If two lines are perpendicular then product of their slopes is – 1

m1 × m2 = – 1

slope of a line perpendicular to a line AB =

15. Similarity

Similar figures: If two figures have same shape but not in size, then they are similar.

Similarity as a Size transformation:

It is the process in which a given figure is enlarged or reduced by a scale factor ‘k’, such that the resulting figure is similar to the given figure.

The given figure is called an ‘object’ and the resulting figure is called its ‘image’

Properties of size transformation:

Let ‘k’ be the scale factor of a given size transformation, then

If k > 1, then the transformation is enlargement.

If k = 1, then the transformation is identity transformation.

If k < 1, then the transformation is reduced.

Note:

• Each side if the resulting figure = k times the corresponding side of the given figure.
• Area of the resulting figure = k2 × (Area of the given figure).
• Volume of the resulting figure = k3 × (Volume of given figure).

Model: The model of a plane figure and the actual figure are similar to one another.

Let the model of the plane figure drawn to the scale 1 : n, then scale factor k =  .

• Length of the model = k × (length of the actual figure)
• Area of the model = k2 × (Area of the actual figure).
• Volume of the model = k3 × (Volume of actual figure).

Map: The model of a plane figure and the actual figure are similar to one another.

Let the Map of the plane drawn to the scale 1 : n, then scale factor k =  .

• Length of the map = k × (Actual length)
• Area of the model = k2 × (Actual area).

16. Similarity of Triangles

Similar triangles: Two triangles are said to be similar If: (i) their corresponding angles are equal and (ii) their corresponding sides are in proportional (in same ratio).

If ∆ABC ~ ∆DEF, then

• ∠A = ∠D, ∠B = ∠E and ∠C = ∠F

Here k is scale factor, (i) if k > 1, then we get enlarged figures (ii) if k = 1, then we get congruent figures (iii) if k < 1, then we get reduced figures.

Axioms of Similar Triangles

SAS – Axiom:

If two triangles have a pair of corresponding angles equal and the corresponding sides including them proportional, then the triangle is similar.

In ∆ABC and ∆DEF, ∠A = ∠D and

AA – Axiom : If two triangles have two pairs of corresponding angles equal, then the triangles are equal.

SSS – Axiom: If two triangles have their three sides of corresponding sides proportional, then the triangles are similar.

Basic proportionality Theorem:

In a triangle a line drawn parallel to one side divides the other two sides in the same ratio (proportional).

In ∆ ABC, DE ∥ BC ⇒

Converse of Basic proportionality Theorem:

If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.

In ∆ ABC,  ⇒ DE ∥ BC.

∎ Ina triangle the internal bisector of an angle divides the opposite side in the ratio of the sides containing the angle.

In ∆ ABC, AD bisects ∠A then

∎ The areas of two similar triangles are proportional to the squares of their corresponding sides.

If ∆ABC ~ ∆DEF, then

∎ The areas of two similar triangles are proportional to the squares of their corresponding medians.

If ∆ABC ~ ∆DEF and AM, DN are medians of   ∆ABC and ∆DEF respectively then

∎ The areas of two similar triangles are proportional to the squares of their corresponding altitudes.

If ∆ABC ~ ∆DEF and AM, DN are altitudes of ∆ABC and ∆DEF respectively then

## 17. Loci

Locus: Locus is the path traced out by a moving point which moves according to some given geometrical conditions.

The plural form of Locus is ‘Loci’ read as ‘losai’

∎ The locus of the point which is equidistance from two given fixed points is the perpendicular bisector of the line segment joining the given fixed       points.

∎ Every point on the perpendicular bisector of AB is equidistance from A and B.

∎ The locus of the point which is equidistance from two intersecting lines is the pair of lines bisecting the angles formed by the given lines.

∎Every point on the angular bisector of two intersecting lines is equidistance from the lines.

## 18. Angle and Cyclic Properties of a Circle

∎ The angle subtended by an arc of a circle is double the angle subtended by it at any point on the circle.

∠AOB = 2 ∠ACB

∎ Angles in a same segment of a circle are equal.

∎ The angle in a semi-circle is 900

∎ If an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is semi-circle.

A quadrilateral is said to be cyclic if all the vertices passing through the circle.

The opposite angles of cyclic quadrilateral are supplementary.

∎ If pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

∎ The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

∎ Every cyclic parallelogram is a rectangle.

∎ An isosceles trapezium is always cyclic and its diagonals are equal.

∎ The mid-point of hypotenuse of a right-angled triangle is equidistance from its vertices.

19. Tangent Properties of Circles

Tangent: A line which intersect the circle at only one point is called Tangent to the circle.

∎ The tangent at any point of a circle and radius through the point are perpendicular to each other.

∎ If two tangents are drawn to a circle from an exterior point, then

• The tangents are equal in length.
• The tangents subtend equal angle at the centre.
• The tangents are equally inclined to the line joining the point and the centre if the circle.

Intersecting Chord and Tangents

Segment of a chord:

If P is a point on a chord AB of a circle, then we say that P divides AB internally into two segments PA and PB.

If AB is a chord of a circle and P is a point on AB produced, we say that P divides AB externally into two segments PA and PB.

∎ If two chords of a circle intersect internally or externally, then the product of the lengths of their segments are equal.

Alternate segments:

In the given figure APB is a tangent to the circle at point at a point P and PQ is a chord

The chord PQ divides the circle into two segments PSR and PSQ are called alternate segments.

The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

∠QPB = ∠PSQ and ∠APQ = ∠PRQ

20. Constructions

21. Volume and Surface Area of Solids

Cylinder:

Solids like circular pillars, circular pipes, circular pencils etc. are said to be in cylindrical shape.

Radius of the base = r

Height of the cylinder = h

Curved surface area = 2πrh sq. units

Total surface area = 2πr (r + h) sq. units

Volume = πr2h cubic. units

Hollow Cylinder:

Height = h

Thickness of the cylinder = R – r

Area of cross section = π (R2 – r2) sq. units

Volume of material = πh (R2 – r2) cubic. units

Curved surface area = 2πh (R+ r) sq. units

Total surface area = 2π (Rh + rh + R2 – r2) sq. units

Cone:

Radius of the base = r

Height of the cylinder = h

Slant height = l

l2 = r2 + h2 ⇒ l =

Curved surface area = πrl sq. units

Total surface area = πr (r + l) sq. units

Volume = πr2h cubic. Units

Sphere:

Objectives like football, throwball, etc. are said to be the shape of sphere.

Surface area = 4πr2 sq. units

Volume = πr3 cubic. Units

Spherical Shell:

The solid enclosed between two concentric spheres is called spherical shell

Thickness of the cylinder = R – r

Volume of the material =  π (R3 – r3) cubic. Units

Hemi sphere:

Curved Surface area = 2πr2 sq. units

Surface area = 3πr2 sq. units

Volume = πr3 cubic. Units

22.Trigonometrical Identities

The word Trigonometry derived from Greek word, tri three, gonia angle and metron to measure.

Angle: – The figure formed by two rays meeting at a common end point is an angle.

Naming the sides in a right-angled triangle:

AB = Perpendicular =opposite side of θ (opp)

AC is hypotenuse (hyp)

Trigonometric ratios:

Quotient relations:

Trigonometric Identities:
(i) sin
2θ + cos2θ = 1          (ii) sec2θ − tan2θ = 1        (iii) cosec2θ − cot2θ = 1

∎ sin2θ = 1 − cos2θ; cos2θ = 1 – sin2θ

∎ sec2θ = tan2θ + 1; sec2θ – 1 =tan2θ

∎ Cosec2θ = Cot2θ + 1; Cosec2θ – 1 = Cot2θ

Trigonometric tables:

A trigonometric Table Consist of three parts:

• A column on the extreme left containing degree from 00 to 890.
• The columns headed by 0’, 6’, 12’, 18’, 24’, 30’, 36’, 42’, 48’ and 54’.
• Five columns of mean differences, headed by 1’, 2’, 3’, 4’ and 5’

Note:

• The mean difference is added in case of ‘sines’, ‘tangents’ and ‘secants’
• The mean difference is subtracted in case of ‘cosines’, ‘cotangents’ and ‘cosecants’

Relation between degrees and minutes:

10 = 60’⇒ 1’ =

Trigonometric Tables Charts:

By clicking below you get Sin, Cosine and Tangent Tables

Taking too long?

| Open in new tab

23. Heights and Distances

Horizontal line: A line which is parallel to earth from observation point to object is horizontal line
Line of sight: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

Angle of elevation: The line of sight is above the horizontal line and angle between the line of sight and horizontal line is called angle of elevation.

