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6 J# k# JOH Q } ` = ~ [~ H < [24-294] Neglect of mathematics works injury to all knowledge. Jx K # ~ [~ H <, XH J^ ` L ^ _, POQ ` ` " `. W` _ ij #, i ^ ` = Y ` = # WK #. XH ^ ~ } = H, O, ^ H }=, ~ x =, ` H ~ =, c = H O h " o # _ Q x, H ZQ = u,kq = ` " ~ " ^ g =~ H " } [ = K # _ ` ## Ja =$k ~ K H Q } ` i* #= J= ~ = Jx ~ [~ H < K #.. ik =.2 #i = ~ =, #+ = = i ~ =_.3 ` = L Q =, Jk H = Q U~ _ W` ~ Y~ (Overhead Expenses) ` _ = i #.4 K H =_.5 ~ =_ H K H =_ H = ^ ^ =.6 ~ [= M ` (Fixed Deposits) = i J# =~ # H P=~ # x M ` (Recurring Deposits).7 q = # ` =.8 H = = i x. ik = u = x+ ` # x` rq` ~ Q '' Q J< < xh K ~ _ xh u H }O ` O K O\ _. ^ x H ~ x K _ #, ~ # O ` # H = ( = = )# qx y < _. ~ u Q O # O_ < \ =~ _ = = i _ q^ #O, L` u = i q =O\ q+ = #= _ x+ u = i J# ` = # L y < _. * H z = Q = # ` * =, ` O* = ~ x $ n ~ P = < \ H ` H PH ~ }` ^ $_ OQ H ~ }O, = # s " \ x~ }O L yoz# ~ # x+ u, " i < } =. =~ = O@ ; + H ~ = P~ Q = ; P^ = Y~ " ^ Q " \ x ''H ~ }= e` = J< O O^ = OK = x J< H q+ ` = _ _ = # k. g\ < q = # ` = JO\ ~. = #O Ja =$k K =_ H J C K _ OQ x, ~ x K _ OQ x K =. = # ~ H }, P~ Q =, q^, q ye# J= ~ # f~ ^ ` # ` " K # k. nx H ~ = #= qq^ ~ H # # ` = # K e < =. D J^ = #O^ = # rq` O` = _ _ O_ q+ = # Q i ` H O^ =. 2

7 J# k# JOH Q } ` =.2 #i = ~ : =, #+ = = i ~ =_ = #= = O^ ` ~ Q u < =, #+ =, ~ =_ # Q i < ~ H < =. XH i [ H ` K O^ =. =, #+ = = i ~ =_ e` = (i) = = J= H ^ ~ H # " (ii) #+ = = H # ^ ~ J= H ^ ~ (iii) ` = = = H # ^ ~ #+ = (iv) #+ ` = = H # ^ ~ J H = ~@ (v) ~ =_ (I) = (vi) " ` = = J + =_ = Pnr.3 ` =, Jk H = Q U~ _ Y~ (Overhead Expenses) = ^ ~ ( H ) ` _ (Discount) = i # (Tax) H L Q =.3.. ` = H L Q = = #= ` = # Q i WO` = O^ ` ~ Q u < < ~ H < =. W C_ = # ` e # ^ xx W K CH O^ =. (i) ~O_ ` = = 2% = 2 (ii) 600 Hr 8% = = 48 kg (ii) 25% = 25 = 5 = 4 4 W _ = #= H x H ` = # Z L yok =K < ` O^ =. L^ ~ }. ~O_ ~ _ k # ` " O`? `2. 70 = ( ) = = 270 H = # ` = = # = = 5 5 %. 9 = 2 = 50 % = 4 = 25 % = 4 3 = 75 %

8 J^ = L^ ~ }.2 XH " ` = 2% J# #k `080 J~ # " ` = # H # Q # =. " ` = _ H x Jx J# H # =. " ` = 2% = `080 2 # x = 080 x = 080 = ` ` " ` = = `9000 L^ ~ }.3 25 = Ok q^ ~ 72% = Ok H = Oz g} = H " ~. H g} = x " i OY H # Q # =? H g} = H " i ` = = 72% H g} = H " i OY = 25 = Ok q^ ~ 72% = = 25 8 = 7 = Ok. H g} = x" ~ L^ ~ } % ` =Q Q OY H # Q # = % = 5 ` = # H = # " ` = = = 204. # = 8 q^ ~ # 240 = 36 L^ ~ }.5 XH WO\ H ^ ~ 5 H # O_ 2 H ` y #k. ` y # ^ ~ ` = # H # Q # =? " = `5,00,000 = i# " = ` 2,00,000 ` y # ^ ~ = 5,00,000 2,00,000 = 3,00,000 ` ` y # ^ ~ ` = = # = 20% * H = LOK H #= # JO = iy# " ` = ~ Q ^ ` = = x[" # ^ ~ ` iy# " ` = ` ~ Q ^ ` = = x[" # ^ ~ 4 4

9 J# k# JOH Q } ` = J =.. i # = ^ #= # Z# H # =. (i) 25 O_ 5 H = O_ H =. H = O_ ` =... (A) 5% (B) 25% (C) 0% (D) 20% (ii) ~ `, ~ ` 5 f = H # (sweets) 20%, 80% Q q lok H < ~. " i W^ ih Zx f = H =K #? = J= = ` # z# ` #= XH Q = OQ ~ O ` H < ~. W C_ h= " ~ uh # K z < " ^ \ ~ A OQ ~ O ^ ~ # Q i OK =. u< OQ ~ O ^ ~ Q i Oz ~ Q ^ ` = # H # Q # =. 2 = %. 25 (A) 25 (B) 4 (C) 8 (D) 5 (iii) XH Q A ` Q 5% J# #k 30 J~ # " ` O Zx? (A) (B) 200 (C) 50 (D) 300 (iv) ^ ~ `34,000. D O=` ~ = ^ x ^ ~ 25% iy#, ` ^ ~ ZO`? (A)` 6,500 (B) ` 8,500 (C) ` 8,000 (D) ` 7,000 (v) XH = H ` # < i r` = `20,000 #O^ `3000 ^ K #. J` x ^ ` " O`? (A) 5% (B) 5% (C) 0% (D) 20% 2. (i) XH " ` = 20% J# #k 40 b@~ # < J~ # " ` O Zx b@~ # < L# k? (ii) 25% }= J# #k 5000H.g. = #" #, " ` O } ^ ~ = ZO`? (iii) XH " ` = 3.5% J# #k `54.25 = #" #, " ` " O`? (iv) 60% J# #k 30 xq + = = #" #, " ` O H = # H # Q # =? (v) XH = = 4% J= H # J# #k `2 = #" # J= H ^ ~ # H # Q # =? 3. ^ ` # r` = P@q_ (Recreation) H = Y~ K `2000 5%# = #" #, P" r` " O`? 4. x K _ ~ # = q _ O_ 25% J#Q 250 O_. J~ x " ` O O_ Zx? = i = Oz O_ Zx? 5 5

