P RAVINDRA
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Coordinate Geometry and Straight Lines detailed synopsis #📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #📖ఎడ్యుకేషన్✍ #🏆పోటీ పరీక్షల స్పెషల్ #👩💻టెట్/DSC ప్రత్యేకం #🎓జాబ్ ప్రిపరేషన్📚
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P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #🏆పోటీ పరీక్షల స్పెషల్ #📖ఎడ్యుకేషన్✍
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MATH Solutions By PHANI#🏆పోటీ పరీక్షల స్పెషల్ #📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #👩💻టెట్/DSC ప్రత్యేకం![FOUR CASES OF VOEINEQUALITIES ABSOLUTE IINONEVARIABLD 071T S0!u CASE1 ` Solve; Ixl < 5 Sct Builczr Notctic n ' IX| {1nಗ/ಗ -5 < * < 5 hhere' Vototion ' و =7=5-4=]- (-5, 5) 02ಡ a Solve: Ix] = x| < @ {:RI, < x $ 4] -4 = *= 4` 'nL[Lnಗ Incetval Movotlont ] 0-4, 4] 3 5U CASE3; ಗu ldrr Hotctien ' Solve: Ix > 3 Ixl {+-7|+< v1> 31 X <7] *> 3 vher 0> LLrultqtlon -6 $ ! 2 1 0 1 2 J 3) 0(3, olumon CASE4 EHAB Set BuJer ctotona Solve: Ixl 2 2 I*| 2 q Or r2 X$=1 X 2 2 VhLre l ಭranalನlon; 2]0[2, ',v' RCMINBIRI பன்பம ப TLon danau Non <rಎsl FOUR CASES OF VOEINEQUALITIES ABSOLUTE IINONEVARIABLD 071T S0!u CASE1 ` Solve; Ixl < 5 Sct Builczr Notctic n ' IX| {1nಗ/ಗ -5 < * < 5 hhere' Vototion ' و =7=5-4=]- (-5, 5) 02ಡ a Solve: Ix] = x| < @ {:RI, < x $ 4] -4 = *= 4` 'nL[Lnಗ Incetval Movotlont ] 0-4, 4] 3 5U CASE3; ಗu ldrr Hotctien ' Solve: Ix > 3 Ixl {+-7|+< v1> 31 X <7] *> 3 vher 0> LLrultqtlon -6 $ ! 2 1 0 1 2 J 3) 0(3, olumon CASE4 EHAB Set BuJer ctotona Solve: Ixl 2 2 I*| 2 q Or r2 X$=1 X 2 2 VhLre l ಭranalನlon; 2]0[2, ',v' RCMINBIRI பன்பம ப TLon danau Non <rಎsl](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_923897_1b7d63f9_1785988838160_sc.jpg?tenant=sc&referrer=feed-service&f=160_sc.jpg)
125- MATH Solutions By PHANI#🏆పోటీ పరీక్షల స్పెషల్ #📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #➗10th గణితం
714 - MATH Solutions By PHANI#🏆పోటీ పరీక్షల స్పెషల్ #📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #👩💻టెట్/DSC ప్రత్యేకం #EAMCET
812 - P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #📖ఎడ్యుకేషన్✍
![Straight Lines VSAQ PRAVINDRA, M.Sc Question : Find the equation of straight line passing through the (at;] 2 2 20t2 २at1 (atz points ) and Solution Q(atz P(0t1 2 2 २at1 Let the given points be 2qt2) ' and Slope of the line PQ: Zat2 20t y2 - y PR m = 3 2 12 - X1 at21 at 2(t2 Za(t2 -t) = t1) 2 (since t2 = t1) F (t2 - t1)(t2 - t1) t2 + t1 2 a(tz Equation of the line (Point-Slope form): 2 Zat1 Using P(at- point 20t1 m(x qt ) PR 2 (x - at,2) 20t1 ] - t2 + t1 Simplifying: 2(1 - 0t12) (12 + 11)(y - 20t1 = IIATHS TUITIOIS 20t1 2ut1 (2 + t1)y - (2 + t1) २x - = Rearranging: WARASIGUDA (2+t1)y + 20(t12 _ t112 - 12) = 0 2 21 - 20t10 (t1 + t2)y 2x - = 0 t2 SECUNDERABAD 20t11 (t1 + 12)y - 2x t2 = 0 is the required equation of the line Straight Lines VSAQ PRAVINDRA, M.Sc Question : Find the equation of straight line passing through the (at;] 2 2 20t2 २at1 (atz points ) and Solution Q(atz P(0t1 2 2 २at1 Let the given points be 2qt2) ' and Slope of the line PQ: Zat2 20t y2 - y PR m = 3 2 12 - X1 at21 at 2(t2 Za(t2 -t) = t1) 2 (since t2 = t1) F (t2 - t1)(t2 - t1) t2 + t1 2 a(tz Equation of the line (Point-Slope form): 2 Zat1 Using P(at- point 20t1 m(x qt ) PR 2 (x - at,2) 20t1 ] - t2 + t1 Simplifying: 2(1 - 0t12) (12 + 11)(y - 20t1 = IIATHS TUITIOIS 20t1 2ut1 (2 + t1)y - (2 + t1) २x - = Rearranging: WARASIGUDA (2+t1)y + 20(t12 _ t112 - 12) = 0 2 21 - 20t10 (t1 + t2)y 2x - = 0 t2 SECUNDERABAD 20t11 (t1 + 12)y - 2x t2 = 0 is the required equation of the line](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_377367_321a21a5_1785462375630_sc.jpg?tenant=sc&referrer=feed-service&f=630_sc.jpg)
