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![🏆పోటీ పరీక్షల స్పెషల్ - Find the sum The series is given by: 32 + 62 3 + 6 3 _ ೧ 3" + 6" = ; 1 4+8 42 + 82 4+8 4" + 8" Solution: Step Factor each term: 32 (]2 _ 22) 33 (13 + 23) 3(1+2) 3" (1"+2") 5 = 42 (12 + 22) 4(1+2) 4" (1" + 2") 43 (13 _ 23) Step 2 Cancel the comnmnon factors: 33 32 = _ 4 4 42 2: Step 3 Multiply equation (1) by 32 4$ = 4 4 3२ 3"+1 5 = 4 4 4"+1 Step 4 Subtract: equation (2) equation (1) 3"+1 $ - 5 [7-[1 3111 3 > 4" Step 5 Solve for S: 3"+1 3 > 4" 4"+1 3"+1 } > 4" _ 3 (4" _ 3 5= $=3 $ Education Reform Centre Find the sum The series is given by: 32 + 62 3 + 6 3 _ ೧ 3" + 6" = ; 1 4+8 42 + 82 4+8 4" + 8" Solution: Step Factor each term: 32 (]2 _ 22) 33 (13 + 23) 3(1+2) 3" (1"+2") 5 = 42 (12 + 22) 4(1+2) 4" (1" + 2") 43 (13 _ 23) Step 2 Cancel the comnmnon factors: 33 32 = _ 4 4 42 2: Step 3 Multiply equation (1) by 32 4$ = 4 4 3२ 3"+1 5 = 4 4 4"+1 Step 4 Subtract: equation (2) equation (1) 3"+1 $ - 5 [7-[1 3111 3 > 4" Step 5 Solve for S: 3"+1 3 > 4" 4"+1 3"+1 } > 4" _ 3 (4" _ 3 5= $=3 $ Education Reform Centre - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_276285_30818dbb_1772616273722_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=722_sc.jpg "🏆పోటీ పరీక్షల స్పెషల్ - Find the sum The series is given by: 32 + 62 3 + 6 3 _ ೧ 3\" + 6\" = ; 1 4+8 42 + 82 4+8 4\" + 8\" Solution: Step Factor each term: 32 (]2 _ 22) 33 (13 + 23) 3(1+2) 3\" (1\"+2\") 5 = 42 (12 + 22) 4(1+2) 4\" (1\" + 2\") 43 (13 _ 23) Step 2 Cancel the comnmnon factors: 33 32 = _ 4 4 42 2: Step 3 Multiply equation (1) by 32 4$ = 4 4 3२ 3\"+1 5 = 4 4 4\"+1 Step 4 Subtract: equation (2) equation (1) 3\"+1 $ - 5 [7-[1 3111 3 > 4\" Step 5 Solve for S: 3\"+1 3 > 4\" 4\"+1 3\"+1 } > 4\" _ 3 (4\" _ 3 5= $=3 $ Education Reform Centre Find the sum The series is given by: 32 + 62 3 + 6 3 _ ೧ 3\" + 6\" = ; 1 4+8 42 + 82 4+8 4\" + 8\" Solution: Step Factor each term: 32 (]2 _ 22) 33 (13 + 23) 3(1+2) 3\" (1\"+2\") 5 = 42 (12 + 22) 4(1+2) 4\" (1\" + 2\") 43 (13 _ 23) Step 2 Cancel the comnmnon factors: 33 32 = _ 4 4 42 2: Step 3 Multiply equation (1) by 32 4$ = 4 4 3२ 3\"+1 5 = 4 4 4\"+1 Step 4 Subtract: equation (2) equation (1) 3\"+1 $ - 5 [7-[1 3111 3 > 4\" Step 5 Solve for S: 3\"+1 3 > 4\" 4\"+1 3\"+1 } > 4\" _ 3 (4\" _ 3 5= $=3 $ Education Reform Centre - ShareChat")