10 J^ = 5. ~ } ` # 2= ` ~ Q u sh f # = ~ H Ok \ H W= _ #q. = ~ ` = # H # Q # =. J` k H ~ O = f # = ~ = ~ ` = = ~ (i) POQ = (ii) ` Q (iii) H (iv) uh = (v) ~ # = (vi) r= = XH ~ H H\ [@ 20 O^ = # " ~ H ~ ` P_ ~. nx O^ " ^ \ ~ 25% QeK ~. J~ # " ^ \ ~ Qez# O^ = Zx? 7. ~ " `0,000 XH O=` ~ = # 8% ~ =_ K C# XH O x K #. J` # 5 O=` ~ = ` ~ ` O^ =_ ZO`? 8. XH = Q i O ^ ~ `200/ ^ xh JOQ _ " _ 5% ` _ ~ z #, J= H ^ ~ # H # Q # =? 9. XH OQ }H (H ) " ~ O #O^ 500 = Ok = M = d sh [~ J~ < ~. JO^ 2% = Okx Z# H xi. J~ ` ZO` = Ok Z# H # _ i? = i ZO` = Ok Zx H H ^? H # Q # =. 0. XH q = = 30% ~ y 40% ` ` < Q = (zinc) = i q ye#k x H. 20 H. Q. q = = x H ZO` = # k?. O_ < = i ` = ~ ~ ^ ~ # Q = OK ~ u Zx H xezi. O_ < 44% J#Q 484 F@, ` = ~ 36% F@ a Oz#q. (i) " ` = Q = = F@ OY ZO`? D W^ ~ J ~ F@ " x " i OY # H # Q # =? 2. XH = H ` # r` = 40% [#= #, 20% WO\ J^ q Q ` Q = ^ K #. ZO` ` = ^ K #? J` x r` = `34,400 J~ # J` # ^ K " ` = # H # Q # =? 3. } ` WOw + #O^ 50 = ~ 35 = ~ Q } ` = #O^ 30 = ~ H 27 = ~ k Oz#k. ~O_ O\ U ~ O = Z = = ~ O^ #? Zx = ~ Z =Q O^ #? 4. XH = H ` # r` = 5% # Q Ok# _ `250 = #" #, J` x r` " O`? 5. q" = L yok ^ (suit) ^ ~ `2 # O_ `2500 iy#k. iy# ^ ~ ` = # H # Q # =? 6 6

11 J# k# JOH Q } ` =. 40% = % % 2. XH ` ~ Q u 25% q^ ~ #_ z 65% q^ ~ k K H " #= (bicycle) # q ye# " ~ ~ #O^ =K K < ~. ~ =K " i ` = # H # Q # =? 3. XH ` ~ Q u 30% = Ok q^ ~ On, 50% = Ok ` Q #, q ye# " ~ OK + # " i k f + Q f < ~. ZO` = Ok OK + # k f + Q f < ~. 4. XH #Q ~ = 30%, 40% = Q " ~, q ye# " ~ L< ~. ` = # H # Q # =? Q } + = i Q qo^ < J< =~ =^ J= p~ # H # k. Q } Q. " O_ # ` Q. H O = # H p~ # < #. Q qo^ < 300 Q " O_ # ` 200 Q. H O = # H e p~ # < #. ~O_ O\ # " O_ ` = # H # Q x Z=i [s = Oz^ x H # Q # =? (Q = xh : Z =Q " O_ H ey L# k - = Ozk).3.2 = = i #+ = L Q = D J^ = = #= #+ = H # k < ~ H O^ =. (i) W C_ J= H ^ ~ H ` = H # Q # =. D H Ok O^ ~ = # O_ Q OK ^ =. ~ * + XH H = # `80 H x ` # ` _ H J" #. J` xh 5% = ~ = ## ^ xx ZO` J= = #? ~ * + H # H = ^ ~ `80, J~ # nxx H # ^ ~ (C.P.) JO^ ~. 5% = H = ## Jq # ^ ~ H # Q ` = = H # ^ ~ 5% = 5 80 = `4 nxx \ = # ` e #k. J= H ^ ~ > H # ^ ~ J= H ^ ~ = H # ^ ~ + = = = ` 84. ~ * + H = # Jq # ^ ~ = ` 84 D H # ` = # = yoz K =K #. { = Jq # ^ ~ = H # ^ ~ = ^+ 5h # = 05 # 80 = ` 84.

12 J^ = (ii) W C_ = #= H # ^ ~ H ` = # H # Q D H Ok O^ ~ = # O_ Q OK ^ = XH JOQ _ " _ XH K u Q _ ~ = # 5% = ` ` 540 J" #. J~ # Q _ ~ = H # " ZO`? JOQ _ " _ 5% = # \ Q _ ~ = J" #. WH _ H # ^ ~ # ~ = ^. JO^ = = #= H # ^ ~ # ` Jx J# O^ =. = H # ^ ~ 5%. ` = = 5% H # " ` nx # O_ = 5 = ` 5. J= H ^ ~ = H # ^ ~ + = = + 5 = ` 05. WH _ J= H ^ ~ `05 J~ # H # ^ ~ ` JQ #. WH _ Q _ ~ J= H ^ ~ (S.P.) `540 J~ # H # " (C.P) = 540# 05 ` Q _ ~ = # `54.29 JOQ _ " _ H < #. D H # ` ^ u k : H # ^ ~ (C.P) = = ` { ` = Jq #^ ~ = = = ` = #= Jq #^ ~, H # ^ ~ H Ok q^ = Q ` = # " =K # 8 8

13 J# k# JOH Q } ` = L^ ~ }.6 g ^ XH H ~ \.q x `5,200 H x 20% #+ = # Jq #, Jq # ^ ~ ZO`? ~ L yoz# ^ u #+ = H # ^ ~ 20% J~ # = 20 # 5200 = ` 3040 J= H ^ ~ = H # ^ ~ #+ = = 5,200 3,040 = ` 2,60 ~ = i ~ + < ~ ^ ~ K # H [" XH >. \.q H Jq # ^ ~ `260 JQ #. L^ ~ }.7 XH \ x 5% #+ = ` `3600 Jq # H # ^ ~ ZO`? ^ q L yoz# ^ u #+ = 5% J#Q, H # " x J# H x# #+ = = `5 JQ #. H # ^ ~ =[-5]= ` 85 J.^. ` 85 J~ # H.^. = `. Jq # ^ ~ ` 3600 J~ # H # ^ ~ = # = ` H # " ` 6,000. ( H ) ( H ) ~ + < L yoz# ^ u H # ^ ~ = ` 5, 200 #+ = = 20% #+ ` = Jq # ^ ~ = H.^. = - 20 # 5200 = 80 # 5200 = ` 2,60 ~=u L yoz# ^ u #+ = = 5%. Jq # ^ ~ = ` 3,600 H # ^ ~ = #+ ` = J.^. = = = ` 6,000 = = H # ^ ~ = /#+ = Jq # ^ ~ " + OQ q + < 6,000 9% = " H X" < 3,500 2% #+ = H \ c~ " 3% #+ = 6,786 2½% = 7,000 U~ H O_ + #~ 32,400 7% = 9 9