1711 - P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #📖ఎడ్యుకేషన్✍ #🏆పోటీ పరీక్షల స్పెషల్ INTEGRATION BY PARTS
![INTEGRATION BY PARTS u [ (u' | v dx) dx Jvdx V dx u = ILATE RULE (Choose u) 8 where Inverse functions 1, 1 are functions of X Logarithmic functions Algebraic functions A according to u and v must be chosen Trigonometric functions ] the ILATE rule, where E Exponential functions MATHS TUITIONS WARASIGUDA, SECUNDERABADK Algebraic Logarithmic 1 Trigonometric Exponential Inverse functions functions functions functions functions 0 EXAMPLE Remember! J x log dx Example: 1 dx Choose u first log x Solution: Let du (as per ILATE rule) 2 Jx dx 0 dx dv > V Find du, V By intesration by parts, Apply the formula Jvdx - J (u Jvdx) dx J x 108 > dx carefully-| (2) - [ (T . 2 ) dx Simplify the result (log x) 12 J (2 log dx 2 22 log Y ೩2 + C २ 2 2 2 د2 ) x 108 > log X dx +C INTEGRATION BY PARTS u [ (u' | v dx) dx Jvdx V dx u = ILATE RULE (Choose u) 8 where Inverse functions 1, 1 are functions of X Logarithmic functions Algebraic functions A according to u and v must be chosen Trigonometric functions ] the ILATE rule, where E Exponential functions MATHS TUITIONS WARASIGUDA, SECUNDERABADK Algebraic Logarithmic 1 Trigonometric Exponential Inverse functions functions functions functions functions 0 EXAMPLE Remember! J x log dx Example: 1 dx Choose u first log x Solution: Let du (as per ILATE rule) 2 Jx dx 0 dx dv > V Find du, V By intesration by parts, Apply the formula Jvdx - J (u Jvdx) dx J x 108 > dx carefully-| (2) - [ (T . 2 ) dx Simplify the result (log x) 12 J (2 log dx 2 22 log Y ೩2 + C २ 2 2 2 د2 ) x 108 > log X dx +C](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_409393_1a35223f_1785234408045_sc.jpg?tenant=sc&referrer=feed-service&f=045_sc.jpg)
1612 - P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #📖ఎడ్యుకేషన్✍ #🏆పోటీ పరీక్షల స్పెషల్ INTEGRATIONS FORMULAE
![INDEFINITE INTEGRATIONS | d Jftx)dx Jxdx INDIRECT FORMULAE [f()]"+1 J f'() [f(x)]"dx 1 +C 3 " n+1 (f(x))2 f'(x) - f(x) dx 2 +C ೆಕ 2 2 f'(x)V f(x) dx [f(x)]3/ 3 +C 3 _f'() V f(x) +C 4 dx 2 Vf(x) MATHS TUITIONS ೫ WARASIGUDA, SECUNDERABAD f@) 5 log | f(x) | + C dx F f'(x) ] ] dx +C 6 F f(x))2 f(x) 0 ex ( f(x) + f'(x) ) dx ex flx) + c = (< f'() + f(x) ) dx 8 f(x) + C 3 . = INDEFINITE INTEGRATIONS | d Jftx)dx Jxdx INDIRECT FORMULAE [f()]"+1 J f'() [f(x)]"dx 1 +C 3 n+1 (f(x))2 f'(x) - f(x) dx 2 +C ೆಕ 2 2 f'(x)V f(x) dx [f(x)]3/ 3 +C 3 _f'() V f(x) +C 4 dx 2 Vf(x) MATHS TUITIONS ೫ WARASIGUDA, SECUNDERABAD f@) 5 log | f(x) | + C dx F f'(x) ] ] dx +C 6 F f(x))2 f(x) 0 ex ( f(x) + f'(x) ) dx ex flx) + c = (< f'() + f(x) ) dx 8 f(x) + C 3 . =](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_378415_3150939f_1785234377154_sc.jpg?tenant=sc&referrer=feed-service&f=154_sc.jpg)
714 - P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #🏆పోటీ పరీక్షల స్పెషల్ #📖ఎడ్యుకేషన్✍
1114 - P RAVINDRA#📘ఇంటర్ సైన్స్ & మ్యాథ్స్ #📖ఎడ్యుకేషన్✍ #🏆పోటీ పరీక్షల స్పెషల్ EAMCET MATHS Circles
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