Maths last 10 year papers

Gi e , ABC PQR A ( ABC) = a ( PQR) T P e: ABC ​≅ ​ PQR = ​a (ABC)AB2​ ​/PQ​2​ = BC2​ /​ QR2​ ​ = AC2​ ​/PR​2 a (P QR) : A Ra i f a ea f imila ia gle i e al he ae f c e di g ide B ,a gi e , a (AB C ) = 1 a (P QR) ​AB2​ ​/PQ2​ ​ = BC2​ /​ QR2​ ​ = AC​2/​ PR​2 =​ 1 AB2​ =​ PQ​2 /​ AB = PQ BC2​ ​ = QR2​ ​/ BC = QR AC​2 ​ = QR2​ ​ / AC = PR : - B SSS c g e ce a i m ABC PQR , he ce P ed. Downloaded from www.padhle.in

A 18). Le a ge PT a d QT a e d a ci cle f ce e O a h i fig e. B h he gi e a ge PT a d QT ch he ci cle a P a d Q e ec i el . We ha e e: le g h f PT = le g h f QT C c i :- d a a li e egme ,f m ce e O e e al i T chi g i f a ge . N POT a d QOT We k a ge make he igh a gle i h he adi f he ci cle. He e, PO a d QO a e adii . S , ∠OPT = ∠OQT = 90 Downloaded from www.padhle.in

POT a d QOT ∠OPT = OQT = 90 C mm h e e OT A d OP = OQ [ OP a d OQ a e adii] S , R - H - S le f imila i POT QOT He ce, OP/OQ = PT/QT = OT/OT PT/QT = 1 PT = QT [ he ce ed] A 19). Gi e , 4 a ​ ​ = 3 Ta ​ = 3 = P 4 B N , P = 3K, B = 4K, We k , a ​ ​ = ( e e dic la / ba e) , af e c m a i g , e e dic la = 3 a d ba e = 4 H e e = (3 + 4 ) = 5 N , i ​ ​ = e e dic la / h e e = 3/5 c = ba e / h e e = 4/5 N , (4 i - c + 1)/(4 i + c - 1) = (4 3/5 - 4/5 + 1)/(4 3/5 + 4/5 - 1) = (12 - 4 + 5)/(12 + 4 - 5) Downloaded from www.padhle.in

= 13/11 A 20). Gi e a 2A = c (A 18 ) c (90 2A) = c (A 18 ) [ a =c (90 )] C m a i g a gle e ge , 90 2A = A 18 90+18=A+2A 3A=108 A= 108/3 A = 36 A 20). Downloaded from www.padhle.in

Gi e , ABCD i a a e f ide = 12 cm. P, Q, R a d S a e he mid i f ide AB, BC, CD a d AD e ec i el . A ea f haded egi = A ea f a e 4 A ea f ad a = a2​ ​ 4 1/4 ​2 = (12)2​ ​ 3.14 (6)2​ =144 3.14 36 =144 113.04 = 30.96 cm2​ A 21). Downloaded from www.padhle.in

Le he igi al heigh f c li de = 10cm Ba e adi f c li de = 3.5cm ba e adi f hemi he e = 3.5cm( ame a ha f c li de ) The al face a ea ld be he m f he c ed face a ea f c li de a d he face a ea f 2 hemi he e . face a ea f c li de = 2 h face a ea f e hemi he e = 2 OR V l me f ice = l me f c e = He e , = 12 [ ∵ diame e = 24 m ] a d h = 3.5m N , l me f ice = 1/3 22/7 (12) 3.5 m = 1/3 22/7 12 12 7/2 m =22 24 = 528 m Agai , a ea f ca a cl h = c e face a ea f c e = l He e l = = 12 + 3.5 = 144 + 12.25 = 156.25 = 12.5 m Downloaded from www.padhle.in

∴ A ea f ca a cl h = 22/7 12 12.5 m = 471.43 m N =​ 280 = 140 2 2 The c m la i e f e e c j g ea e ha 140 i 182. Media cla i 10 15. l = 10, h = 5, N = 280, c.f. = 49 a d f = 133 MEDIAN = l + N c.f . h 2 f = 10 + 140 4 5 133 = 10 +​ 1*5 133 = 10 + 455/133 = 13.42 A 13.42 Downloaded from www.padhle.in