14 J^ = L^ ~ }.8 H = H # ^ ~ 0 H = Jq # ^ ~ H = #= Q = # k. nx = H #+ = H # Q x ` = # H # Q # =? H = J= H ^ ~ # x Jx J# H # =. 0 H = J= H ^ ~ = `0x H = J= H ^ ~ = `x H = H # ^ ~ =0 H = Jq #^ ~ = `0x WH _ Jq # ^ ~ > H # ^ ~ ` = = Jq # ^ ~ H # ^ ~ ` = = = H # ^ ~ = x 0x = ` x 0 0 = x 0 x = 0%. L^ ~ }.9 XH = H ~O_ K u Q _ ~ = # `594 J" #. ^ x XH ^ xx 0% = # ~O_ = ^ x 0% #+ = H ey# " ` = J` xh = H #+ = H # Q x ` = # H # Q # =? " ^ \ K u Q _ ~ J= H ^ ~ = `594 = = 0% ` " ^ \ K u Q _ ~ H # ^ ~ = { ` = J= H ^ ~ = # 594 ^+ 0h = # 594 = `540 0 J^ q^ = Q ~O_ = K u Q _ ~ #+ = 0% J~ # = #+ ` = J= H ^ ~ = # 594 ^- 0h = 90 # 594 = `660 " ` = = # = H #+ = H # Q #= ## = #= ~O_ O\ H # ^ ~, Jq #^ ~ # H # Q #= #. ~O_ Q _ ~ = H # " " ` = = = `,200. ~O_ Q _ ~ = Jq #" " ` = = = `,88. " ` = #+ = =,200,88 = `2 #+ ` = = = #+ = H # " # = %.

15 J# k# JOH Q } ` =.3.3. Jk H = Q U~ _ W` ~ Y~ ( H ) ~ " } Y~ (Overhead Expenses) = " ` O _ ` H * ~ " o XH U~ ~ # `8000 H < ~. " i W JOQ _ H ^ Q ~ ^. K ^ ~ = = " ^ xx WO\ H f H x " ~ H O_ x (~ " } ) = \ _ i. ^ x ~ " } Y~ Q # `500 J~ #k. W C_ P Z~ ~ ^ ~ `8,000 = ` = H ^, ^ x` ~ " } Y~ # (Overhead) `500 K i # ^ x< ~ " } Y~ (Overhead Expenses) Jx = _ #. W C_, U~ ~ H # " = J ^ ~ + ~ " } Y~ = 8, = ` 8,500 H ~ ` O _ XH ` = ~ f H ~ # K < _ ~ =^ `2,75,000 H < ~. H # ` ~ " ` ^ xx ~ OQ `25,000, ` # O` = ih f H x Q # `2,000 Y~ K #. D H Ok [" e=. (i) H ~ H " ` = ^ ~ ZO`? (ii) H ~ H J ^ ~ (real cost)zo`? (iii) WH _ K _ # W` ~ Y~ Uq? H #O^ ~ OQ " Y~, ~ " } Y~ # H e W` ~ Y~ (overhead expenses) JQ #. H ~ H # ^ ~ = J ^ ~ + W` ~ Y~ = 2,75,000 + (25, ,000) = 2,75, ,000 = `3,02,000. nxx \ = #= ` H # k U= #Q XH = = # H # # _ ^ ~ = ` = H H H x Y~ _ H H = O^ U~ _ #. D Jk H Y~ # H # ^ ~ ` H = #. nx< W` ~ Y~ (Overhead Expenses) JO^ ~. nx Q (repairs), e(labour charges), ~ " } (transportation)" ^ Q Y~ H e LO_ #. L^ ~ }.0 ~ A XH " \ ~ H # `36,000 H x H x Q = * _ Oz (Extra fitting) JO^ = Q = H O` _ Y~ K #. J` _ ^ xx 0% = \ Jq # `44,000 =K #. J` _ H ` Q = * _ Q # ZO` Y~ K #?

16 J^ = H # ^ ~ x ` J# H # =. = = 0%, J= H ^ ~ = `0 J= H ^ ~ `0 J~ # H # ^ ~ ` Jq # ^ ~ `44000 J~ # H # ^ ~ = # = ` 40,000 0 H ` Q = # * _ J~ # Y~ = 40,000 36,000 = ` 4,000. J =.2. Jq # ^ ~, H # ^ ~ H # Q # =. H # ^ ~ Jq # ^ ~ = #+ = (i) ` 7,282 `208 (ii) ` 572 ` 72 (iii) ` 9,684 ` 684 (iv) `,973 ` 273 (v) ` 6,76,000 `8, M m # io = Jq # H # ^ ~ ^ ~ (i) ` 320 ` 384 (ii) ` 2,500 ` 2,700 (iii) ` 380 ` 36 = & ` = #+ = & #+ ` = (iv) ` 40 (v) ` 5,000 ` 500 = 2 2 ` 2 #+ = 3. H Ok " \ H 5% = ~ # J= H ^ ~ # H # Q # =. (i) k K H " #= " `700 = i W` ~ Y~ `50. (ii) XH OQ }H " * " `,50 = i ~ " } Y~ `50. (iii) XH O_ ~ O ` = (table-top wet grinder) " `2,560 Wz H x ^ xx Q # `40 Y~ K i.

17 J# k# JOH Q } ` = 4. XH J= H ^ ~ _ XH " * # 0% = ` `,320 J" #. J~ # H # ^ ~ ZO`? 5. 6 < \ H = H # " 2 < \ H = Jq # " H = #" # ` = # H # Q # =? 6. XH = H ~O_ = = # `375 J" #. ^ x " ^ \ = = x 25% = # ~O_ = ^ xx 25% #+ = # J" #. " ` = J` xh = H #+ = H # Q x, ` = # H # Q # =? 7. J# ~ < XH WO\ x `7,75,000 H < #. ^ xx JO^ = Q = Q # (interior decoration) `,25,000 Y~ K #. ` ~ " ` ^ xx 20% = \ J" #. WO\ x Jq # ^ ~ # H # Q # =? 8. J= XH ` W J~ " " ~ Y~ \ H ` Q (remodelling) = ~ #. ^ xx 20% = \ # ~O_ H J" #. P W H # ^ ~ ZO`? 9. [ = ~ XH = # #Q ~ = # ^ ~ = Q `2,00,000 H x Q Q # `,45,000 Y~ K #. J` # ^ xx `25,00,000 J= = #x ` Oz " ~ uh #O^ Q # `5,000 Y~ K #. J` x ` " O`? 0. XH = H ` # ^ Q ~ = # ~O_ " ~ ~ * u H # XH H \ `3,605 J" #. ^ x XH ^ xx 5% = #, ~O_ =^ xx 9% #+ = # J" #. ^ x J` xh H ey#k = H #+ =? (Q = xh : H # ^ ~ (C.P) x H # Q #= # ).3.4 = ^ ~ H ` _ (Discounts) L Q = x# \ k#= [ ` # ` e ^ O _ ` H O H Ou O_ H JOQ _ H #. P" K JOQ H Ok # K K #. ^ x J~ = P" ^ _ ^. nx< = # ` K H # K XH JOQ _ " toz XH Q # H < #. P Q # ^ ~ `550 Jx Q i Oz = # k. nx< Q i O ^ ~ (Marked Price) JO^ ~. P" JOQ _ " _ H `550 ~ z #k. J~ ` JOQ _ " _ uiy q ye# z ~ ~ K #. JO` H H 20% = ^ ~ ~ K K < = x K #. WH _ 20% = ^ ~ H ` _ J#Q Q i O ^ ~ 20% ` y O Jx J~ =. ` _ = =` 0. Q i O ^ ~ # O_ ` y OK " ` = # ` _ ( H ) = ^ ~ Jx JO^ ~. ^ ~ }= Q XH = = Q i Oz# ^ ~ < Q i O ^ ~ H * a` ^ ~ JO^ ~. 3 3