Angle of depression: The line of sight is below the horizontal line and angle between the line of sight and horizontal line is called angle of depression.

Solving procedure:

∎All the objects such as tower, trees, buildings, ships, mountains etc. shall be consider as linear for mathematical convenience.

∎The angle of elevation or angle of depression is considered with reference to the horizontal line.

∎The height of observer neglected, if it is not given in the problem.

∎To find heights and distances we need to draw figures and with the help of these figures we can solve the problems.

# 24. Graphical Representation of Statistical Data

Data: A set of given facts in numerical figures is called data.

Frequency: The number of times an observation occurs is called its frequency.

Frequency Distribution: The tabular arrangement of data showing the frequency of each observation is called its frequency distribution.

Class interval: Each group into which the raw data is condensed is called a class interval.

Class limits: Each class interval is bounded by two figures is called Class limits.

Left side part of class limit is called ‘Lower limit’

Right side part of class limit is called ‘Upper limit’

Inclusive form:  In each class, the data related to both the lower and upper limits are included in the same class, is called Inclusive form.

Ex: 1 – 10, 11 – 20, 21 – 30 etc.

Exclusive form: In each class, the data related to the upper limits are excluded is called Exclusive form.

Ex: 0 – 10, 10 – 20, 20 – 30 etc.

Class size = upper limit – lower limit

Class mark =  [lower limit + upper limit]

Note:

In an inclusive form, Adjustment factor =  [lower limit of one class – upper limit of previous class]

Histogram:  A histogram is a graphical representation of a frequency distribution in an exclusive form, in the form of rectangles with class interval as bases and the corresponding frequencies as heights

Method of drawing a Histogram:

Step-1:  If the given frequency distribution is in inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Example:

Frequency polygon:

Let x1, x2, x3, …, xn be the class marks of the given frequency distribution and f1, f2, f3, …, fn be the corresponding frequencies, then plot the points (x1, f1), (x2, f2), …. (xn, fn) on a graph paper and join these points by a line segment. complete the diagram in the form of polygon by taking two or more classes.

Example:

Cumulative Frequency curve or Ogive:

In order to represent a frequency distribution by an Ogive, we mark the upper class along X– axis and the corresponding cumulative frequencies along Y – axis and join these points by free hand curve, called Ogive.

Example:

25. Measures of Central Tendency

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Mean

Mean for Un Grouped data:

The mean of ‘n’ observations x1, x2, x3, …, xn is

Mean =

The Symbol Σ is called ‘sigma’ stands for summation of the data.

Note:

If the mean of a data x1, x2, … xn is m, then

• Mean of (x1+k), (x2 + k), …. (xn + k) = m + k
• Mean of (x1−k), (x2 − k), …. (xn − k) = m −k
• Mean of (k x1), (k x2), …. (k xn) = k m

If x1, x2, …. xn are of n observations occurs f1, f2, …. fn times respectively then mean is

Mean of grouped data:

Methods of finding mean:

Class mark (mid value) =

Direct method: ;  xi is class mark of ith class, fi is frequency of class.

Assumed mean method: ;  di = xi – a and a is assumed mean.

Step – deviation method:  ; 𝛍i =  , h is class size.

26. Median, Quartile and Mode

Median

Median is the middle most observation of given data.

For un grouped data:

First, we arrange given observations into ascending or descending order.

If n is odd median =  observation.

If n is even median =

For grouped data:

Median = , where

l is the lower boundary of median class

f is the frequency of median class

c.f is the preceding cumulative frequency of the median class

h is the class size

Quartiles

The observations which divides the whole set of observations into 4 equal parts are known as Quartiles.

Lower Quartile (First Quartile): If the variates are arranged in ascending order, then the observations lying midway between the lower extreme and the median is called the Lower Quartile. It is denoted by Q1.

If n is Even Q1 observation

If n is Odd Q1 =  observation

Middle Quartile: The middle Quartile is the median, denoted by Q2.

Upper Quartile (Third Quartile): If the variates are arranged in ascending order, then the observations lying midway between and the median the upper extreme is called the Upper Quartile. It is denoted by Q3.

If n is Even Q =  observation

If n is Odd Q1 = observation

Range: The difference between the biggest and the smallest observations is called the Range.

Interquartile Range: The difference between the upper quartile and Lower quartile is called the inter quartile.

Range = Q3 – Q2

Semi – interquartile range:

Semi – interquartile Range = ½ [ Q3 – Q2]

Estimating median:

Step 1: If the given frequency distribution is not continuous, convert into the continuous form.

Step 2: Prepare the cumulative frequency table.

Step 3: Draw Ogive for the cumulative frequency distribution given above

Step 4: Let sum of the frequencies = N.

Step 5: Mark a point A on Y- axis corresponding to

Step 6: From A draw Horizontal line to meet Ogive curve at P. From P draw a vertical line PM to meet X – axis at M. Then the abscissa of M gives the Median.

Estimating Q1 and Q2:

To locate the value of Q1 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

To locate the value of Q3 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

Mode

The value of a data which is occurred most frequently is called Mode.

Modal class: The class with maximum frequency is called the Modal class.

Estimation of Mode from Histogram:

Step 1: If the given frequency distribution is not continuous, convert it into a continuous form.

Step 2: Draw a histogram to represent the above data.

Step 3: from the upper corner of the highest rectangle, draw line segments

To meet the opposite corners of adjacent rectangles, diagonally

Let these line segments intersect at P.

Step 4: Draw PM perpendicular to X-axis at M, Then the abscissa of M is The Mode

27. Probability

J Cordon Italian mathematician wrote the first book on probability named “the book of games and chance”.

Probability:

It is the concept which numerically measures the degree of certainty of the occurrence of an event.

Some words in probability:

Experiment: A repeatable procedure with a set of possible results.

Trial: By a trial, we mean experimenting.

Outcome: a possible result of an experiment.

Sample space: All the possible outcomes of an experiment.

Sample point: Just one of the possible outcomes.

Event: One or more outcomes of an experiment.

Probability of occurrence of an Event (Classical definition):

In a random experiment Let S be the sample space and E be the event, then E ⊆ S. The probability of occurrence of E is defined as:

P(E) =

Deck of cards: A deck of playing cards consists of 52 cards which are divided into 4 suits of 13 cards each. They are black spade     , black clubs, red heart     and red diamond       . The cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9 ,10, Ace, Jack, Queen and King. Jack, Queen and King are called face (picture) cards.

Impossible event: If there is no probability of an event to occur then it is impossible event. Its probability is zero.

Sure or certain event: If the probability of an event is 1 then it is sure or certain event.

Complimentary event: Let E denote the event, ‘not E’ is called complimentary event of E. It is denoted by  . P ( ) = 1 – P(E) ⟹ P ( ) +P(E) = 1.

0 ≤ P(E) ≤ 1

## CBSE 10th Class Maths Concept

CBSE 10th Class Maths: This concept note is designed for CBSE 10th class maths students, This concept notes is to help students for the CBSE board examination and other competitive exams also…

### 1. REAL NUMBERS

Rational number: The number, which is written in the form of is called a rational number. It is denoted by Q.

Irrational number: – the number, which is not rational is called an irrational number. It is denoted by Q’ or S.

Prime number: – The number which has only two factors 1 and itself is called a prime number. (2, 3, 5, 7 …. Etc.)

Composite number: – the number which has more than two factors is called a composite number. (4, 6, 8, 9, 10… etc.)