![➗10th గణితం - REALMATHS MATHS CHAPTER - 2 INTEGERS Choose the best answer only 7added to -1 gives 1 (d) (b) (c) 8 3 added to _ 3 gives (b) 1 subtracted from 1 gives (b) (a) =1 Sum of 10 5 and 12 is (೩) 27 _27 (c) _3 following statements is false Which ofthe (c) 4 < - 5 (d) 4> - 5 a)-4 > - 5 (b) - 4 < 5 Which of the following is in increasing order (a) 0 , 1 , - 1 (b)-1 , - 2 , - 3 (c)-1 , 0 , 1 (d)-1,1, - 2 Which ofthe following is correct (೩) _ 8 > - 7 (c) _ 1 < 0 ()-2>4 (b)1 <0 following number forms Which of the a pattern (b)~5 ,- 3 (c) 0 , 2 , 3 , 4 (d) 1 , 2 , 4 , 6 6 , - 3 - 2 0 3 Sum of _ 36 and 29 is] a) -651 (b) 65 (d) Which ofthe following will give answer with negative sign] 10 48 + 79 40 + 40 48 + 30 48 + ( - 39 ) (b) ೩) What will be the additive inverse of-1 ? 11 (a) -2 (b) - ] Sum of two positive integers is always-| 12 positive] (a) Negative () Sum of a negative and a positive integer is - 13 either positive] negative (c) always positive Always negative] Zero ) 01 ofintegers whose sum is -5 14. The pair -3 , -2 5, _1 ; a) (0) 15. 39 _ 50 1s a) Not possible (b) -89 10 -11 Math's online class what's up:+251900032087 Page repared by: ATTMEDOMER 47 REALMATHS MATHS CHAPTER - 2 INTEGERS Choose the best answer only 7added to -1 gives 1 (d) (b) (c) 8 3 added to _ 3 gives (b) 1 subtracted from 1 gives (b) (a) =1 Sum of 10 5 and 12 is (೩) 27 _27 (c) _3 following statements is false Which ofthe (c) 4 < - 5 (d) 4> - 5 a)-4 > - 5 (b) - 4 < 5 Which of the following is in increasing order (a) 0 , 1 , - 1 (b)-1 , - 2 , - 3 (c)-1 , 0 , 1 (d)-1,1, - 2 Which ofthe following is correct (೩) _ 8 > - 7 (c) _ 1 < 0 ()-2>4 (b)1 <0 following number forms Which of the a pattern (b)~5 ,- 3 (c) 0 , 2 , 3 , 4 (d) 1 , 2 , 4 , 6 6 , - 3 - 2 0 3 Sum of _ 36 and 29 is] a) -651 (b) 65 (d) Which ofthe following will give answer with negative sign] 10 48 + 79 40 + 40 48 + 30 48 + ( - 39 ) (b) ೩) What will be the additive inverse of-1 ? 11 (a) -2 (b) - ] Sum of two positive integers is always-| 12 positive] (a) Negative () Sum of a negative and a positive integer is - 13 either positive] negative (c) always positive Always negative] Zero ) 01 ofintegers whose sum is -5 14. The pair -3 , -2 5, _1 ; a) (0) 15. 39 _ 50 1s a) Not possible (b) -89 10 -11 Math's online class what's up:+251900032087 Page repared by: ATTMEDOMER 47 - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_803874_bb9e613_1772598801645_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=645_sc.jpg "➗10th గణితం - REALMATHS MATHS CHAPTER - 2 INTEGERS Choose the best answer only 7added to -1 gives 1 (d) (b) (c) 8 3 added to _ 3 gives (b) 1 subtracted from 1 gives (b) (a) =1 Sum of 10 5 and 12 is (೩) 27 _27 (c) _3 following statements is false Which ofthe (c) 4 < - 5 (d) 4> - 5 a)-4 > - 5 (b) - 4 < 5 Which of the following is in increasing order (a) 0 , 1 , - 1 (b)-1 , - 2 , - 3 (c)-1 , 0 , 1 (d)-1,1, - 2 Which ofthe following is correct (೩) _ 8 > - 7 (c) _ 1 < 0 ()-2>4 (b)1 <0 following number forms Which of the a pattern (b)~5 ,- 3 (c) 0 , 2 , 3 , 4 (d) 1 , 2 , 4 , 6 6 , - 3 - 2 0 3 Sum of _ 36 and 29 is] a) -651 (b) 65 (d) Which ofthe following will give answer with negative sign] 10 48 + 79 40 + 40 48 + 30 48 + ( - 39 ) (b) ೩) What will be the additive inverse of-1 ? 