18 J^ = [ JOQ _ " _ H ~ z # " ` = `440 = `550 - `0 = Q i O ^ ~ - = ^ ~ nxx \ = #= ` H # k = ^ ~ = Q i O ^ ~ J= H ^ ~ J= H ^ ~ = Q i O ^ ~ = ^ ~ ( H ) ` _ Q i O ^ ~ = J= H ^ ~ + = ^ ~ ( H ) ` _ L^ ~ }. XH H Q i O ^ ~ `500 J~ # ^ xx `350 Jq i. nx ` _ ` " O`? Q i O ^ ~ (M.P) = ` 500, J= H ^ ~ (S. P) = ` 350 ` _ = Q i O ^ ~ Jq # ^ ~ = = ` 50 `500 Q # ` _ = ` 50 ` Q # ` _ = ` _ ` = = 0%. # L^ ~ }.2 XH Q # * a` ^ ~ `220 ^ xh 20% ` _ ~ z # J= H ^ ~, ` _ H # Q # =. Q # H Q i O ^ ~ (* a` ^ ~ ) `220 ` _ = 20% 20 ` _ = # 220 = ` 44 J= H ^ ~ = Q i O ^ ~ ` _ = = ` 76. O_ Q O^ ~ = H J > H, M n = i W` ~ ^ H }= qq^ = = 0% H 20% H 30% = ^ ~ ~ = # ` e #. ` q < 'P_ #O^ " ~ ` = " ~ = =$k = i H ` L` ` # ik = Q # J< H = = g ^ 0% # O_ 50% =~ = ^ ~ ~ = # ` e #. ` _ J# #k Q i O ^ ~ W < ~. H \ Q i O ^ ~ < P^ ~ = Q f H #= # 4 4

19 5 5 J# k# JOH Q } ` = L^ ~ }.3 XH c~ " 5% ` _ ~ z # ` ~ " ` `5225 Jq # Q i O ^ ~ # H # Q # =? H $+ L yoz# ^ u D H #O^ = ^ ~ ` = ~ = _ #k JO^ = Q i O ^ ~ x ` J# H # =. ` _ ` = = 5% ` _ = 5 = ` 5. J= H ^ ~ =Q i O ^ ~ ` _ = 5 = ` 95 J= H ^ ~ 95 ~ J~ # Q i O ^ ~ J= H ^ ~ = `5225 Q i O ^ ~ = ` c~ " Q i O ^ ~ = ` 5,500. L^ ~ }.4 XH JOQ _ " _ ` # " _ H ^ ~ (customers) 0% ` = = ^ ~ ~ z _ J` # 20% = # O^ K < _. XH = = H J= H ^ ~ # `450 J~ # Q i O ^ ~ # H # Q # =. =x` L yoz# ^ u: Q i O ^ ~ # ` J# H # =. ` _ = 0% Q.^ ~ = 0 Q.^ ~ = 0 # = ` 0 J.^. = Q i O ^ ~ - ` _ = - 0 = ` 90 = = 20% H.^. = 20 # 450 = Rs. 90 J.^. = H.^ + = = = ` 540. J= H ^ ~ 90 J~ # Q i O ^ ~ J.^. ` 540, Q.^ ~ = 540# = ` ` Q i O ^ ~ = ` 600 ` = = H Q i O ^ ~ =` 600 [ H ] [ H ] q + L yoz# ^ u J= H ^ ~ (S.P) = `5225 ` _ = 5% Q i O ^ ~ (M.P) =? Q.^ ~ = J= H ^ ~ ` _ % = ` j# = # = ` 5,500. q= ` = # = yok ^ u ` _ = 0%, = = 20%, H.^ ~ = ` 450, Q.^ ~. =? { % Q.^ ~ = ` _ % H # ^ ~ ^+ 20h = # 450 ^ - 0 h = 20 # = ` 600

20 J^ = L^ ~ }.5 XH _ ~ 0% ` _ ~ z 0% = # O^ K < _. XH H Q i O ^ ~ `220 J~ # J= H ^ ~ ZO`? Q O^ < L yok ^ u = O^ < L yok ^ u Q i O ^ ~ = `220 ` _ = Q i O ^ ~ 0% ` _ = 0% = 0 # 220 = Rs. 22 = = 0% Jq # ^ ~ (S.P)= Q.^ ~ ` _ Q i O ^ ~ = ` 220 = = ` 98 { ` _ % H # ^ ~ (C.P) = `0 J~ # H # ^ ~ = % Q.^ ~ = = H # ^ ~ 0% [ H ] = 0 # = Rs. 0 J= H ^ ~ = H # ^ ~ + = = + 0 = `0 J.^. `0 H # ^ ~ ` JQ # J= H ^ ~ `98 J~ # 98 H # ^ ~ = # 0 = ` 80. L^ ~ }.6 XH \.q x 0% = i 20% =~ Q = ^ ~ ~ z `4,400 H Jq #, Q i O ^ ~ # H # Q # =? J= H ^ ~ = ` 4,400 Q i O ^ ~ # ` J# H # =. " ` _ = 0% = " ^ \ J= H ^ ~ = 0 = `90 ~O_ = ` _ = 20% = 90 ~O_ = ` _ ` ~ " ` J= H ^ ~ = 90 8 = `72 J= H ^ ~ ` 72. J~ # Q i O ^ ~ ` JQ #. J= H ^ ~ ` 4,400, J~ # Q i O ^ ~ = Q i O ^ ~ = `20, # = `0 20 # = `8 4400# = `20, = - 0 # = 90 # 220 = ` 80. 0