Co-prime numbers: – Two numbers are said to be co-prime numbers, if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7) …etc.]

Euclid division lemma: – For any positive integers a and b, then q, rare integers exist uniquely satisfying the rules a = bq + r, 0 ≤ r < b.

### To find H.C.F by using Euclid’s division lemma:

• For any two integers a and b (a > b).  Apply Euclid division lemma, to a and b, we find whole numbers q and r such that a = bq + r, 0≤r<b.
• If r = 0, b is the H.C.F of a and b. If r≠ 0, apply Euclid division lemma, to b and r.
• Continue the process till the remainder is zero. The divisor at this stage is the required H.C.F.

#### Note: –

• Euclid division lemma also called a division algorithm.
• Euclid division lemma is stated for only positive integers, it can be extended for all integers except 0.

### The fundamental theorem of arithmetic:

Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which prime factors occur.

Ex: – 12 = 2 ×2× 3, 15 = 3× 5 and so on.

CBSE 10th Class Maths Concept

#### To find LCM and HCF by using the prime factorization method:

H.C.F = product of the smallest power of each common prime factor of given numbers.

L.C.M = product of the greatest power of each prime factor of given numbers.

• ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a.
• Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals.

#### Decimal expansion of rational numbers:

Decimal expansion of rational numbers is either terminating or non-terminating repeating (recurring)decimals.

Ex: – 1.34, 2.345, 1.2222… and so on.

In p/q, if the prime factorization of q is in form 2m 5n, then p/q is a terminating decimal. Otherwise non-terminating repeating decimal.

##### Decimal expansion of irrational numbers:

Decimal expansion of irrational numbers is non-terminating decimals.

Ex: – 1.414…., 1.314….

TS 10th class maths concept (E/M)

Ts Inter Maths IA Concept

PDF Files || Inter Mathematics 1A and 1B

## 12 th Class Maths Concept

12 th Class Maths: This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 12th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the CBSE 12th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of 12thMath problems asked in the CBSE board and entrance examinations.

12 th Class Maths

#### 1. RELATIONS AND FUNCTIONS

Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

Cartesian product: Let A and B are two non- empty sets. The Cartesian product of A and B is denoted by A × B and is defined as set of all ordered pairs (a, b) where a ϵ A and b ϵ B.

Relation: Let A and B are two non-empty sets the relation R from A to B is subset of A×B.

⇒ R: A→B is a relation if         R⊂ A × B

Types of relations:

Empty relation: – A relation in a set A is said to be empty relation, if no element of A is related to any element of A.

R = ∅ ⊂ A × A

Universal Relation: – A relation in a set A is said to be universal relation, if each element of A is related to every element of A

R = A × A

Both empty relation and universal relation are sometimes called trivial relations.

Reflexive relation:  A relation R in a set A is said to be reflexive,

if ∀a ∈ A ⇒ (a, a) ∈ R.

Symmetric relation:  A relation R in a set A is said to be symmetric,

if ∀a, b ∈ A ⇒, (a, b) ∈ R ⇒ (b, a) ∈ R.

Anti-Symmetric relation:  A relation R in a set A is said to be Anti-symmetric,

if ∀a, b ∈ A; (a, b) ∈ R (b, a) ∈ R ⇒ a = b.

Transitive relation:  A relation R in a set A is said to be Transitive,

if ∀a, b, c∈ A; (a, b) ∈ R (b, c) ∈ R ⇒ a = c.

Equivalence relation: A relation R in a set A is said to equivalence, if it is reflexive, symmetric and transitive.

Function: A relation f: X → Y is said to be a function if ∀ xϵ X there exists a unique element y in Y such that (x, y) ϵ f.

(Or)

A relation f: A → B is said to be a function if (i) x ϵ X ⇒ f(x) ϵ Y

(ii)  x1, x2 ϵ X, x1 = x2 in X ⇒ f(x1) = f(x2) in Y.

TYPES OF FUNCTIONS

One– one Function (Injective): – A function f: X→ Y is said to be one-one function or     injective if different elements in X have different images in Y.

(Or)

A function f: X→ Y is said to be one-one function if f(x1) = f(x2) in Y ⇒x1 = x2 in X.

On to Function (Surjection): – A function f: X→ Y is said to be onto function or surjection if for each yϵ Y there exists x ϵ X such that f(x) = y.

Bijection: – A function f: X→ Y is said to be Bijection if it is both ‘one-one’ and ‘onto’.

Composite function:  If f: A→B, g: B→C are two functions then the composite relation gof is a function from A to C.

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).

## 11th Class Maths Concept

11th Class Maths: This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 11th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the CBSE 11th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of11thMath problems asked in the CBSE board and entrance examinations.

## 1. SETS

Well-defined objects:

1. All objects in a set must have the same general similarity or property.
2. Must be able to confirm whether something belongs to the set or not.

Set: – A collection of well-defined objects is called a set.

∗ Sets are usually denoted by capital English alphabets like A, B, C, and so on.

∗ The elements in set are taken as small English alphabets like a, b, c, and so on.

∗ Set theory was developed by George canter.

• If any object belongs to a set, then it is called an object/element. We denote by ∈ to indicate that it belongs to. If it does not belong to the set then it is denoted by ∉.

Ex: – 1 ∈ N, 0 ∈ W, −1 ∈ Z, 0 ∉ N, etc.

Methods of representing sets:

Roster or table or listed form: –

In this form all the elements of the set are listed, and the elements are separated by commas and enclosed within braces { }.

Ex: – set of vowels in English alphabet = {a, e, I, o, u},

set of even natural numbers less than 10 = {2, 4, 6, 8} etc.

Note: – In roster form, an element is not repeated.  We can list the elements in any order.

Set builder form:

Pointing an element in a set to x (or any symbols such as y, z, etc.) followed by a colon(:), next to write the properties or properties of the elements in that set and placed in flower brackets is called the set builder form.: Or / symbols read as ‘such that’

Ex: – {2, 4, 6, 8} = {x / x is an even and x ∈N, x< 10},

{a, e, i, o, u} = {x : x is a vowel in English alphabet}.

Null set: – (empty set or void set) the set which has no elements is called as a null set. It is denoted by ∅ or { }.

Finite and infinite sets: – If a set contains a finite no. of elements then it is called a finite set. If a set contains an infinite no. of elements then it is called an infinite set.

Ex: – A = {1, 2,3, 4} → finite set

B = {1, 2, 3, 4….}

Equal sets: – two sets A and B are said to be equal sets if they have the same elements., and write as A = B

Ex: – A = {1, 2, 3, 4}, B = {3, 1, 4, 2}

⟹ A = B.

Subset: – for any two sets A and B, if every element of set A is in set B, then we can say that A is a subset of B. It is denoted by A ⊂ B.

Ex: – If A = {1, 2, 3, 4, 5, 6, 7, 8}, subsets of A are {1}, {1, 3, 5}, {1,2,3,4}, and so on.

Power set: – set of all the subsets of a set A is called the power set of A. It is denoted by p(A).

Ex: – A = {1,2,3}

P(A) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, ∅}.

Intervals:

∗ Open interval: – (a, b) = {x: a< x <b} → set of rational numbers lies between a and b.

∗ Closed interval: – [a, b] = {x: a≤ x ≤b} → set of rational numbers lies between a and b, including a and b.

∗ Open – closed: – (a, b] = {x: a< x ≤b} → set of rational numbers lies between a and b, excluding a and including b.

∗ Closed-open: -[a, b) = {x: a≤ x <b} → set of rational numbers lies between a and b, including a and excluding b.

Universal set: – A set that contains all the subsets of it under our consideration is called a universal set.

Cardinal number of a set: – Number of elements in a set A is called the cardinal number of that set A. It is denoted by n(A).

• If a set has n elements, then no. of elements of that set has 2n

Equivalent sets: – two set A and B are said to be equivalent sets if n(A) = n(B) (they have the same cardinal number).

Ex: – A = {1, 2, 3}, B = {a, b, c}

n(A) = 3 and n(B) = 3

∴ A = B.