A 23). Gi e , eed f m b a i ill a e = 18 km/h . Le eed f eam = km/h S eed f b a d eam=(18 + ) km/h . A d eed f b a eam=(18 ) km/h . Time f he eam j e = 24 18 Time f he eam j e = 24 18 + N , Acc di g he e i , 24 - 24 =1 18 18 + 24( 18 + ) 24 (18 ) =​ 1 (18 ) (18+ ) 24 * 18 + 24 24 * 18 24 =1 324 2 48 / 324 2​ ​ = 1 48 = 324 ​2 2​ ​ + 48 324 = 0 Downloaded from www.padhle.in

2​ ​ + 54 6 324 = 0 ( + 54) 6 ( + 54) = 0 ( + 54) ( 6) = 0 Ei he + 54 = 0 , = 54 N , a he eed ca be ega i e, - 6 = 0 , = 6 A S , he eed f he eam i 6 km/h OR Le igi al eed f he ai be km/h. The , ime ake a el 63 km = 63/ h Ne eed = ( + 6) km/h Time ake a el 72 km = 72/( + 6) h Acc di g he e i , 63 + 72 =3 +6 ​ O l i g, 135 +378 = 3 ​2​ + 18 ​2​ - 39 - 126 = 0 Downloaded from www.padhle.in

( - 42)( + 3) = 0 = -3 = 42 A he eed ca be ega i e, = 42 Th , he a e age eed f he ai i 42 km/h . A The a e age eed f he ai i 42 km/h . A 24). Le he f c ec i e mbe i AP be (a 3d),(a d),(a+d) a d (a+3d) S , acc di g he e i . a 3d+a d+a+d+a+3d=32 4a=32 a=32/4 a=8......(1) N , (a 3d)(a+3d)/(a d)(a+d)=7/15 15(a 9d )=7(a d ) 15a 135d =7a 7d 15a 7a =135d 7d 8a =128d P i g he al e f a=8 i ab e e ge . 8(8) =128d 128d =512 d =512/128 Downloaded from www.padhle.in

d =4 d=2 S , he f c ec i e mbe a e 8 (3 2) 8 6=2 8 2=6 8+2=10 8+(3 2) 8+6=14 F c ec i e mbe a e 2,6,10 a d 14. A 25). Downloaded from www.padhle.in

​ABC i a e ila e al ia gle , he e D i ide BC i ch a a ha BD = BC/3 . Le E i he i ide BC i ch a a ha AE BC . ​N , ABE a d AEC ∠AEB = ∠ACE = 90 AE i c mm ide f b h ia gle , AB = AC [ all ide f e ila e al ia gle a e e al ] ​F m R - H - S c g e ce le , ABE ACE ∴ BE = EC = BC/2 N , f m P hag a he em , ADE i igh a gle ia gle ∴ AD = AE + DE ------(1) ABE i al a igh a gle ia gle ∴ AB = BE + AE ------(2) F m e a i (1) a d (2) AB - AD = BE - DE = (BC/2) - (BE - BD) = BC /4 - (BC/2) - (BC/3) = BC /4 - (BC/6) = BC /4 - BC /36 = 8BC /36 = 2BC /9 ​ AB = BC = CA Downloaded from www.padhle.in

S , AB = AD + 2AB /9 9​ AB - 2AB = 9AD H​ e ce, 9AD = 7AB OR Gi e : A igh a gled ABC, igh a gled a B T P e- AC =AB +BC C c i : d a e e dic la BD he ide AC . P f: We k ha if a e e dic la i d a f m he e e f a igh a gle f a igh a gled ia gle he h e e, ha ia gle b h ide f he e e dic la a e imila he h le ia gle a d each he . We ha e, ADB∼ ABC. (b AA imila i ) Downloaded from www.padhle.in

The ef e, AD/ AB=AB/AC i al) (I imila T ia gle c e di g ide a e AB =AD AC ..(1) i al) Al , BDC∼ ABC The ef e, CD/BC=BC/AC (i imila T ia gle c e di g ide a e O , BC =CD AC ..(2) Addi g he e a i (1) a d (2) e ge , AB +BC =AD AC+CD AC AB +BC =AC(AD+CD) Downloaded from www.padhle.in