21 J# k# JOH Q } ` = L^ ~ }.7 XH " ~ _ XH = = # `,200 H x 30% Z =Q Q i O ^ ~ # x~ ~ Oz 20% = ^ ~ ~ z J" #. Jq # ^ ~ = i ` = ZO`? = = H H # ^ ~ (C.P.) ` J# H # =. Q i O ^ ~ (M.P) H # ^ ~ (C.P) Z =Q \ # = `30 H # ^ ~ ` J~ # Q i O ^ ~ `30 JQ # H # ^ ~ = `200. Q i O ^ ~ (M.P)= 200 # 30 = ` # 560 = ` 32 ` _ = % = J= H ^ ~ (S.P) = Q i O ^ ~ (M.P) ` _ = = `248 ` = = J= H ^ ~ (S.P) H # ^ ~ (C.P) = = ` 48. ` = = = H # ^ ~ = # = 4% XH JOQ _ " _ 20% ` _ WK #. H Ok " \ H J= H q =# H # Q # =. (i) XH Q i O ^ ~ = `20. (ii) XH Oz H Q i O ^ ~ = `250. (iii) XH [` ^ ~ H (shoes)q i O ^ ~ = ` # (Tax) L Q =, = #= \.q. = i " ~ uh '' i # H = =k # = J< K < =. # J#Q < q? = # ` = = # [ # O_ '' # ZO^ = K < ~? = #= = # ` " = # H = # J` = ~ " # ~ ^ ~, ~ = u, h\ ~ ^, q^ `, ` Q h~ " ^ Q = ` H e Oz ^ x Ja =$k K #. g@x O\ H ~ ` = # ''_ J= ~ =. D J= ~ " # _ # [ # O_ # ~ = = K ^ ~. 7 7

22 J^ = # ~O_ ~ H = :. ` H # (Direct Tax) [ < ~ Q ` = # K e OK # # ` H # JO^ ~. P^ # (Income Tax), P # (Property Tax), =$u # (Professional Tax), h\ # (Water Tax) " ^ Q #q ` H # (Direct Tax). 2. ~ H # (Indirect Tax) [ < ~ Q H H ~ H = Q # # ~ H # JO^ ~. HO ^ ` # (Excise Tax) ^ = L` u JQ H x = = " # < HO ^ ` # JO^ ~. nxx ~ ` ` = = K #. " # (Service x= K ~ _ JH O> \, > e # a, " ^ Q " \ K e OK # < " # JO^ ~. D # # = # JOkOK " ~ = K ~ ` ` = # K e OK ^ ~. P^ # (Income Tax) = # ` = # Jk H P^ = =K # Wk Ju = Y " #k. u ~ _ XH O=` ~ = # x~ ~ Oz# P^ = # K P^ # # = #. x[" # ^ ~ _ Yz ` = Q i # = = P^ # # = #. J= H # (Sales Tax) / q = P^ i` # (Value Added Tax) J= H # XH = = # q H ~ OK O^ ~ = J= H ^ ~ _ K J= H = g ^ = K _ # # J= H # JO^ ~. H # Q ^ ~ _ " ~ = = # H # Q K # C_ = = H ^ ~ # K e OK #. XH = = H J= H ^ ~ ` = J= H # x~ ~ OK #. ^ xx ~ ^ K ~ ^ ~. D ~ A = = ^ ~ K ~ _ = # # # q = P^ i` # JO^ ~ (Value Added Tax (VAT)). = #= K e OK ^ ~ < # _ K ~ _ = # k. 8 8 XH = = # Jq # _ ^ x ` = qk OK # # J= H # JO^ ~.

23 J# k# JOH Q } ` = 20 O=` ~ = #O^ e qq^ " ~ = = J= H # # H # Q # =.. q^ ` ih ~ = % 2. \ % 3. _ [ % 4. Q $ H ~ }= % 5. ~ #= % J= H # Q } J= H # = J= H # J= H # ` = = J= H # ` = 9 9 = = ^ ~ = = ^ ~ ~ ^ #O^ e " ` = ^ ~ = = = ^ ~ + J= H # L^ ~ }.8 q< ^ XH Ow` " ~ ^ = # `2,000 H < #. P = = # 8% J= H # qk Oz# J` _ ZO` " ` = K e OK = #. J= H # ZO`? Ow` " ~ ^ ^ ~ = ` 2,000 J= H # ` = = 8% J= H # = 8 # 2000 = ` 960 q< ^ K e Oz# " ` = = 2, = `2,960 L^ ~ }.9 XH j` bh ~ } = H ^ ~ J= H # ` H e ` 4,355 ^ x Q i O ^ ~ ` 3,050 J~ # J= H # ` = H # Q # =? ~ j^ #O^ j` bh ~ } = Jq # ^ ~ = `4355 Q i O ^ ~ = `3050 J= H # = ~ ^ " ` = = = ^ ~ J= H # ` = = = 4,355 3,050 = `,305 J= H # = = ^ ~ = 305 # 3050 = 0% H x " ~ = = #, K H ~, a =, L C, H =, x, H = =O\ = = = # ` = J= H # # O_ q # ~ O Wz #k.

24 J^ = L^ ~ }.20 XH \ H x `2,730 H < #. nx # 5% VAT # qk OK _ # k. VAT " H = O^ ^ ~ ZO`? = i VAT ZO`? q = P^ i` # = 5%. VAT O_ ^ ~ ~ # VAT 5% ` = H e ` 05/ ~ Q #. x[" # ^ ~ ` J~ # VAT H e # ^ ~ `05 JQ #. ^ ~, VAT # H e ` 2,730 S` \ H ^ ~ = 05 \ H ^ ~ = `2,600 # 2730 = `2,600 VAT = 2,730 2,600 = ` 30. H Ok " \ H J= H ^ ~ H # Q # =. u ^ xh 5% J= H # qk OK _ #k. (i) XH ` ko_ `60 J~ # (ii) ~O_ ~ XH H \ ` % VAT H e J= H ^ ~ ~ = _ #k. J ^ ~ # H # Q # =. (i) XH q^ `4,500 H # _ #k. (ii) H > `200 H # _ #k. J =.3. i # = ^ #= # Z# H # =. (i) ` _ J# #k Z C_ nx # P^ ~ _ #.. (A) Q i O ^ ~, (B) H # ^ ~ (C) J= H ^ ~ (D) =_ (ii) Q i O ^ ~ = ` 40, J= H ^ ~ = ` 05, J~ # ` _ =. (A) ` 245 (B) ` 25 (C) ` 30 (D) ` 35 (iii) = Q i O ^ ~ - ` _. (A) H # ^ ~ (B) J= H ^ ~ (C) ~ j^ #O^ e ^ ~ (D) # = (iv) XH = = q = # " _ # #... Jx JO^ ~. (A) " # (B) VAT (C) HO ^ ` # D) " # (v) XH = = # `28 ` _ Wz J= H ^ ~ 240 ~ # Q i O ^ ~. (A) `22 (B) `228 (C) `268 (D) ` XH H = # `450 Q i OK _ #k. JOQ _ " _ ^ x 20% ` = ` _ Wz H ^ ~ # Jq #, J= H ^ ~ H # Q # =. 3. XH ^ ~ ^ i x 0% = i 20% =~ Q ` _ Wz # ` ~ " ` Jq #" `5,760 J~ # Q i O ^ ~ H # Q # =