Venn diagrams:

U = {1, 2, 3, 4, 5, 6}
the relationship between sets is usually represented by means of diagrams which are known as ‘Venn diagrams. These diagrams consist of rectangles and circles. A universal set is represented by rectangles and subsets by circles.

U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {1, 2}

## TS inter 1st year

TS inter 1st year: These blueprints were designed by ‘Basics in Maths’ team. These to-do’s help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear.

These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams.

TS inter 1st year

## TS Inter second year

TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

### TS Inter second year

#### 1. CIRCLES

Circle: In a plane, the set of points that are at a constant distance from a fixed point is called a circle.

∗ The fixed point is called the centre (C) of the circle and the constant distance is called the radius(r) of the circle

Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle.

Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle.  •

∗ The equation of the circle with centre (h, k) and radius r is

(x – h)2 + (y – k)2 = r2

∗ The equation of the circle with centre origin and radius r is

x2 + y2 = r2

⇒ x2 + y2 = r2 is called standard form of the circle.

The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0

∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0

It’s centre c = (– g, – f) and radius

∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0.

∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0.

∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0.

∗ The circles which have the same centre are called concentric circles.

∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is

x2 + y2 + 2gx + 2fy + k = 0.

∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 on

• x -axis is   if g2 – c > 0
• y -axis is if f2 – c > 0

Note: –

(a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point.

(b)  if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point.

(c) if g2 – c < 0, then the circle does not meet the x- axis.

(d) if f2 – c < 0, then the circle does not meet the y- axis.

∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is

(x – x1) (x – x2) + (y – y1) (y – y2) = 0.

∗ Let A, B be any two points on a circle then,

• The line is called the secant line of the circle.
• The line segment is called the chard of the circle.
• AB is called the length of the chord.

∗ A chord passing through the centre is called the diameter of the circle.

∗ The angle subtended by a chord on the circumference of at any point is equal.

The perpendicular bisector of a chord of a circle is asses through the centre of the circle.

∗ The angle in a semicircle is 900.

∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is

Where ci = − (x2 + y2) and i = 1,2,3

∗ centre of the circle is

#### Parametric form:

If P (x, y) is a point on the circle with centre (h, k) and radius r, then

X = h + r cosθ, y = k + r sinθ  0 ≤ θ ≤ 2π.

⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ.

Note:

1.  If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π.
2. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r.

Notations:

S = x2 + y2 + 2gx + 2fy + c

S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c

S11 = x12 + y12 + 2gx1 +2fy1 + c

S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c

Position of a point with respect to the circle:

A circle divides the plane into three parts.
1. The interior of the circle

2. The circumference which is the circular curve.

3. The exterior of the circle.

Power of point:

Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0.

• The power of point P (x1, y1) w.r.t. S = 0 is S11.

•Let S = 0 be a circle in a plane and P (x1, y1) be any point in the same plane. then

1. P lies in the interior of the circle ⇔ S11 < 0.
2. P lies on the circle ⇔ S11 = 0.
3. Plies in the exterior of the circle ⇔ S11 > 0.

Secant and tangent of a circle:

Let P be any point on the circle and Q be neighbourhood point of P lying on the circle. join P and Q, then the line PQ is the secant line.

The limiting position of the secant line PQ when Q is approached to the point P along the circle is called a tangent to the circle at P.

Length of the tangent:

If P is any point on the circle S = 0 and T is any exterior point of the circle, then PT is called the length of the tangent.

∗ If S = 0 is a circle and P (x1, y1) is an exterior point with respect o S = 0, then the length of the tangent from P (x1, y1) to S =0 is

Condition for a line to be a tangent:

• A straight-line y = mx + c (i) meet the circle x2 + y 2 = r2 in two distinct points if
• Touch the circle x2 + y 2 = rif
• Does not touch the circle x2 + y 2 = r2 in two distinct points if

Note:

1. For all real values of m, the straight line is a tangent to the circle x2 + y2 = r2 and the slope of the line is m.
2. A straight-line y = mx + c is a tangent to the circle x2 + y2 = r2 if c = .
3. The equation of a tangent to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 having the slope m is  where r is the radius of the circle.
4. The circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 touches (i) x – axis if g2 = c (ii) y – axis if f2= c.

Chord joining two points on a circle:

If P (x1, y1) and Q (x2, y2) are two points on the circle S = 0 then the equation of secant line PQ is S1 + S2 = S12.

Equation of tangent at a point on the circle:

The equation of the tangent at the point (x1, y1) to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is S1 = 0.

The equation of the tangent at the point (x1, y1) to the circle x2 + y2 = r2 is xx1 + yy1 – r2 = 0.

Point of contact:

If a straight-line lx + my + n = 0 touches the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 at P (x1, y1), then this line is the tangent to the circle S = 0. And the equation of the tangent is

(x1 + g) x + (y1 +f) y + (gx1 + fy1 + c) = 0.

∗ The equation of the chord joining two points θ1, θ2 on the circle x2 + y2 + 2gx + 2fy + c = 0 is

∗ The equation of the chord joining the points θ1, θ2 on the circle x2 + y2 = r2 is

∗ The equation of the tangent at P(θ) on the circle x2 + y2 + 2gx + 2fy + c = 0 is

∗ The equation of the tangent at P(θ) on the circle x2 + y2 = r2 is

x cosθ + y sinθ = r.

Normal:

The normal at any point P of the circle is the line which is passing through P and is perpendicular to the tangent at P.

• The equation of the normal at P (x1, y1) of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

(x – x1) (y1 + g) – (y – y1) (x1 + g) = 0.

• The equation of the normal at P (x1, y1) of the circle x2 + y 2 = r2 is xy1 – yx1 = 0.

Chord of contact and Polar:

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then there exists two tangents from P to the circle S = 0.

Chord of contact: –

If the tangents are drawn through P (x1, y1)

to a circle S = 0 touch the circle at points A and B then the secant line AB is called the chord of contact of P with respect to S = 0

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact of P with respect to S =0 is S1 = 0.

Note:

1. If the point P (x1, y1) is on the circle S = 0, then the tangent itself can be defined as the chord of contact.
2. If the point P (x1, y1) s an interior point of the circle S = 0, then the chord of contact does not exist.

Pole and Polar: –

Let S = 0 be a circle and P be any point if any line is drawn through the point Pin the plane other than the centre of S = 0. then the points of intersection meet the circle in two points A and B, of tangents drawn at A and B lie on a line called polar of P and P, is called Pole of polar.

∗ The equation of the polar of P (x1, y1) with respect to the circle S = 0 is S1 = 0.

Note: –

1. If Plies outside the circle S = 0, then the polar of P meets the circle in two points and the polar becomes the chord f contact of P.
2. If P lies on the circle S = 0, then the polar P becomes the tangent at P o the circle.
3. If P lies inside the circle S = 0, then the polar of P does not meet the circle.
4. If P is the centre of the circle S = 0, then the polar of P does not exist.
5. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 = r2 is
6. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is
7. The polar of P (x1, y1) with respect to the circle S = 0 passes through Q (x2, y2)  ⟺ polar of Q passes through P.

Conjugate points: Two P and Q are said to be conjugate points with respect to the circle S = 0, if the polar of P with respect to S = 0 passes through Q.

⇒ The condition for the points P (x1, y1), Q (x2, y2) to be conjugate with respect to the circle S = 0 is S12 = 0.

Conjugate lines: Two lines L1 = 0 and L2 = 0 are said to conjugate lines with respect to the circle S = 0 if the pole of L1 = 0 is lies on L2 = 0.

The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

r2 (l1l2 + m1m2) = (l1g + m1f – n1) (l2g + m2f – n2)

⟹ The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 = r2 is  r2 (l1l2 + m1m2) = n1n2

Inverse points: Let S = 0 be a circle with centre C and radius r. two points P and Q are said to be inverse points with respect to the circle S = 0 if

1. C, P, Q are collinear.
2. P, Q lies on the same side of C.
3. CQ = r2.

⟹  If lies inside of the circle S = 0, then Q lies outside of the circle.

⟹  If P lies on the circle S = 0, then P =Q.