( F m he fig e AD + CD = AC) AB +BC =AC . AC The ef e, AC =AB +BC A 26). Se fc c i :- (i) D a a li e egme BC i h mea eme f 6 cm. (ii) N c c a gle 60 f m i B a d d a AB = 5 cm. (iii) J i he i C i h i A. Th ABC i he e i ed ia gle. (i ) D a a li e BX hich make a ac e a gle i h BC a d i ie f e e A. ( ) C f e al a f li e BX amel BB1, BB2, BB3, BB4. ( i) N j i B4 C. D a a li e B3C' a allel B4C. ( ii) A d he d a a li e B'C' a allel BC. He ce AB'C' i he e i ed ia gle. Downloaded from www.padhle.in

A 27). L.H.S. = ( Si A - 2Si ​3A​ ) / ( 2c 3​ A​ - c A) = c A (1 - 2 i ​2A​ ) / [ 2 ( 1- i ​2A​ ) - 1] A =c A (1 - 2 i ​2​A ) / [ 2 - 2 i 2​ A​ - 1] A A =c A (1 - 2 i ​2​A ) / [ 2 - 2 i 2​ ​A - 1] =c A (1 - 2 i ​2​A ) / (1 - 2 i ​2​A ) A = Ta A ( He ce P ed ) Downloaded from www.padhle.in

A 28). G​ iven:      D​ iameter of upper end of bucket = 30 cm  R​ adius of the upper end of the frustum of cone, r1 = 15 cm  ​ D​ iameter of lower end of bucket = 10 cm  R​ adius of the lower end of the frustum of cone, r2 = 5 cm  ​Height of the frustum of Cone, h = 24 cm  ​Slant height of bucket, L = √[(h² + (r1 - r2)²]  L = √[24² + (15 - 5)²]  L = √(576 + 10²)  L = 26 cm  A​ rea of metal sheet = Curved Surface Area of bucket + area of lower end  = π(r1 + r2)L + πr2²  = 3.14(15 + 5) × 26 + π(5)²  = 3.14 × 20 × 26 + 25 × 3.14  = 1711.3 cm²  Hence, the Area of metal sheet used to make the bucket is 1711.3 cm².  ii).  We  should  avoid  bucket  made  by  ordinary  plastic  because  it  is  non-biodegradable. It is harmful for our environment.      Downloaded from www.padhle.in

Ans 29).    ​ H​ eigh f ligh h e = 100 m The a gle f de e i f 2 hi a e 30 a d 45 deg ee. T​ Fi d: If e hi i e ac l behi d he he he ame ide f he ligh h e, he fi d he di a ce be ee he 2 hi Sl i : Refe he a ached fig e I ABC AB = Heigh f e = Pe e dic la = 100 m BC = Ba e ∠ACB = 45 We ill e ig me ic a i fi d he le g h f ba e : I ABD e = Pe e dic la = 100 m AB = Heigh f BD = Ba e Downloaded from www.padhle.in

∠ADB = 30 me ic a i fi d he le g h f ba e : We ill e ig U e 3 = 1.732 ​ BD= 100 * 1.732 ​ BD = 173.2 m N e a e e i ed fi d he di a ce be ee he 2 hi i.e. CD = BD-BC. S , CD = BD-BC. CD = 173.2 m - 100 m CD= 73.2 m Th he di a ce be ee he 2 hi i 73.2 m. A 30). Downloaded from www.padhle.in

⟹ 720 704 20​f​ 18​f ⟹ 2​f ​ 16 ⟹ ​f ​ 8 OR Downloaded from www.padhle.in

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CBSE Ma he a ic 2017 G aI c: . ()A ac ( )T ac 30 d dd c A, B, C a d D. ( )S c Ac a 6 1 a ac . S c Bc a 6 2 a ac , S c C c a 10 3a ac . S c D c a 8 4 a ac . ( )T ac c .H ,a a c c a b dd a 3a ac a d 3 4a ac . Y a . aa ac ( ) U a ca c a d. Downloaded from www.padhle.in