25 J# k# JOH Q } ` = 4. Y~ XH OQ }H O ` = # `38,000 = ^ } O ` = (printer) # `8,000 H < #. D ~O_ O_ # 7% J= H # K e Oz# H # ^ ~ # H # Q # =? 5. XH =O@ = y J= H ^ ~ = i q = P^ i` # (VAT) H e `9,60 J~ #k. VAT 6% J~ # P =O@ = y Q i O ^ ~ # H # Q # =? 6. ik ~ `5,000 Q i O ^ ~ H ey # XH \ # 0% ` _ H < #. ^ x 0% J= H # K e Oz# J` _ JOQ _ " _ H ZO` K e OK #? 7. 9% ` = J= H # K e Oz XH j` bh ~ } ^ #= # `,70 H < #. J ^ ~ # H # Q # =? 8. XH " ~ _ ` # ^ Q ~ = # = = H # ^ ~ H O> 40% Z =Q Q i O ^ ~ # x~ ~ OK #. J` _ 5% ` _ Wz Jq # =? #+ =? H # Q x ` = # H # Q # =? 9. XH \ g `,500 Q i O ^ ~ # = Oz 0% ` _ ^ ~ H \ OK ~. O_ Q = = JOQ _ " _ uiy 5% ` _ Wz J" #. ^ xh J= H ^ ~ H # Q # =? 0. `3500 # Q i Oz# ~ # XH _ `2,800 H < #. ` _ ` = H # Q # =?. n 5 K H # `200 H x 5% = \ J" #. qq Q ^ ~ _ 4% J= H # K e Oz# XH H K H " ZO`? 2. H Ok " \ H ` _, ` _ ` =, J= H ^ ~, Q i O ^ ~ H # Q # = =~ OY = = Q i O ^ ~ ` _ ` = ` _ " ` = J= H ^ ~ (i) p~ ` 2,300 20% (ii) H = [` ` `05 (iii) _ xoq > % --- `6,000 (iv) " + OQ " + < ` 4, `3,775 H > ` 3,224 2½% " O@ " O@< ` _ H ey# 20% = i 5% = Oz^? H X H i 25% Wz # ` _ = Oz^? H ~ }= ` =. 2 2

26 J^ =.4. K H =_ 7= ` ~ Q u #O^ h= ~ =_, ~ =_ Q } ` = #, " ` = # H # Q < ~ H < ~. W C_ K H =_, K H =_ Q } ` = #, " ` = H # Q < ~ H O^ ~. q# `50,000 XH OH # O_ 4% =_ ~@ ` ~O_ O=` ~ = J Cf < _. " ^ \ O=` ~ = q# K e Oz#k. ~ =_ = P # n# r = # # 4 = ` 2,000 D `2000 " ^ \ O=` ~ = = yo = #. XH " H ~ # =_ J ` K i = #. P + I = ` 52,000, ~ k H ` Q U~ _ #k. Wk ~O_ = O=` ~ J JQ #. ~O_ = O=` ~ = =_ S.I. = = P # n# r # # 4 = ` 2,080 ~O_ = O=` ~ = # q# Z = =_ K e OK = # k. D q^ = Q Q } K H =_ JO^ ~. = = Q O #O^, c = H O h, ` H ~ = xk x~ } O Jx O\ # K H =_ x Q } O` ~. L^ ~ }.2 ~ " `8000 # 3 O=` ~ = 5% =_ ~@ K C# XH O (finance ) x K #. 3 O=` ~ = ` ~ " ` ~ " O^ K H =_ ZO`? : : " ^ \ O=` ~ J = `8,000 " ^ \ O=` ~ =_ = = P # n# r 8000 # # 5 = `,200 " ^ \ O=` ~ " ` = = J + =_ = P+I = XH J = ` = Q } OK =_ x ~ =_ JO^ ~. J H =_ x, =_ H =_ x H e H OK ^ u< K H =_ JO^ ~ = 8,000 +,200 = `9,

27 J# k# JOH Q } ` = 2: " ^ \ O=` ~ " ` " ~O_ = O=` ~ J JQ #. ~O_ = O=` ~ J = `9,200 ~O_ = O=` ~ =_ = P # n# r = 9200 # # 5 = `,380 ~O_ = O=` ~ " ` = = P + I = 9,200 +,380 = `0,580 3: ~O_ = O=` ~ " ` " = _ = O=` ~ J JQ #. = _ = O=` ~ J = `0,580 P # n# r =_ = = _ = O=` ~ " ` = = P + I = 0580 # # 5 = `,587 = 0,580 +,587 = `2,67 = _ O=` ~ = ~ " O^ =_ K H =_ H ` = # k A P = 2,67 8,000 = `4,67. # K H =_ Q } Oz# ^ u K _ = Q # ^ = H xk Q # = # k. Z = O=` ~ = ~ z #K XH H O=` ~ = # H = #. JO^ = # = #= K = Q K H =_ " ` = # Q } ` = # < ~ H O^ =. J (P) Q #, =_ ~@ ` = # (r) % Q #, H = # (n) O=` ~ = Q # f H x# K H =_ H ` = : H Ok q^ = Q : " ^ \ O=` ~ J = P " ^ \ O=` ~ =_ = " ^ \ O=` ~ =_ = P + I = P # n# r P# # r = Pr = P + Pr = P r ` + j 23 23

28 J^ = 2 : ~O_ = O=` ~ J = P ~O_ = O=` ~ =_ = ~O_ = O=` ~ " ` = = P + I r ` + j P r ` + # # j ( ~ =_ ` = < L yoz#) = P r r ` + # j 3 : = _ = O=` ~ J = P+ r = _ = O=` ~ =_ = = _ = O=` ~ " ` = = P + I n ' O=` ~ = ` ~ " ` " ` = (A) = P K H =_ H # Q = i n O=` ~ = K H =_ (i. e.) K H =_ = P = P r r r ` + P # j+ ` + j = P r r ` + j` + j = P r 2 ` + j ` j P r ` + j 2 2 # # r ( ~ =_ ` = < L yoz#) = P r r ` + 2 # j = P r r r ` + 2 P 2 # j + ` + j = r 2 P r ` + + j ` j = P r 3 ` + j r n ` + j = A P r n ` + -P j O^ ~ = : O=` ~ = # XH i H (Compounded Annually) u O=` ~ = yo =_ x J ` H ^ u< O=` ~ = # XH i =_ H OK ^ u JO^ ~. WH _ A P = ` + r n j, K H =_ = A P r 24 24