⟹  Let S = 0 be a circle with centre C and radius r. The polar of P meets the line CP in Q iff  P, Q is inverse points.

⟹  f P, Q are inverse points with respect to S = 0, then P, Q are conjugate points with respect to the circle S = 0.

⟹  If P, Q are inverse points with respect to S = 0, then Q is the foot of the perpendicular from P on the polar of P with respect to the circle S = 0.

Equation of the chord with the given middle point:

The equation of the chord of the circle S = 0 having P (x1, y1) as its midpoint is S1 = S11.

Common tangents to the circle:

⟹ A straight line L = 0 is said to be a common tangent to the circle S = 0 and S= 0 if it is a tangent to both S = 0 and S’ = 0.

Two circles are said to touch each other if they have only one common tangent.

The relative position of two circles:

Let C1, C2 centres and r1, r2 be the radii of two circles S = 0 and S’ = 0respectively.

1.If C1C2 > r1+ r2, then two circles do not intersect.

⟹2 direct common tangents and

2 transverse common tangents

Total 4 common tangents

⟹P divides C1C2 in the ratio r1: r2 internally

Here P is called the internal centre of similitude (I.C.S)

⟹ Q divides C1C2 in the ratio r1: r2 externally

Here Q is called the external centre of similitude (E.C.S)

2.If C1C2 = r1+ r2, two circles touch each other

⟹ Q divides C1C2 in the ratio r1: r2 externally

⟹ two direct common tangents and one common tangent. Total 3 tangents

Here Q is called the external centre of similitude (E.C.S)

3.

⟹ two direct common tangents

Here Q is called the external centre of similitude (E.C.S)
Q divides C1C2 in the ratio r1: r2 externally

⟹ internal centre of similitude does not exist.

4.

⟹ only one common tangent

internal centre of similitude does not exist.

5.

no. of common tangents zero.

Note: the combined equation of the pair of tangents drawn from an external point P (x1, y1) to the circle S = 0 is S S11 = S12.

## 2.SYSTEM OF CIRCLES

A set of circles is said to be a system of circles if it contains at least two circles.

The angle between two intersecting circles:

If two circles S = 0 and S’ = 0 intersect at P then the angle between the tangents of two circles at P is called angle between the circles at P.

⟹ If two circles S = 0 and S’ = 0 intersect at P and Q then the angle between the tangents of two circles at P and Q are equal.

⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2 and θ is the angle between the circles then.

⟹ If θ is the angle between the circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 then

⟹ Two intersecting lines are said to be Orthogonal if the angle between the circles is a right angle.

Condition for the orthogonality:

⟹ The condition that the two circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 cut each other orthogonally is 2gg’ + 2ff’ = c + c’.

⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2. Two circles cut orthogonally if d2 = r12 + r22.

∎  If S = 0, S’ = 0 are two circles intersecting at two distinct points, then S – S’ = 0 represents a common chord of these two circles.

∎ If S = 0, S’ = 0 are two circles touch each other, then S – S’ = 0 represents a common tangent of these two circles.

∎ If S ≡ x2 + y2 + 2gx + 2fy + c =  0 and L ≡ lx + my + n = 0 are the equation of the circle and  a line respectively intersecting each other, then S + λ L = 0 represent a circle passing through the point intersection of  S = 0 and L = 0 ∀ λ ∈ ℛ.

The radical axis of two circles s defined as the locus of the point which moves so that its powers with respect to the two circles are equal.

(OR)
The locus of a point, for which the powers with respect to given non-concentric circles are equal, is a straight line is called Radical axis of the given circles.

∎ The equation of Radical axis f the circles S = 0 and S’ = 0 is S – S’ = 0.

The radical axis of any two circles is perpendicular to the line joining their centres.

The lengths of tangents from a point on the radical axis of two circles are equal if exist.

Radical axis of two circles bisects all common tangents of the two circles.

∎ If the centres of any three circles are non-collinear then the radical axis of each pair of circles chosen from these three circles re concurrent.

Radical centre:  The point of concurrence of radical axes of each pair of three circles is called radical centre (see above figure).

∎ If the circle S = 0 cuts the each of the two circle S’ = 0 and S’’ =0 orthogonally then the centre of S =0 lies on the radical axis of S’ = 0 and S’’ = 0.

∎ Radical axis of two circles is

• The ’common chord’ if the two circles intersect at two distinct points.
• The ‘common tangent’ at the point of contact if the two circles touch each other.

The radical axis of any two circles bisects the line joining the points of contact of common tangents to the circles.

Let S = 0, S’ = 0 and S’’ =0 be three circles whose centres are non- collinear and no two circles of these are intersecting then the circles having

• Radical centre of these circles as the centre of the circle.
• Length of the tangent from the radical centre to any one of these three circles cuts the given three circles orthogonally.

### CONIC SECTIONS

Conic: The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line is in the constant ratio is called Conic.

OR

The locus of a point moving on a plane such that its distance from a fixed point and a fixed line on the plane are in a constant ratio ‘e’, is called a Conic.

Focus: The fixed point is called focus and it is denoted by S.

Directrix: The fixed straight line is called the directrix.

Eccentricity: The constant ratio is called eccentricity and it is denoted by ‘e’.

Conic is the locus of a point P moving on a plane such that SP/PM = e, PM is the perpendicular distance from P to directrix at M.

If e = 1, then the conic is parabola.

if 0 < e < 1, then the conic is Ellipse.

if e > 1, then the conic is Hyperbola.

## 3.PARABOLA

If e = 1, then the conic is parabola.

• The standard form of parabola is y2 = 4ax.
• Focus S = (a. 0).
• Equation of directrix is x + a = 0.
• Vertex A = (0, 0) and A is the mid-point of SZ.

• Equation of the parabola with focus (α, β) and directrix lx + my + n = 0 is

• If the focus is situated on the left side of the directrix, the equation of the parabola with vertex as the origin and the axis is X-axis is y2 = – 4ax.
• The vertex being the origin, if the axis of the parabola is taken as Y – axis, equation of the parabola is x2 = 4 ay or x2 = – 4 ay according to the focus is above or below the X-axis.

Nature of the curve:

The nature of the parabola f the equation y2 = 4 ax (a>0)

• F y = 0, then 4 ax = 0 and x = 0

∴ the curve passes through the origin.

• If x = 0, then y2 = 0. Which gives y = 0. Y – axis is the tangent to the parabola at origin.
• Let P(x, y) be any point on the parabola (a>0) and y2 = 4 ax, we have x ≥ 0 and

∴ for any positive real value of x, we obtain two value of y of equal magnitude but opposite in sign. This shows that the curve is symmetric about X-axis and lies in the first and fourth quadrants.

The curve does not exist on the left side of the Y-axis since x ≥ 0 for any point (x, y) on the parabola.

Chord: The line segment joining two points on a parabola is called a chord.

Focal chord: A chord which is passing through focus is called Focal Chord.

Double ordinate: A chord through a point P on the parabola, which is perpendicular to the axis of the parabola is called Double ordinate.

Latus rectum: The double ordinate passing through the focus is called Latus rectum.

⟹ Length of Latus rectum = 4a.

Various forms of the parabola

1. y2 = 4ax

focus: (a, 0)

equation of directrix: x + a = 0

axis of parabola: y = 0

vertex: (0. 0)

2. y2 = −4ax

equation of directrix: x − a = 0

focus: (−a, 0)

axis of parabola: y = 0

vertex: (0. 0)

3. x2 = 4ay

focus: (0, a)

equation of directrix: y + a = 0

axis of parabola: x = 0

vertex: (0. 0)

4. x2 = −4ax

focus: (0, −a)

equation of directrix: y − a = 0

vertex:  (0. 0)    axis of parabola: x = 0

5. (y – k) 2 = 4a (x – h)

focus: (h + a, k)

equation of directrix: x – h + a = 0

axis of parabola: y – k = 0

vertex: (h. k)

6. (y – k) 2 = −4a (x – h)

focus: (h – a, k)
equation of directrix: x – h – a = 0

axis of parabola: y – k = 0

vertex: (h. k)

7. (x – h) 2 = 4a (y – k)

focus: (h, k + a)

equation of directrix: y – k + a = 0

axis of parabola: x – h = 0

vertex: (h. k)

8. (x – h)2 = −4a (y – k)

focus: (h, k – a)

equation of directrix: y – k – a = 0

vertex: (h. k)
axis of parabola: x – h =0

9.

focus: (α, β)

equation of directrix: lx + my + n = 0

axis of parabola: m (x – α) – l (y – β) = 0

vertex: A

Note:

1. Equation of the parabola whose axis parallel to X – axis is x = ly2 + my + n.
2. Equation of the parabola whose axis parallel to Y – axis is y = lx2 + mx + n.