11 (a) -2 (b) - ] Sum of two positive integers is always-| 12 positive] (a) Negative () Sum of a negative and a positive integer is - 13 either positive] negative (c) always positive Always negative] Zero ) 01 ofintegers whose sum is -5 14. The pair -3 , -2 5, _1 ; a) (0) 15. 39 _ 50 1s a) Not possible (b) -89 10 -11 Math's online class what's up:+251900032087 Page repared by: ATTMEDOMER 47 REALMATHS MATHS CHAPTER - 2 INTEGERS Choose the best answer only 7added to -1 gives 1 (d) (b) (c) 8 3 added to _ 3 gives (b) 1 subtracted from 1 gives (b) (a) =1 Sum of 10 5 and 12 is (೩) 27 _27 (c) _3 following statements is false Which ofthe (c) 4 < - 5 (d) 4> - 5 a)-4 > - 5 (b) - 4 < 5 Which of the following is in increasing order (a) 0 , 1 , - 1 (b)-1 , - 2 , - 3 (c)-1 , 0 , 1 (d)-1,1, - 2 Which ofthe following is correct (೩) _ 8 > - 7 (c) _ 1 < 0 ()-2>4 (b)1 <0 following number forms Which of the a pattern (b)~5 ,- 3 (c) 0 , 2 , 3 , 4 (d) 1 , 2 , 4 , 6 6 , - 3 - 2 0 3 Sum of _ 36 and 29 is] a) -651 (b) 65 (d) Which ofthe following will give answer with negative sign] 10 48 + 79 40 + 40 48 + 30 48 + ( - 39 ) (b) ೩) What will be the additive inverse of-1 ? 11 (a) -2 (b) - ] Sum of two positive integers is always-| 12 positive] (a) Negative () Sum of a negative and a positive integer is - 13 either positive] negative (c) always positive Always negative] Zero ) 01 ofintegers whose sum is -5 14. The pair -3 , -2 5, _1 ; a) (0) 15. 39 _ 50 1s a) Not possible (b) -89 10 -11 Math's online class what's up:+251900032087 Page repared by: ATTMEDOMER 47 - ShareChat")
![📖ఎడ్యుకేషన్✍ - Basic Algebra Formula (a+b)2 02 _ 20b +62 3 (a - b)2 200 +62 >02 'ಲ (a +b) (a-b) = a2_62] (x+a) (x+b) =22 + (+b) +ab (X+a) (X-b) = 12 + (0-b)T-0b (x-a) (x+b) =22 + (6-a) €-abl (x-a) (x-৮) = ~2_(@+৮) গ + @b = @3 +3ab (a + 6) (a + ৮)3 + 63 = @3-53_3@b (@~6) (a-b)3 (2 +y + z)2 = ೩2 _ ೈ2 +22 _ 2 2y2 | 239 + 212 _ (X+y-z)2 = 12 + 12 + 22 _ 2ல~~ 2~9 232 (x-y+2)2 = ~2 +~2_ 29 2y> + २r२ + ५२ (X-y-z)2 = 12 + 12 + 22_2Tg + 292- 22 T3 _ y3 (2+y+2)(22 + y2 _3xy2 +23_ + 22_ ೩2) ல 3~ = a " x2 +y2 = % [(c+y)2 + (v-y)2]] (x + a)(x + b)(x + c) = .3 + (a + b + c)२ + (ab + bc + ca )a + abc + y३ (2 + y)(22_ry+y2) = + (3-!) (೩2 _ ೩y + 12) x? " ~ys 2[(೩ _ !)2 _ (y _ 2)2 _ (೭ _ ೩)21 +y2 +22 Xyyz zX = ಲ + Basic Algebra Formula (a+b)2 02 _ 20b +62 3 (a - b)2 200 +62 >02 'ಲ (a +b) (a-b) = a2_62] (x+a) (x+b) =22 + (+b) +ab (X+a) (X-b) = 12 + (0-b)T-0b (x-a) (x+b) =22 + (6-a) €-abl (x-a) (x-৮) = ~2_(@+৮) গ + @b = @3 +3ab (a + 6) (a + ৮)3 + 63 = @3-53_3@b (@~6) (a-b)3 (2 +y + z)2 = ೩2 _ ೈ2 +22 _ 2 2y2 | 239 + 212 _ (X+y-z)2 = 12 + 12 + 22 _ 2ல~~ 2~9 232 (x-y+2)2 = ~2 +~2_ 29 2y> + २r२ + ५२ (X-y-z)2 = 12 + 12 + 22_2Tg + 292- 22 T3 _ y3 (2+y+2)(22 + y2 _3xy2 +23_ + 22_ ೩2) ல 3~ = a x2 +y2 = % [(c+y)2 + (v-y)2]] (x + a)(x + b)(x + c) = .3 + (a + b + c)२ + (ab + bc + ca )a + abc + y३ (2 + y)(22_ry+y2) = + (3-!) (೩2 _ ೩y + 12) x? ~ys 2[(೩ _ !)2 _ (y _ 2)2 _ (೭ _ ೩)21 +y2 +22 Xyyz zX = ಲ + - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_533131_a05db6a_1772517410548_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=548_sc.jpg "📖ఎడ్యుకేషన్✍ - Basic Algebra Formula (a+b)2 02 _ 20b +62 3 (a - b)2 200 +62 >02 'ಲ (a +b) (a-b) = a2_62] (x+a) (x+b) =22 + (+b) +ab (X+a) (X-b) = 12 + (0-b)T-0b (x-a) (x+b) =22 + (6-a) €-abl (x-a) (x-৮) = ~2_(@+৮) গ + @b = @3 +3ab (a + 6) (a + ৮)3 + 63 = @3-53_3@b (@~6) (a-b)3 (2 +y + z)2 = ೩2 _ ೈ2 +22 _ 2 2y2 | 239 + 212 _ (X+y-z)2 = 12 + 12 + 22 _ 2ல~~ 2~9 232 (x-y+2)2 = ~2 +~2_ 29 2y> + २r२ + ५२ (X-y-z)2 = 12 + 12 + 22_2Tg + 292- 22 T3 _ y3 (2+y+2)(22 + y2 _3xy2 +23_ + 22_ ೩2) ல 3~ = a \" x2 +y2 = % [(c+y)2 + (v-y)2]] (x + a)(x + b)(x + c) = .3 + (a + b + c)२ + (ab + bc + ca )a + abc + y३ (2 + y)(22_ry+y2) = + (3-!) (೩2 _ ೩y + 12) x? \" ~ys 2[(೩ _ !)2 _ (y _ 2)2 _ (೭ _ ೩)21 +y2 +22 Xyyz zX = ಲ + Basic Algebra Formula (a+b)2 02 _ 20b +62 3 (a - b)2 200 +62 >02 'ಲ (a +b) (a-b) = a2_62] (x+a) (x+b) =22 + (+b) +ab (X+a) (X-b) = 12 + (0-b)T-0b (x-a) (x+b) =22 + (6-a) €-abl (x-a) (x-৮) = ~2_(@+৮) গ + @b = @3 +3ab (a + 6) (a + ৮)3 + 63 = @3-53_3@b (@~6) (a-b)3 (2 +y + z)2 = ೩2 _ ೈ2 +22 _ 2 2y2 | 239 + 212 _ (X+y-z)2 = 12 + 12 + 22 _ 2ல~~ 2~9 232 (x-y+2)2 = ~2 +~2_ 29 2y> + २r२ + ५२ (X-y-z)2 = 12 + 12 + 22_2Tg + 292- 22 T3 _ y3 (2+y+2)(22 + y2 _3xy2 +23_ + 22_ ೩2) ல 3~ = a x2 +y2 = % [(c+y)2 + (v-y)2]] (x + a)(x + b)(x + c) = .3 + (a + b + c)२ + (ab + bc + ca )a + abc + y३ (2 + y)(22_ry+y2) = + (3-!) (೩2 _ ೩y + 12) x? ~ys 2[(೩ _ !)2 _ (y _ 2)2 _ (೭ _ ೩)21 +y2 +22 Xyyz zX = ಲ + - ShareChat")
![👩💻టెట్/DSC ప్రత్యేకం - (ix j).k+(j > k) i [Foreign 2012] Write the value of Ex. 6 (i x j) k + (j > k) i = k k +i i =1+1 = 2 Sol. +3k If0 is the angle between two vectors i - 2j Ex 7 [CBSE 2018] and 3i - 2j + k, find sin 0. - २j + k a = i _ 2j + 3k and b = 3ಗ Sol Let Ia X b | sin 0 |q |b | ; ; a Xb = 1 -2 1 3 !(4) - j(~8) + k(4) 4i + ৪j + 4k = la X b| = 4+1+4+1 =4+6 1q| = 1] + 4 + 9 Also V14 3 19+4+1 = 114 b ~ 216 416 sin 0 1141]4 (ix j).k+(j > k) i [Foreign 2012] Write the value of Ex. 6 (i x j) k + (j > k) i = k k +i i =1+1 = 2 Sol. +3k If0 is the angle between two vectors i - 2j Ex 7 [CBSE 2018] and 3i - 2j + k, find sin 0. - २j + k a = i _ 2j + 3k and b = 3ಗ Sol Let Ia X b | sin 0 |q |b | ; ; a Xb = 1 -2 1 3 !