Qe i Sec i -A (1 Ma Each) 1. Wha i he c diffe e ce f a A.P. i hich a​21​ a​7​ = 84 ? 2. If he a g e be ee a ge d a f a e e a i P a ci c e f adi a a d ce e O, i 60 , he fi d he e g h f OP 3. If a e 30 high, ca a had 10 3 g he g d, he ha i he a g e f e e a i f he ? 4. The babi i f e ec i g a e a ead f a hea f 900 a e i 0 18. Wha i he be f e a e i he hea ? Sec i - B (2 Ma Each) 5. Fi d he a e f , f hich e f he ad a ic e a i ​2​ 14 + 8 = 0 i 6 i e he he . 6. Which e f he ge i 20, 19 1 , 18 1 , 17 3 , ... i he fi ega i e e ? 4 2 4 7. P e ha he a ge d a a he e d i f a ch d f a ci c e a e e a a g e i h he ch d Downloaded from www.padhle.in

8. A ci c e che a he f ide f a ad i a e a ABCD. P e ha AB + CD = BC + DA 9. A i e i e ec he -a i a d -a i a he i P a d Q e ec i e . If (2, 5) i he id i f PQ 10. If he di a ce f P( , ), f A(5, 1) a d B( 1, 5) a e e a , he e ha 3 = 2 . Sec i - C (3 Ma Each) 11. If ad bc, he e ha he e a i (a2​ ​ + b2​ ​ ) 2​ ​ + 2 (ac + bd) + (c​2​ + d2​ ​ ) = 0 ha ea . 12. The fi e f a A.P. i 5, he a e i 45 a d he fa i e i 400. Fi d he be f e a d he c diffe e ce f he A.P 13. O a aigh i e a i g h gh he f fa e , iC a d D a e a di a ce f 4 a d 16 f he a g e f e e a i f C a d D f he he f e ec i e . If c e e a , he fi d he heigh f he f he e a e e. 14. A bag c ai 15 hi e a d e b ac ba . If he babi i f d a i g a b ac ba f he bag i h ice ha f d a i g a hi e ba , fi d he be f b ac ba i he bag. 15. I ha a i d e he i ( 24/11 , ) di ide he i e eg e j i i g he i P(2, 2) a d Q(3, 7) ? A fi d he a e f . Downloaded from www.padhle.in

16. Th ee e ici c e each f dia e e 3 c , a ci c e f dia e e 4 5 c a d a e ici c e f adi 4 5 c a e d a i he gi e fig e. Fi d he a ea f he haded egi . 17. I he gi e fig e, c ce ic ci c e i h ce e O ha e adii 21 c a d 42 c . If ∠ AOB = 60 , fi d he a ea f he haded egi . [U e 22/7 ] 18. Wa e i a ca a , 5 4 ide a d 1 8 dee , i f i g i h a eed f 25 /h . H ch a ea ca i i iga e i 40 i e , if 10 c f a di g a e i e i ed f i iga i ? Downloaded from www.padhle.in

19. The a heigh f a f f a c e i 4 c a d he e i e e a d 6 c . Fi d he c ed face a ea f i ci c a e d a e 18 c f he f . 20. The di e i f a id i c b id a e 4 4 2 6 10 .I i e ed a d eca ie adi a d hic e i ah c i d ica i e f 30 c 5 c . Fi d he e g h f he i e. Sec i - D (4 Ma Each) 21. S e f : 1 +​ ​ 3 ​=​ 5 ,​ ​ 1, 1 , 4 +1 5 +1 +4 5 22. T a i g ge he ca fi a a i 3 1 h . If e a 13 ch i e a e 3h i each a e ha he he fi he a , he h a e fi he a ? 23. If he a i f he f he fi e f A.P. i (7 + 1) : (4 + 27), he fi d he a i f hei 9 h e 24. P e ha he e g h f a ge d a f a e e a i a ci c e a e e a . 25. I he gi e fig e, XY a d X'Y' a e a a e a ge a ci c e i h ce e O a d a he a ge AB i h i f c ac C, i e ec i g XY a a d X'Y' a B. P e ha ∠ AOB = 90 . Downloaded from www.padhle.in