29 J# k# JOH Q } ` = O^ ~ = 2: J~ O=` ~ =_ (Semi - Annually) u J~ O=` ~ = K H =_ x H # Q #= ## O=` ~ = # ~O_ J~ OII Q K = #. u 6 < H OK = ## ~@ # Q = Q r ` j f H #= #. 2n J~ O=` ~ =_, A = P8 + 2 ` jb = i C.I=A P O^ ~ = 3: = _ < K H =_ u = _ < K H =_ Q } OK = ## XH O=` ~ = # 4 ~ H OK = #. =_ ~@ ` r 4 j =O` JQ #. 4n D O^ ~ =, A P r r = ` jb = i C.I.=A P O^ ~ = 4: H = a # ~ = L# C_ =_ H H = a # ~ = = O_ #, L^ ~ } 5 4 O=` ~ = Jx Wz # H Ok q^ = Q " ` = # (A) H # Q # =K #. 2 r 5 A = P r ` + j ` jb = i C.I.=A P.. 5 OII ¼ OII L^ ~ }.22 `5,625 8% K C# O=` ~ = # XH i =_ H OK ^ u 3 OII K H =_ H # Q # =? = _ O=` ~ = " ` = = P r 3 ` + j = 5625 ` + j = 5625 ` + 25 j = 5625 ` j = 5625 # 27 # 27 # = ` 9,683 K H =_ = A P = 9,683 5,625 = ` 4,

30 J^ = XH O=` ~ = # = i J~ O=` ~ = # K H =_ x H # Q =. O. O=` ~ = # P~ < P = ` ` = =0% O=` ~ = # P = ` ` = = 0% K H =_ O=` ~ = #, K H =_ P~ < 2 H = = XH O=` ~ = H = = 6 < H ½ O=` ~ = 3 I = # 0# = `0 # 0 # I = 2 = `5 4 A = + 0 = `0 A = + 5 = `05 ` ~ " ` P~ < H P = `05 = = ` # 0 # I = 2 = `5.25 A = = ` A = `0 A = `0.25 P~ < = #= =_ Q } OK = ## ~O_ ~ (P~ < ) = i =_ ` = Q = f H x Q } OK = #. L^ ~ }.23 J `0 XH O=` ~ = # 0% K C# P~ < =_ H OK ^ u 8 < JQ K H =_ x H # Q # =? P = `0 r = 0% OII# n = 8 < = 8 2 O=` ~ = = 3 2 O=` ~ = = 2 O=` ~ = 2n 8 < " ` = = P r K H =_ = ` jb = 0 ` j = 0 ` j 2# = 0 # 2 # 2 # = ` = ` = A P ` jb XH " ` = # 8% K C# XH O=` ~ = # JQ K H =_ x 3 < H XH = ~ =_ x Zx ~ H O = #?

31 J# k# JOH Q } ` = L^ ~ }.24 ` 20,000 J OII 5% ` = K C# 2 O=` ~ = JQ K H =_ ZO`? 3 P = ` 20,000, r = 5% OII n = 2 3 OII 2 3 O=` ~ = ` ~ " ` = A = r 2 r P ` + j ` jb 2 = ` + j ` jb 2 = ` j ` j 20 2 = ` 20 j ` 20 j = # 23 # 23 # = 27, C.I. = A P = 27, ,000 = 7, K H =_ q = H (Inverse Problems) = #= ~ O` ~ " D ` = # < ~ H < =., A = P r ` + n j, nx O^ A, P, r,n J< < Q JO = = # q. D < yo\ = _ ` e # < = q =# = #= H # Q #=K #. L^ ~ }.25 J ` 640 U =_ ~@ ` 2 O=` ~ = ` " ` = Q = ~ #. K H =_ XH O=` ~ xh XH i H OK = #. P = 640, A = , n = `2, r =? A = P r n ` + j = 640 r ` + j = r 2 ` + j = r ` + j 2 = r 2 ` + j

32 J^ = ` 2 0 j = r ` + j 0 = + r r = - 0 r = r = 0 r = 0 ~@ = r = 0% ( O=` ~ = # ) L^ ~ }.26 K H =_ 5% K C# ZO` H = # ` 600, ` Q = ~ #? P = `600, A = `852.20, r = 5% OII, n =? A = P r ` + j = 600` + j = ` 05 n j = ` 2 j n = ` 2 j n 20 ` j = ` 2 n 20 j 2 n 5 n n = 3 O=` ~ =.5 ~ =_ H K H =_ H Q ^ = ( H ) = ` = P = J n = ~O_ OII R = ~@ ` = 2 OII ~ =_, K H =_ H Q ^ = = r H Ok " \ H H = ~@ x H # Q # =.. XH " ` = # 2 OII 8% K C# K H =_ x P~ < H XH = ~ Q } OK =. 2. XH " ` = ½. OII 4% K C# K H =_ x P~ < H Q } OK =. 2 P` j L^ ~ }.27 J ` 8000 ~@ 0%, H = 2 OII J~ # ~ =_ K H =_ H Q ^ = # H # Q # =. WH _, P = `8000, n = 2 O=` ~ =, r = 0% O=` ~ = # 28 28

33 J# k# JOH Q } ` = ~ =_ K H =_ ^ = = r 2 P` j = 8000 ` 0 2 j = 8000 ` 2 0 j = 8000 # # 0 0 = `80 J =.4. H Ok " \ H " ` = #, K H =_ x H # Q # =. =. OY J OII# ~@ % H = (i) 0 5% 3 (ii) % 2 (iii) 8,000 0% 2. Ow` J H =^ 8000 ~ # 2% K C# 2 OII J CQ f H < #. O=` ~ = # XH i =_ H OK ^ u Ow` ZO` K H =_ x K e OK #? 3. " < XH " ~ = ` _ Q > #. 5% K H =_ K C# O=` ~ = # XH i =_ K e OK ^ u (i) 2= O=` ~ = yo P" M ` = O_ " ` = ZO`? (ii) 3= O=` ~ =_ ZO`? 4. 24,000 ~ J O=` ~ = # 0% K C# ½ O=` ~ = # ZO` =_ JQ #. u 6 < =_ H OK =? 5. O=` ~ = # 2 ½ % =_ ~@ ` 8 < ^ q_ 8,92 ~ # x K # 6 < XH = ~ =_ x H OK ^ u =_ x H # Q # =? 6. O=` ~ = # 6% =_ ~@ ` 9 < 5,625 ~ 3 < H OK ^ u ZO` =_ JQ #? 7. 4% =_ ~@ ` 2 O=` ~ =,632 ~ # K H =_ Q Ok# J H # Q # =. 8. qh OII 5% =_ ~@ ` `26,400 O J CQ f H < #. O=` ~ = # XH i =_ H OK ^ u J` # 2 O=` ~ = 4 < ` ~ " ` J C " ` = f~ = ## ZO` " ` = = #? 9. 0% O=` ~ =_ ~@ K C# Ps XH O # O_ 80,000 ~ # J CQ f H < #. ½ O=` ~ = ` ~ " ` K e Oz# (i) O=` ~ K H =_ (ii) u 6 < K H =_ H Oz# ~O_ O\ H " ` = x ^ = # H # Q # =. 0. 5% O=` ~ =_ ~@ K C# 2,400 ~ 2 O=` ~ = ~ =_, K H =_ ^ = # H # Q # =.. 6% O=` ~ =_ ~@ K C# 6,400 ~ 2 O=` ~ = ~ =_ K H =_ ^ = # H # Q # =