(4) - j(~8) + k(4) 4i + ৪j + 4k = la X b| = 4+1+4+1 =4+6 1q| = 1] + 4 + 9 Also V14 3 19+4+1 = 114 b ~ 216 416 sin 0 1141]4 - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_514724_1d55b92f_1772517308770_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=770_sc.jpg "👩💻టెట్/DSC ప్రత్యేకం - (ix j).k+(j > k) i [Foreign 2012] Write the value of Ex. 6 (i x j) k + (j > k) i = k k +i i =1+1 = 2 Sol. +3k If0 is the angle between two vectors i - 2j Ex 7 [CBSE 2018] and 3i - 2j + k, find sin 0. - २j + k a = i _ 2j + 3k and b = 3ಗ Sol Let Ia X b | sin 0 |q |b | ; ; a Xb = 1 -2 1 3 !(4) - j(~8) + k(4) 4i + ৪j + 4k = la X b| = 4+1+4+1 =4+6 1q| = 1] + 4 + 9 Also V14 3 19+4+1 = 114 b ~ 216 416 sin 0 1141]4 (ix j).k+(j > k) i [Foreign 2012] Write the value of Ex. 6 (i x j) k + (j > k) i = k k +i i =1+1 = 2 Sol. +3k If0 is the angle between two vectors i - 2j Ex 7 [CBSE 2018] and 3i - 2j + k, find sin 0. - २j + k a = i _ 2j + 3k and b = 3ಗ Sol Let Ia X b | sin 0 |q |b | ; ; a Xb = 1 -2 1 3 !(4) - j(~8) + k(4) 4i + ৪j + 4k = la X b| = 4+1+4+1 =4+6 1q| = 1] + 4 + 9 Also V14 3 19+4+1 = 114 b ~ 216 416 sin 0 1141]4 - ShareChat")
![👩💻టెట్/DSC ప్రత్యేకం - TY T ExamSiDE] MATH/ 0 MATHS WITH | Solutions MATHEMATICS MAIHS PHANEENDRA By PHANI ByPHANI a rad sin & tan 0 cr0 coS 0 0 16+ 12 6- 12 2-13 2+~3 15 5-1 5- 215 10 V10+215 15+215 18 5 5+1 5+1 110 _ 2v5 V10 _ 215 T 36 V10-215 15 + ] 5 V10 _ 215 5+1 5+1 V10-2V5 37 54 5+1 110 _ 215 10 5~2v5 5-1 110+215 21 15+2/5 72 5 5 16 + v2 16 - v2 51 2-3 2+13 75 12 TY T ExamSiDE] MATH/ 0 MATHS WITH | Solutions MATHEMATICS MAIHS PHANEENDRA By PHANI ByPHANI a rad sin & tan 0 cr0 coS 0 0 16+ 12 6- 12 2-13 2+~3 15 5-1 5- 215 10 V10+215 15+215 18 5 5+1 5+1 110 _ 2v5 V10 _ 215 T 36 V10-215 15 + ] 5 V10 _ 215 5+1 5+1 V10-2V5 37 54 5+1 110 _ 215 10 5~2v5 5-1 110+215 21 15+2/5 72 5 5 16 + v2 16 - v2 51 2-3 2+13 75 12 - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_434232_187316f1_1772429395703_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=703_sc.jpg "👩💻టెట్/DSC ప్రత్యేకం - TY T ExamSiDE] MATH/ 0 MATHS WITH | Solutions MATHEMATICS MAIHS PHANEENDRA By PHANI ByPHANI a rad sin & tan 0 cr0 coS 0 0 16+ 12 6- 12 2-13 2+~3 15 5-1 5- 215 10 V10+215 15+215 18 5 5+1 5+1 110 _ 2v5 V10 _ 215 T 36 V10-215 15 + ] 5 V10 _ 215 5+1 5+1 V10-2V5 37 54 5+1 110 _ 215 10 5~2v5 5-1 110+215 21 15+2/5 72 5 5 16 + v2 16 - v2 51 2-3 2+13 75 12 TY T ExamSiDE] MATH/ 0 MATHS WITH | Solutions MATHEMATICS MAIHS PHANEENDRA By PHANI ByPHANI a rad sin & tan 0 cr0 coS 0 0 16+ 12 6- 12 2-13 2+~3 15 5-1 5- 215 10 V10+215 15+215 18 5 5+1 5+1 110 _ 2v5 V10 _ 215 T 36 V10-215 15 + ] 5 V10 _ 215 5+1 5+1 V10-2V5 37 54 5+1 110 _ 215 10 5~2v5 5-1 110+215 21 15+2/5 72 5 5 16 + v2 16 - v2 51 2-3 2+13 75 12 - ShareChat")