26. C c a ia g e ABC i h ide BC = 7 c , ∠ B = 45 , ∠ A = 105 . The c c a he ia g e h e ide a e 3 4 i e he c e di g ide f he ABC 27. A ae a e i f i g a a heigh f 300 ab e he g d. F i g a hi heigh , he a g e f de e i f he ae a e f i b h ba fa i e i i e di ec i a e 45 a d 60 e ec i e . Fi d he id h f he i e . [U e 3 = 1 732] 28. If he i A( + 1, 2 ), B(3 , 2 + 3) a d C(5 1, 5 ) a e c i ea , he fi d he a e f . 29. T diffe e dice a e h ge he . Fi d he babi i ha he be b ai ed ha e (i) e e , a d (ii) e e dc 30. I he gi e fig e, ABCD i a ec a g e f di e i 21 c 14 c . A e ici c e i d a i h BC a dia e e . Fi d he a ea a d he e i e e f he haded egi i he fig e. Downloaded from www.padhle.in

31. I a ai - a e ha e i g e , he ai - a e f a f f 22 20 d ai i a c i d ica a ha i g a dia e e f ba e 2 a d heigh 3.5 . If he a i f , fi d he ai fa i c . W i e ie a e c e a i . A e A 1). Gi e , a21 a7 = 84 (​ a + 20d) (a + 6d) = 84 a​ + 20d a 6d = 84 ​20d 6d = 84 ​14d = 84 d=6 S , he c diffe e ce i 6 Downloaded from www.padhle.in

A 2). Gi e , ∠ ​A B ​= 60 a d ​∠ ​A ​= 30 I​ igh a g e ​ A ​, OP =c ec30​ OA OP = 2 OP = 2a He ce, he e g h f OP i 2a. A 3). Gi e : Heigh f he e - 30 Le g h f had he g d - 10 3 S i: I A​ BC,​ a = AB BC Ta = 30 10 3 Ta = Ta 60 Downloaded from www.padhle.in

= 60 He ce, he a g e f e e a i i 60 A 4). e a ead f a hea f Gi e : P babi i f e ec i g a 900 a e i 0 18 T a a e = 900 P(E) = 0.18 N. = 0.18 T. N. = 0.18 900 N . f e a e = 900 0 18 = 162 A : He ce, he . f e a e i 162. A 5). Gi e : Q ad a ic e a i - ​ 2​ ​ 14 + 8 = 0 ​S , e e =​ ​ he i be = 6​ S​ f = ( 14) ​ ​ +​ 6​ = Downloaded from www.padhle.in

7​ = 14 =2 W​ e , P d c f = ​ * 6​ = 8 6​ ​2​ = 8 O P i g a e faf e . (i) 6 ( 2 )2​ ​ ​= ​ 8 ​6* 4/ 2​ ​ = ​ 8 24 = 8 ​2 =0 8 2​ ​ 24 = 0 8 ( 3) = 0 Ei he 8 = 0 3=0 =3 O F = 0, he gi e c di i i be a i fied SO, a e i be = 3 Downloaded from www.padhle.in

A 6). Gi e , A.P. i 20, 19 1 , 18 1 , 17 3 . 4 2 4 S0, hi i e a 20, 77 , 37 , 71 . 4 2 4 He e, a = 20, d = 77 - 20 = 77 80 = 3 4 4 4 Le a​ ​ i fi ega i e e a​ ​ + ( 1)d < 0 20 + ( 1) ( 3 )<0 4 20 ( 3 ) + 3 <​ 0 4 4 20 + 3 < 3 4 4 83 < 3 4 4 > 83 3 4 4 > 83 27.66 3 ( SI ce e ca be i f ac i , d i ff) A : 28​ h​ e i be he fi ega i e e f he gi e A.P. Downloaded from www.padhle.in

A 7). Gi e , a ci c e f adi OA a d ce ed a O i h ch d AB a d a ge PQ & RS a e d a f i A a d B e ec i e . Le AB be a ch d f a ci c e i h ce e O a d e AP a d BP be he a ge a A a d B. Le he a ge ee a P.J i OP S e OP ee AB a C. T e : ∠PAC = ∠PBC P f : I PAC a d PBC PA = PB [Ta ge f a e e a i a ci c e a e e a ] ∠APC = ∠BPC [PA a d PB a e e a i c i ed OP] PC = PC [C ] PAC PBC [SAS C g e ce] ∠PAC = ∠PBC [C.P.C.T] He ce P ed. A 8). Gi e : A ci c e che a he f ide f a ad i a e a ABCD T P e: AB + CD = BC + DA Downloaded from www.padhle.in