Focal distance:  The distance of a point on the parabola from its focus is called Focal distance.

⟹ Focal distance of parabola s x1 + a

Parametric equations of a parabola:

The point (at2, 2at) satisfy the equation y2 = 4ax, the parametric equations of parabola are  x = at2, y = 2at. The point P(at2, 2at) is generally denoted by the point ‘t’ or P(t).

Notation:

1. S ≡ y2 – 4 ax
2. S1 ≡yy1 – 2a (x + x1)
3. S12 ≡ y1y2 – 2a (x1 + x2)
4. S11 ≡ y12 – 4 ax1

Equation of a tangent and normal at a point on the parabola:

∎ y = mx + c is a tangent to the parabola y2 = 4ax, then c = a/m or a =cm, and the point of contact is (a/m2, 2a/m).

∎ if m = 0, the line y = c is parallel to the axis of the parabola (i.e., x – axis)

y = c ⟹ c2 = 4ax ⟹ x = c2 /4a

∴ point of contact is (c2/4a, c).

∎ if m ≠ 0 and c = 0, then

Y = mx ⟹ x = 4a/m2 and y = 4a/m

∴ point of contact is (4a/m2, 4a/m).

∎ The equation of the chord joining the points (x1, y1) and (x2, y2) is S1 + S2 = S12.

∎ The equation of the tangent at P (x1, y1) to the parabola S = 0 is S1= 0.

∎ The equation of the normal at P (x1, y1) is (y – y1) = – y1/2a (x – x1).

Parametric form:

∎ The equation of the tangent at a point ‘t’ on the parabola y2 = 4ax is x – yt + at2 = 0.

∎ Equation of the normal at a point ‘t’ on the parabola y2 = 4ax is y + xt = 2at + at3.

∎ The condition for the straight-line lx + my + n = 0 to be a tangent to the parabola y2 = 4 ax is

am2 = nl and point of contact is (n/l, –2 am/l).

∎ common tangent to the parabolas y2 = 4 ax and x2 = 4 by is x a1/3 + y b1/3 + a2/3 b2/3 = 0.

∎ The equation of the chord of contact of the external point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The equation of the polar of the point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The pole of the line lx + my + n = 0 w.r.t. parabola y2 = 4ax is (n/l, -2am/l).

∎ If two points P (x1, y1), Q (x2, y2) are conjugate points w.r.t. parabola S = 0, then S12 = 0.

∎ The lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate lines with respect to the parabola y2 = 4 ax, then l1n2 + l2n1 = 2a m1m2.

## 4.ELLIPSE

Ellipse: A conic with eccentricity less than unity s called Ellipse.

∎ Equation of Ellipse in standard form is

⇒b2 = a2 (1 – e2) ⇒ e2 – 1 = -b2/a2

Major and Minor axis:

⟹ The line segment AA’ and BB’ of length 2a and 2b respectively are axes of the Ellipse.

⟹ If a > b AA’ is called Major axis and BB’ is called Minor axis and vice-versa if a<b.

Various form of Ellipse:

1.

Major axis: along the x-axis

Length of Major axis:2a

Minor axis: along y – axis

Length of Minor Axis:2b

Centre: (0, 0)

Foci: S = (ae, 0) and S’ = (–ae, 0)

Equation of directrices: x = a/e and x = –a/e

Eccentricty:

2.

Major axis: along y – axis

Length of Major axis:2b

Minor axis: along x – axis

Length of Minor Axis:2a

Centre: (0, 0)

Foci: S = (0, be) and S’ = (0, –be)

Equation of directrices: x = b/e and x = –b/e

Eccentricty:

Centre not at the origin

3.

Major axis: along with y = k

Length of Major axis:2a

Minor axis: along x = h

Length of Minor Axis:2b

Centre: (h, k)

Foci: S = (h +ae, k) and S’ = (h – ae, k)

Equation of directrices: x = h + a/e and x = h – a/e

Eccentricty:

4.

Major axis: along x = h

Length of Major axis:2b

Minor axis: along with y = k

Length of Minor Axis:2a

Centre: (h, k)

Foci: S = (h, k + be) and S’ = (h, k – be)

Equation of directrices: xy = k + b/e and y = k – b/e

Eccentricty:

Chord: The line segment joining two points on a parabola is called a  chord of Ellipse.

Focal chord: A chord which is passing through one of the foci is called Focal Chord.

Latus rectum: A focal chord perpendicular to the major axis of the Ellipse is called Latus Rectum. Ellipse has two latera recta.

Length of the Latus rectum:

1.The coordinates of the four ends of the latera recta of the ellipse

L = (ae, b2/a), L’ = (ae, -b2/a) and L1 = (-ae, b2/a), L1’= (-ae, -b2/a).

length of the Latus rectum = 2b2/a.

2.length of the Latus rectum of an ellipse   is 2a2/b and the coordinates of the four ends of the latera recta are

L = (a2/b, be), L’ = (-a2/b, be) and L1 = (a2/b, -be), L1’ = (-a2/b, -be).

3. The equation of the Latus rectum of the Ellipse   through S is x = ae and through S’ is x = -ae.

4. The equation of the Latus rectum of the Ellipse   through S is y = be and through S’ is y = -be.

5. If P (x, y) is any point on the Ellipse  whose foci are S and S’, then SP +S’P is constant.

Auxiliary circle: The circle described on the major axis of an Ellipse as the diameter is called the Auxiliary circle of the Ellipse. The Auxiliary circle of the Ellipse is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Ellipse  are x = a cosθ and y = b sinθ.

Notation:

Equation of Tangent and Normal

The equation of any tangent to the Ellipse can be written as

The condition for a straight-line y = mx + c to be a tangent to the Ellipse       is  c2 = am2 + b2.

∎ The point of contact of two parallel tangents to the Ellipse are (-a2m/c, b2/c) and (a2m/c, -b2/c)

∎ The equation of the chord joining two points (x1, y1) and (x2, y2) on the Ellipse S = 0 is S1 + S2 = S12.

∎ The equation of the Normal at P (x1, y1) to the Ellipse is

∎ Equation of the tangent at P(θ) on the Ellipse

∎ Equation of the normal at P(θ) on the Ellipse S = 0 is

∎ When θ = 0, π; equation of Normal is y =0.

∎ When θ = π/2, 3π/2; equation of Normal is x =0.

∎ The condition for the line lx + my + n = 0 to be a tangent the Ellipse S = 0 is a2l2 + b2m2 = n2.

∎ The condition for the line x cosα + y sinα = p to be a tangent the Ellipse S = 0 is a2 cos2 α + b2 sin2 α = p2.

∎ The pole of the line lx + my + n = 0 with respect to the Ellipse S = 0 is (-a2l/n, -b2m/n).

∎ The condition for the two lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the Ellipse S = 0 is a2l1l2 + b2m1m2 = n1n2. a2l1l2.

## 5.HYPERBOLA

Hyperbola: Hyperbola is a conic in which the eccentricity is greater than the unity.
Standard form of Hyperbola:

Equation of Hyperbola in standard form is

Centre: C (0, 0)

Foci: (± ae, 0)

Directrix: x = ± a/e.

Ecdntricity:

Notation:

Rectangular Hyperbola:

If in a Hyperbola the length of the transverse axis (2a) is equal to the length of the conjugate axis(2b), then the hyperbola is called rectangular hyperbola.