![👩💻టెట్/DSC ప్రత్యేకం - Show that the points A, B, C with position vectors 19 2i - j + k, i - 3j - Sk 4j _ 4k and 3i _ respectively, are the vertices of a right-angled triangle Hence find the area ofthe triangle [AI2017] Given points are -j+1), B(i - 3j 501 5k) A(2i and _ (31 4j - 4k) (i - 3j - 51) - (2i _ j + k) = -i - 2j - 6k AB = BC 4j - 4k) - ({ - 3j - 5k) = 21 - j + k (3i _ (2i _ j + k) - (3i _ 4j - 4k) = -i + 3j + 5k CA Me notice BC.CA = _ 2 _ 3 + $ = 0 BC LCA triangle ABC is right-angled at C. K i j 1 1 CB | 3 5 Area of CA -1 X २ 2 2 _1 2/8i 11j - 5k | [ + v210 sq units 64 + 121 + 25 2 2 Show that the points A, B, C with position vectors 19 2i - j + k, i - 3j - Sk 4j _ 4k and 3i _ respectively, are the vertices of a right-angled triangle Hence find the area ofthe triangle [AI2017] Given points are -j+1), B(i - 3j 501 5k) A(2i and _ (31 4j - 4k) (i - 3j - 51) - (2i _ j + k) = -i - 2j - 6k AB = BC 4j - 4k) - ({ - 3j - 5k) = 21 - j + k (3i _ (2i _ j + k) - (3i _ 4j - 4k) = -i + 3j + 5k CA Me notice BC.CA = _ 2 _ 3 + $ = 0 BC LCA triangle ABC is right-angled at C. K i j 1 1 CB | 3 5 Area of CA -1 X २ 2 2 _1 2/8i 11j - 5k | [ + v210 sq units 64 + 121 + 25 2 2 - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_534004_a68e7a7_1772381532717_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=717_sc.jpg "👩💻టెట్/DSC ప్రత్యేకం - Show that the points A, B, C with position vectors 19 2i - j + k, i - 3j - Sk 4j _ 4k and 3i _ respectively, are the vertices of a right-angled triangle Hence find the area ofthe triangle [AI2017] Given points are -j+1), B(i - 3j 501 5k) A(2i and _ (31 4j - 4k) (i - 3j - 51) - (2i _ j + k) = -i - 2j - 6k AB = BC 4j - 4k) - ({ - 3j - 5k) = 21 - j + k (3i _ (2i _ j + k) - (3i _ 4j - 4k) = -i + 3j + 5k CA Me notice BC.CA = _ 2 _ 3 + $ = 0 BC LCA triangle ABC is right-angled at C. K i j 1 1 CB | 3 5 Area of CA -1 X २ 2 2 _1 2/8i 11j - 5k | [ + v210 sq units 64 + 121 + 25 2 2 Show that the points A, B, C with position vectors 19 2i - j + k, i - 3j - Sk 4j _ 4k and 3i _ respectively, are the vertices of a right-angled triangle Hence find the area ofthe triangle [AI2017] Given points are -j+1), B(i - 3j 501 5k) A(2i and _ (31 4j - 4k) (i - 3j - 51) - (2i _ j + k) = -i - 2j - 6k AB = BC 4j - 4k) - ({ - 3j - 5k) = 21 - j + k (3i _ (2i _ j + k) - (3i _ 4j - 4k) = -i + 3j + 5k CA Me notice BC.CA = _ 2 _ 3 + $ = 0 BC LCA triangle ABC is right-angled at C. K i j 1 1 CB | 3 5 Area of CA -1 X २ 2 2 _1 2/8i 11j - 5k | [ + v210 sq units 64 + 121 + 25 2 2 - ShareChat")