A i' ha he e g h f a ge d a f aee a i he ci c e a e e a , hich ea , AP=AS BP=BQ CR=CQ DR=DS (AP+PB)+(CR+RD)=(AS+SD)+(BQ+QC) AB+BC=AD+BC A 9). Le he i e a i , =0 a d a i , =0 ) P( ,0) a d Q(0, ) (beca a d id i ​(2, ) N , fi d id i 0+ =2 2 0+ ​= -5 2 Downloaded from www.padhle.in

S , =4, = 10 ​A - The c di a e f P: (4,0) ​ :​ (4,0) a d (0, 10) A 10). f P( , ), f A(5, 1) a d B( 1, 5) a e e a ​Gi e : Di a ce T​ e: 3 = 2 A i i gi e , PA=PB ( + 1)2 + ( 5)2 ( 5)2 + ( 1)2 = ​O a i g b h he ide : ( 5)2 + ( 1)2 = ( + 1)2 + ( 5)2 ​ 2​ ​ + 25 -10 + 2​ ​ +1 - 2 = 2​ +​ 1 + 2 + ​2​ + 25 - 10 10 2 = 2 10 ​ 10 2 = 10 + 2 ​ 12 = 8 ​3 =2 He ce, P ed! A 11) Gi e , ad bc (a​2​ + b2​ ​) 2​ ​ + 2(ac + bd) + (c​2​ + d​2)​ = 0 D = b2​ ​ 4ac Downloaded from www.padhle.in

= [2 (ac + bd)]​2 -​ 4(a2​ ​ + b​2​)(c2​ ​ + d​2)​ = 4 [a​2 ​c2​ ​ + b​2​ d​2​ + 2abcd] 4 (a​2​ c2​ ​+ a​2​ d2​ ​ + ​ ​b​2​ c2​ ​+ b​2​ d2​ ​) = 4 [a​2 ​c​2 ​ + b2​ ​ d​2​ + 2abcd - a​2​ c2​ ​- a2​ ​ d2​ ​ - ​ ​b2​ ​ c​2 ​- b​2​ d2​ ]​ = 4 [ a​2d​ 2​ ​ b2​ ​c2​ ​ + 2abcd] = 4 [a​2​d​2​ + b​2c​ 2​ ​ 2abcd] = 4 [ad bc]2​ H​ e ce, D i ega i e! A 12). Gi e : a = 5, a​ ​ = 45, S​ ​ = 400 S ,a e , S​ ​ = /2 [a + ] ​ he e i he a e 400 = ​ /2 [5 + 45 ] 400 = /2 [50] 25 = 400 = 400/25 = 16 N ,a e , a​ ​ = a + ( 1) d 45 = 5 + (16 1) d 45 5 = 15d 15d = 40 d = 8/3 He ce, = 16 a d d = 8/3 A 13). Gi e : Di a ce f 4 a d 16 Le heigh AB f e = h Downloaded from www.padhle.in

N , a e ca ee, i ABC AB ​= = a (90 ) AC 4 ​= c I ABD, AB = a AC 16 = a e . (i) a d (ii) Mi 4 16 = c a h2​ /​ 64 = 1 [c a = 1/ a a = 1] h2​ ​/ 64 h = 8 Heigh f e = 8 Downloaded from www.padhle.in

A 14). Gi e , . f hi e ba = 15 Le . f b ac ba = T a ba = (15 + ) Acc di g he e i , P (B ac ba ) = 3 P (Whi e ba ) (15 + ) =3 ​ 15 ) (15 + = 45 He ce, f b ac ba i he bag = 45 A 15). Le i R di ide PQ i he a i : 1 We ha ,​ R​ = (​ 1 2+ 2 1 , 1 2+ 2 1 ) 1+ 2 1+ 2 (24/11 , ) = ( (3) + 1(2) , (7) + 1( 2) ) +1 +1 =​ 3 +2 , 7+ 2 +1 +1 = ​ 3 + 2 = 24 11 (3 + 2) = 24 ( + 1) 33 + 22 = 24 + 24 33 24 = 24 22 9 =2 = 2/9 Downloaded from www.padhle.in