Its equation is x2 – y2 = a2 and eccentricity is .

Auxiliary circle: The circle described on the transverse axis of hyperbola as diameter is called the auxiliary circle of the hyperbola.

The equation of the auxiliary circle of S = 0 is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Parabola S = 0 are x = a secθ and y = b tanθ.

Conjugate Hyperbola:

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called a conjugate hyperbola.

The equation of hyperbola conjugate to   S ≡  is S’ ≡

∎ For

The transverse axis lies on along X-axis and its length is 2a.

The conjugate axis lies on along Y-axis and its length is 2b.
∎ For

The transverse axis lies on along Y-axis and its length is 2b.

The conjugate axis lies on along X-axis and its length is 2a.

Various form of Hyperbola:

Let  and

1.Hyperbola

The transverse axis along X-axis: y =0

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = 0

Length of the conjugate axis: 2b

Centre: (0, 0)

Foci: (± ae, 0)

Equation of the directrices: x = ± a/e

Eccentricity:

2.Conjugate Hyperbola

The transverse axis along Y-axis: x = 0

Length of the transverse axis:2b

The conjugate axis along X-axis: y = 0

Length of the conjugate axis: 2a

Centre: (0, 0)

Foci: (0, ± be)

Equation of the directrices: y = ± b/e

Eccentricity:

Centre not at the origin:

3.Hyperbola

The transverse axis along X-axis: y = k

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = h

Length of the conjugate axis:2b

Centre: (h, k)

Foci: (h± ae, k)

Equation of the directrices: x = h± a/e

Eccentricity:

4.Hyperbola

The transverse axis along Y-axis: x = h

Length of the transverse axis:2b

The conjugate axis along X-axis: y = k

Length of the conjugate axis: 2a

Centre: (h, k)

Foci: (h, k ± be)

Equation of the directrices: y = k± b/e

Eccentricity:

Equation of tangent and normal at a point on the Hyperbola:

∎ The equation of the tangent at P (x1, y1) to the hyperbola S = 0 is S1 = 0.

∎ The equation of the tangent at P(θ) on the hyperbola S = 0 is

∎ The equation of the Normal at P (x1, y1) to the hyperbola S = 0 is

∎ Equation of the normal at P(θ) on the Hyperbola S = 0 is

∎ The condition for a straight-line y = mx + c to be a tangent to the hyperbola S = 0 is c2 = am2 − b2.

Asymptotes of a hyperbola:

The equations of asymptotes of hyperbola S = 0 are      and the joint equation of asymptotes is

## 6. INTEGRATION

∎ Let E be a subset of R such that E contains a right or left the neighbourhood of each of its points and let f: E → R be a function. If there is a function F on E such that F’(x) = f(x) ∀ x ∈ E, then we call F an anti-derivative of f or a primitive of f.

Indefinite integral: Let f: I→R. Suppose that f has an antiderivative F on I. Then we say that f has an integral on I and for any real constant c, we call F + c an indefinite integral of over I, denote it by    ∫f(x) dx and read it as ‘integral’ f(x) dx.

∫f = ∫f(x) dx = F(x) + c. here c is called constant of integration.

In the indefinite integral ∫f(x)dx, f is called ‘integrand’ and x is called the variable of integration.

⟹

⟹ if f: I⟶R is differentiable on I, then ∫f’(x)dx = f(x) + c.

Standard forms:

Properties of integrals:

∎ ∫ (f ±g) (x) dx = ∫f(x) dx ± ∫g(x) dx + c

∎ ∫(af) (x) dx = a ∫f(x) dx + c

∎ ∫ (f1 + f2 + … + fn) (x) dx = ∫f1(x) dx + ∫f2(x) dx + … +∫fn(x) dx +c

∎ ∫f(g(x)) g’(x) dx = F(g(x)) + c

∎ ∫f(ax + b) dx = 1/a F(ax +b) + c

Some important formulae:

1. ∫eax dx = 1/a eax + c
2. ∫sin (ax + b) dx = -1/a cos (ax + b) + c
3. ∫cos (ax + b) dx = 1/a sin (ax + b) + c
4. ∫sec2 (ax + b) dx = 1/a tan (ax + b) + c
5. ∫ cosec2 (ax + b) dx = 1/a cot (ax + b) + c
6. ∫cosec (ax + b) cot (ax + b) dx = -1/a cosec (ax + b) + c
7. ∫sec (ax + b) tan (ax + b) dx = 1/a sec (ax + b) + c

Integration by parts:

Let u, v real valued differentiable functions in I. Suppose that u,v has an integral on I, then uv’ has an integral on I and

∫(uv’) (x) dx = (uv) – ∫(u’v) (x) dx + c or ∫(uv) dx = u ∫v dx – ∫ [u’ ∫v dx] dx + c

Integration of exponential functions:

∫ex dx = ex + c; ∫x ex dx = (x – 1) ex + c

∫ ex [f(x) +f’(x)] dx = ex f(x) + c

Integration of logarithmic functions:

∫log x dx = x log x – x + c

Integration of inverse trigonometric functions:

Evaluation of integrals form  :

Working rule:  reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate using the substitution t = x + α.

Evaluation of integrals form

Working rule:

Case(i) if a >0 and b2 – 4ac < 0, then reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate.

Case(ii) if a <0 and b2 – 4ac >0, then reduce ax2 + bx + c to the form of (-a) [ β – (x + α)2 +] and then integrate.

Evaluation of integrals form

Working rule:  write px + q in the form of A (ax2 + bx +c)’ + B, then integrate.

Evaluation of integrals form

Working rule:  write cos x =cos2(x/2) – sin2(x/2) and sin x = 2 sin(x/2) cos (x/2)

Put t = tan(x/2), then dt = ½ sec2 (x/2) dx

Cos x = 1 – t2 / 1 + t2, sin x = 2t/1 + t2 then integrate.

Evaluation of integrals form

Working rule:  t = sqrt. (px + q and then integrate.

Evaluation of integrals form

Working rule:  we find real numbers A, B and C such that

(a cos x + b sin x + c) = A(d cos x + e sin x +f)’ + B(d cos x + e sin x +f) + C then by substituting this expression in the  integrand, evaluate the integral.

Integration – partial fraction method:

Let R(x) = f(x) / g(x), g(x) ≠ 0 where f, g are polynomials. If degree of f(x) ≥degree of g(x), then divide f(x) by g(x) by synthetic division method and find polynomials.

Q(x) and h(x) such that f(x) = Q (x) g(x) + h(x) here h = 0 or h ≠ 0 and degree h(x) < degree of g(x). Then R(x) =Q(x) + h(x)/g(x)

We get solution of h(x) / g(x) using partial fractions and then integrate.

Partial fractions:

∎ If R(x) = f(x) / g(x) is proper fraction, then

Case(i): – For every factor of g(x) of the form (ax + b) n, there will be a sum of n partial fractions of the form:

Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form:

∎ If R(x) = f(x) / g(x) is improper fraction, then

Case (i): – If degree f(x) = degree of g(x), then f(x)/g(x) = k + h(x)/g(x) where k is the quotient of the highest degree term of f(x) and g(x).

Case (ii): – If f(x) > g(x)

R(x) =f(x) /g(x) = Q(x) + h(x)/g(x)

Reduction formulas:

## 7.DEFINITE INTEGRATION

Partition: Let a, b∈ R be such that a < b. Then, a finite set P = {x0, x1, …, x i- 1, xi, xi + 1, …, xn} of elements of [a, b] is called to be a partition of [a, b] if a =  x0 <  x1 < … < x i- 1 <  xi <  xi + 1 < … < xn = b.

Norm: if {x0, x1, …, xn} is a partition of [a, b], then the norm of the partition P, denoted by ∥P∥, is defined by ∥P∥ = max {x1 – x0, x2 – x1, …, xn – xn-1}. We donate the set of all partitions of [a, b] by 𝒫 ([a, b]).