S , :1 = 2:9 = 7 2 = 7 ( 2 ) 2 1 9 + 2 +1 9 O i g a d i g he a e f = 4 11 A : Li e PQ di ide i he a i 2 : 9 a d a e f = 4 11 A 16). Gi e : Radi f a ge e i-ci c e = 4 5 c A ea f a ge e i-ci c e = 1 R2​ =​ 1 22 45 45 2 2 7 Dia e e f i e ci c e = 4 5 c , = 4.5/2 c A ea f i e ci c e = 2​ ​ = 22 4.5 4.5 7 2 2 Dia e e f a e i-ci c e = 3 c , = 3 2 ​ A ea f a e i-ci c e = 1 2​ 2 ​ 1 22 3 3 3 2 7 2 2 2 A ea f haded egi = A ea f a ge e i ci c e + A ea f 1 a e ici c e A ea f i e ci c e A ea f 2 a e i ci c e Downloaded from www.padhle.in

( 1 22 4.5 4.5) ​ + ( 1 22 3 3 - 22 4.5 2 7 2 7 2 2 7 2 4.5 1 22 3 3 2 ) - ( ​2 2 7 2 2 ) O i g, = 990 643.5 =​ 346.5 28 28 A - 12.37 c 2​ ​a i ae A 17). S i :A gef haded egi = 360 60 = 300 A ea f haded egi = 360 (R​2​ - ​2​) = 22 300 [42​2​ - 21​2]​ 7 360 = 22 5 ​63 21 7 6 Downloaded from www.padhle.in

He ce, he a ea f he haded egi = 3465 c 2​ A 18). Wid h f ca a =5.4 De h f ca a =1.8 I 60 i ., 25 f a e f h gh i . Which ea . i 40 i , he e i be 25 ​ ​40 ​= 50 , i be a ai ab e f i iga i . 60 3 He ce, he e f a e i he ca a = e f a d i iga ed. V e f ca a = b h . 1. 50000 3​ 3 A ea f a d i iga ed if 1​ 0​cm​ a di g a e i e i ed = ​ . 1. 50000 1 3​ 3 0.1 = 16,20,000 2​ A 19). S i : () =4c S a heigh f f Downloaded from www.padhle.in

Pe i e e f e = 18 c 2 R = 18 c ; R = 9 c Pe i e e f e b =6c 2 = 6 ; = 3/ c C ed S.A. f f = [R + ] = 4 [ 9+3 ] = 4 12 = 48 c ​2 He ce, he c ed face a ea f he f i 48 c ​2 A 20). Gi e : I e adi f i e = 30 c Thic e f i e = 5 c O e adi = 30 + 5 ;R = 35 c N , acc di g he e i , a id i c b id i e ed a d eca i ah c i d ica i e. S ,V . fh i e = V . f c b id h(R2​ ​ 2​ ​ ) = b h 22 h[ 35​2​ - 30​2​ ] = 4.4 2.6 1 100 100 100 7 22 7 h 65 5 = 44 26 1 100 100 O i g h, h = 44 26 100 100 7 = 11200 c / 112 22 65 5 A 21). Downloaded from www.padhle.in

Gi e : 1 + 3 = 5 +1 5 +1 +4 1 5 3 +1 - +4 = -5 +1 ( + 4) 5( + 1) =-5 3 ( + 1)( + 4) +1 ( + 4 5 5) =-5 3 ( 2 + 5 +4) +1 (4 1) =-5 3 ( 2+5 +4) +1 (4 + 1) (5 + 1) = 3( 2​ ​ + 5 + 4) 20 ​2​ + 4 + 5 + 1 = 3 ​2​ + 15 + 12 17 ​2 ​ 6 11 = 0 17 2​ ​ 17 + 11 11 = 0 17 ( 1) + 11 ( 1) = 0 ( 1) (17 + 11) = 0 A Ei he = 1 = 11/ 17 A 22). Le a fi b e a = h Le a fi b he a = ( + 3) h A gi e , ge he he fi b 40 h 13 Which ea 1 + 1 = 13 +3 40 +3+ = 13 ( ) ( +3) 40 (2 + 3) / ( ​2​ + 3 )= 13 40 Downloaded from www.padhle.in