Definite integral:
Riemann sum:
Let f: [a, b] → R be a bounded function for all x in [a, b]. Let P = {x0, x1, …, x i- 1, xi, xi + 1, …, xn} be partition of [a, b], and t ∈ [xi-1, xi], for I = 1, 2, …, n. A sum of the form is called Riemann sum of f relative to P.

Let f is Riemann integrable on [a, b]. if there exists a real number A such that S (P, f) approaches A as ∥P∥ approaches to ‘0’. In other words, given ϵ > 0, there is a δ > 0 such that  for any partition P of [a, b] with ∥P∥ < δ irrespective of the choice of ti in [xi-1, xi]. Such an A, if exists, is unique and is denoted by , it is read as the definite integral of f from a to b. an a is called the lower limit and b is called the upper limit. The function f in  is called ‘integrand’.

if f: [a, b] → R is continuous, then is exists.

∎ If f is continuous on [0, p] where p is a positive integer then

The fundamental theorem of integral calculus:

If f is integrable on [a, b] and if there is a differentiable function F on [a, b] such that F = f, then

we write

properties of definite integrals:

∎ Let f: [a, b] → R be bounded. Let c ∈ (a, b). then f is integrable on [a, b] if and only if it is integrable on [a, c] as well as on [c, b] and in this case

Method of substitution:  Let g: [c, d] → R have continuous derivative on [c, d]. Let f: g([c, d]) → R be continuous. Then (fog) g’ is integrable on [c, d] and

∎ Let f be integrable on [a, b]. Then the function h, defined on [a, b] as h(x) = f (a + b – x)

for all x in [a, b] and

∎ Let f be integrable on [0, a]. Then the function h, defined on [0, a] as h(x) = f (a – x)

for all x in [a, b] and

∎ Let f: [-a, a] → R be integrable on [0, a]. Suppose that f is either odd or even. Then f is integrable on  [-a, a] and

∎ Let f: [0,2 a] → R be integrable on [0, a].

• If f (2a – x) = f(x) for all x in [a, 2a] then f is integrable on [0, 2a] and
• If f (2a – x) = – f(x) for all x in [a, 2a] then f is integrable on [0, 2a] and

∎ If f and g are integrable on [a, b], then their product fig is integrable on [a, b].

Integration by parts:

∎ Let f: R→ R be a continuous periodic function and T be the period of it. Then any positive integer n

Reduction formulae:

∎ Let n≥2 be an integer, then

∎ Let m and n be positive integers, then

Areas under curves:
(i)If f: [a, b] → [0, ∞) is continuous, then the area A of the region bounded by the curve y = f(x), the X-axis and the line x = a and x =b is given by

A =

(ii) If f: [a, b] → (−∞, 0] is continuous, then the graphs of y = f(x)and y = − f(x) on [a, b] are symmetric about the X-axis. So, the area bounded by the graph of y = f(x), the X-axis and the lines x = a, x =b is same as the area bounded by the graph of y = – f(x), the X-axis and the lines x = a and y = b which is given by A =

From (i) and (ii) A =

(iii) Let f: [a, b] → R be continuous and f(x) ≥ 0 ∀ x ∈ [a, c] and f(x) ≤ 0 ∀ x∈ [c, b] where a < c < b. Then the area of the region bounded by the curve y= f(x), the X – axis, and the lines x = a and x = b is given by

Area of region =A =

(iv) Let f: [a, b] → R and g: [a, b] → R be continuous f(x) ≤ g(x) ∀ x∈ [a, b]. Then the area f the region bounded by the curve y = f(x), y = g(x) and the lines x = a, x = b is given by

(v) Let f and g be wo continuous real value functions on [a, b] and c ∈ (a, b) such that f(x) < g(x) ∀ x∈ [a, c) and g(x) < f(x) ∀ x∈ (c, b] with f (c) = g (c). area of the region bounded by y = f (x), y = g(x), and the lines x = a, x = b is given by

(vi) Let f: [a, b] → R and g: [a, b] → R be continuous functions. Suppose that, there exist points x1, x2 ∈ (a, b) with x1< x2 such that f(x1) = g(x1) and f(x2) = g(x2) and f(x) ≥ g(x) ∀ x ∈ (x1, x2). Then the area of the region bounded by the curves by y = f (x), y = g(x), and the lines x = x1, x = x2 is given by and if f(x) ≤ g(x) ∀ x ∈ (x1, x2). ThenIn either case, area is .

## 8. DIFFERENTIAL EQUATIONS

Differential equation: An equation involving one dependent variable and its derivative with respect to one or more independent variables is called a ‘Differential equation’.

If a differential equation contains only one independent variable, then it is called ‘an ordinary differential equation and if it contains more than one independent variable, then it is called ‘a partial differential equation’.

Degree of the differential equation:  If a differential can be expressed as a polynomial equation in the derivatives occurring in it using the algebraic operations such that the exponent of each of the derivatives is the least, then the large exponent of the highest order derivative in the equation is called the degree of the differential equation.

Otherwise, the degree is not defined for a differential equation.

Order of differential equation: The order of the differential equation is the order of the highest derivative occurring in it.

Note: The general form of an ordinary differential equation of nth order is

Formation of the differential equation: suppose that an equation y = ϕ (x, α1, α2, …, αn) where α1, α2, …, αn are parameters, representing a family of curves is given. Then successively differentiating the above equation, a differential equation of the form

We know that y = mx is a straight line passing through the origin

m = dy/ dx ⟹

Solving differential equations:

1. Variable separable method:

If a given differential equation can be put in the form of f(x) dx + g(y) dy = 0 then its solution can be obtained by integrating each of them. This method is called the variable separable method.

Ex: xdy – y dx = 0 can be written as dx/x =dy/y

By integrating we get ∫dx/x = ∫dy/y

⇒ logx = logy + logc

⇒ logx = log yc

∴ x = yc is the required solution

2. Homogeneous Differential Equation:

Homogeneous function: – A function f (x, y) of two variables x and y is said to be a homogeneous function of degree n, if f(kx, ky) = kn f(x, y)  for all values of k for which both sides of the above are meaningful.

Homogeneous Differential Equation: – A differential equation of the form where f (x, y) and g (x, y) are homogeneous functions of x and y of the same degree is called a homogeneous differential equation.

Method of solving the homogeneous differential equation: –

Consider the homogeneous equation   …… (1)

where f (x, y) and g (x, y) are homogeneous functions of x and y of the same degree.

f (x, y) = xn ϕ (y/x) and g(x) xn ψ (y/x)

eqn (1) becomes  ……. (2)

put y = vx. Then  ……. (3)

from (2) and (3)

This can be solved by the variable separable method.

3. Non-Homogeneous Differential Equations:

The differential equation of the form where a, b, c, a’, b’, c’ are constants and c and c’ are not both zero are called non-homogeneous equations. Reduce the above equation to a homogeneous equation by suitable substitution for x and y.

Case(i): –

Suppose that b = – a’. then    becomes

⇒ (a’x + b’y + c’) dy – (ax – a’y + c) dx = 0

⇒ a’(x dy + y dx) + b’ y dy – ax dx + c’ dy – cdx = 0

By integrating we get

a’ xy + b’ y2/2 – a x2/2 + c’y – cx = k

which is a required solution.

Case(ii): –

Suppose that  Then  becomes

Put ax + by = v, then

this can be solved by the variable separable method.

Case(iii): –

Suppose that b ≠ – a’ and a/a’ ≠ b/b’, then taking x = X + h, y = Y + k, where X and Y are variables and h, k are constants. We get  . Hence…..(i)    becomes

Now choose constants h and k such that

ah + bk + c = 0

a’h + b’k +c’ = 0

by solving above equations, we get h. k values

Hence, equation (1) becomes

This is the homogeneous equation in X and Y and then solve by the homogeneous method by putting Y = VX.

3. Linear Differential Equations:

A differential equation of the form   = R,  where P1, P2, …, Pn and R are constants or functions x only, is said to be a linear differentiable equation of nth order.

Method of solving the linear differentiable equation of 1st order: –

The linear differentiable equation of the first order is

Multiplying both sides of (1) by, we get

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