13 2​ ​ + 39 = 80 + 120 13 2​ ​ 41 120 = 0 13 2​ ​ 65 + 24 120 = 0 13 ( 5) + 24( 5) = 0 (​ 5) (13 + 24) = 0 Ei he 5 = 0 13 + 24 = 0 = 5, = 24/13 ​ ca be 24/13 i ce e a e a i g ab i e. ​S , = 5 O​ e a fi he a i 5 h he a fi he a i 5 + 3 = 8 h A 23). We , R​ a i f f fi e f A.P. a e: 2 2 + ( 1) 2 2A + ( 1)D =​ 7 +1 4 + 27 ​N , P = 17 2 + 16 = 120 = 24/19 2A + (16)D 95 O i if i g, A +8 = 24/19 + 8D Downloaded from www.padhle.in

A ​ Ra i f 9 h e f A.P. i 24 : 19 A 24). aee a i a ci c e a e S a e e : The a ge d a f e a. T e : PA = PB C ​ ​ : J i adi OA a d OB a j i O P. P f : I OAP a d OBP OA = OB (Radii) ∠A = ∠B (Each 90 ) OP = OP (C ) AOP BOP (RHS c g.) PA = PB (c c ) A He ce, he a e e i ed. A 25). Gi e : Downloaded from www.padhle.in

XY a d X'Y' a e a a e a ge a ci c e i h ce e O a d ac C, i i e ec i g XY a A a d a he a ge AB i h i f c X'Y' a B Le j i i O C I OPAa d OCA OP=OC (Radii f he a e ci c e) AP=AC (Ta ge f i A) AO=AO (C ide) OPA OCA (SSS c g e ce c i e i ) S , ∠POA=∠COA.........(1) Si i a , ∠QOB ∠OCB ∠QOB=∠COB.........(2) Si ce,POQ i he dia e e f he ci c e, i i a aigh i e. The ef e, ∠POA+∠COA+∠COB+∠QOB=180 Downloaded from www.padhle.in

∘ S , f e a i (1) a d e a i (2) 2∠COA+2∠COB=180∘ ∠COA+∠COB=90∘ ∠AOB=900​ ​ He ce, P ed. A 26). ​ ​ G : BC = 7 c , ∠ B = 45 , ∠ A = 105 . ∠A + ∠B + ∠C = 180 [A g e e] Downloaded from www.padhle.in

105 + 45 + ∠C = 180 150 + ∠C = 180 ∠C = 180 - 150 ∠C= 30 Se fC ci : D a a i e eg e BC= 7 c . Ma e a a g e f 45 a i B a d 30 a i C, hich i e ec each he a i A. Th , ABC i a gi e T ia g e. N f B d a a a BY b a i g a ac e ∠CBY i h ba e BC he ide i e he e e A. N , ca e 4 i B1 ,B2 , B3 ,B4 BY ch ha BB1= B1B2= B2B3 =B3B4. J i B3C, a d d a a i e h gh B4M B3 C i e ec i g he e e ded i e BC a M. F M d a OM AC i e ec i g he e e ded i e BA a O. A Th , OBM i he e i ed ia g e h e ide a e (​ ) ​ i e f he c e di g ide f ABC. Downloaded from www.padhle.in

A 27). high f a i e .Ca dDa e ie Le ae a e i a A, 300 ba f i e . S ,I igh ABC, BC = c 60 AB 300 = 1/ 3 = 300 3 = ​100 3 3 3 = 100 1 732 = 173 2 Downloaded from www.padhle.in

I igh ABD, BD =c 45 AB 300 = 1 = 300 Wid h f i e = + = 173 2 + 300 = 473 2 A Wid h f i e i 473 2 . A 28). Gi e : A( + 1, 2 ), B(3 , 2 + 3) a d C(5 1, 5 ) a e c i ea T fi d: he a e f . if A ( 1, 1) , B( 2, 2), C( 2, 2) a e h ee c i ea i he A ea f ia g eABC = 0 1 [ 1( 2- 1) + ( 3- 1) + 3 ( 1- 2) ] =0 2 Gi e A( +1,2 ) ,B(3 ,2 +3) ,C(5 -1,5 ), he ef e 1 ( +1)[2 +3-5 ]+3 [5 -2 ]+(5 -1)[2 -(2 +3)] =0 2 ( +1)[3-3 ]+3 *3 +(5 -1)(-3)=0 Downloaded from www.padhle.in