34 J^ = 2. 5% =_ ~@ K C# XH " ` = # 2 O=` ~ = ~ =_ K H =_ ^ = 5 ~ J~ # J ZO`? 3. 2% =_ ~@ K C# * ` 2,500 ~ 3 O=` ~ = ~ =_ H J C K #. ~ k H J^ J J^ H = # 0% =_ ~@ K C# K H =_ H f H < #. W^ i Z=~ ZO` Z = =_ x K e Ozi? 4. ZO` _ Q \ # Jk 4% O=` ~ =_ ~@ K C#, ½ O=` ~ = #, 6 < XH = ~ =_ H OK ^ u,32,65 ~ JQ #? 5. Q u 2,000 ~ # 5% O=` ~ =_ ~@ K C# n O=` ~ H = # K H =_ _ Q > #. P" n O=` ~ = ` ~ " ` Ok# " ` = 3,230 ~ J~ # n x H # Q # =? 6. ` 640 J 2 O=` ~ = H = ` " ` = ~ # =_ ~@ # H # Q # =? ~ J, u 6 < =_ H Oz# /2 O=` ~ = ` ~ " ` ` " ` = J~ #k. =_ ~@ # H # Q # =?.5. ~ Q ^, ` ~ Q ^ (Appreciation and Depreciation) a) ~ Q ^ (Appreciation) ~ Q ^ J#Q [# ~ Q ^, H s ~ Q ^, H x = = q = ~ Q ^, P q = ~ Q ^ " ^ Q " \ x zok #. ~ Q ^ H ` =. A = P r ` + j b) ` ~ Q ^ (Depreciation) ` ~ Q ^ J#Q H x = = q = ` y " #= q =, _ = q =, H x O ` = q = " ^ #q ` Q #. ` ~ Q ^ ` =. A = P r ` - j L^ ~ }.28 XH Q = = H [< u O=` ~ = 7% K C# ~ Q K # k. Q = [< 90,000 J~ # ~O_ O=` ~ = ` ~ " ` [< H # Q # =? [< P = 90,000 ~ Q ^ ~@ = 7% H = = 2 O=` ~ =. n O=` ~ = ` ~ " ` [< = n n P r n ` + j. 2 O=` ~ = ` ~ " ` [< = ` + j. OK [< O=` ~ = - [< ,000, ,000, ,500,000, ,000,000,000 3= ` = [< 0 ~@ Q iy#k.

35 J# k# JOH Q } ` = = ` 07 2 j = # 07 # 07 = O=` ~ = ` ~ " ` [< =,03,04 L^ ~ }.29 u O=` ~ = 5% K C# XH O ` = (q + < ) q = ` ~ Q K # k. (depreciates) XH _ ` 30,000 ~ z P O ` = # H x# 3 O=` ~ = ` ~ " ` ^ x q " O`? O ` = (q + < ) H ` ^ ~ P = ` 30,000, ` ~ Q ^ ~@ R = 5%, H = = 3 OII n O=` ~ = ` ~ " ` O ` = q = = P r ` - j = _ O=` ~ = ` ~ " ` O ` = q = = ` n j 3 3 = 30000` 95 3 j = # 95 # 95 # = ` = _ O=` ~ = ` ~ " ` O ` = q = = `25,72.25 L^ ~ }.30 XH Q = [< u O=` ~ = 5% K C# ~ Q K # k. ` [<,04,832 J~ # ~O_ O=` ~ = ~ = [< ZO`? ~O_ O=` ~ = ~ = [<, 5 2 ` P` + j = P 05 ` 2 j = P # 05 # 05 = P = # # 05# 05 = = 95,086 ( ~ OH = # =ioz#) ~O_ O=` ~ = = O^ [< = 95,

36 J^ = J =.5. XH ~ K i# q^ ~ OY u O=` ~ = 5% K C# q^ ~ K ih iy# ~O_ O=` ~ = ` ~ " ` q^ ~ OY ZO`? 2. XH H ~ " `3,50,000 ^ x u O=` ~ = 0% ` ~ Q ^ L# K 3 O=` ~ = ` ~ " ` ^ x ^ ~ ZO`? 3. XH " \ ~ H # `50,000 H < ~. ^ x q = u O=` ~ = 8% ` y #, XH O=` ~ = ` ~ ` ^ ~ ZO`? 4. XH Q u Q O@H 2.5% K C# H i OY Jk H = Q K # k. ~ O = H i OY 5,06,000 J~ # ~O_ Q O@ ` ~ " ` " \ OY # H # Q # =. 5. XH Q = = # O_ [ x~ ^ Q = = # H #= # #Q ~ = # = (migration) = K < ~. ~O_ O=` ~ = = O^ Q = [< 6,000. u O=` ~ = = ` = 5% J~ # ` Q = [< ZO`? 6. XH P~ WO[< ` q = `4,580 u O=` ~ = 0% ` ~ Q ^ L# K = _ O=` ~ = ~ = ^ x ^ ~ ZO`? 7. XH Q = = L^ Q = u H ~ }OQ O=` ~ = # 9% K C# [< ~ Q K # k. ` [<,88 J~ # ~O_ O=` ~ = ~ [< # H # Q # =?.6 ~ [= M ` (Fixed Deposits) J# =~ # x (Recurring Deposits) OH, ` H ~ =, W` ~ xk x~ } O (financial institutions) [ qq^ =_ ` = x K ^ ~. [ ^ K # _ # =_ ` H e H = =k = y # " O@< uiy f H O^ ~. # K # " \ Z< ^ ` x K =K #. " \ H x, Deposit) (i) ~ x (Fixed Deposit) (ii) J# =~ # x (Recurring (i) ~ [= M ` nx O^ XH " ` = # H O` H = # x K ^ ~. nx< F. D. JO^ ~. Q = xh : x ` = H = # H Z = H = # x K =K #. x K H = # \ =_ ` = # x~ ~ OK ^ ~. (ii) J# =~ # x J# =~ # x # ~ x # ZO` ^ = H ^. J# =~ # x x R.D. JO^ ~. D x O^ M ` ^ ~ _ ` # ^ = ~ x \ ZO` H O` _ # u < H O` H = =k =~ ` H ~ = H O #O^ ` Q H \ =K #

37 J# k# JOH Q } ` = O J~ < ` H ~ " < x K # " ` = # =_ ` H e H = =k = y # " O@< uiy WK ^ ~. nx< J# =~ # x JO^ ~. Q = xh : J# =~ # x O^ =_ x ~ =_ ^ u H OK ^ ~. W C_ = #= =_ x H = =k (maturity) ` ~ " ` =K " ` = # H ` = # H # Q O^ =. R J# #k =_ ` =, P J# #k < i " ~ ^ " ` =. N J# #k < OY =_ = PN r nn, N ^ + h = ; E 2 O=` ~ = 2 H = =k ` ~ " ` =K " ` = A = Pn + PNr L^ ~ }.3 ` ~ < XH O #O^ 2 H # 8% O=` ~ =_ ~@ K C# 5 O=` ~ = x K #. H = =k ` ~ " ` O^ " ` = ZO`? x K # J P = ` 2,00,000, n = 5 O=` ~ =, r = O=` ~ = =_ ~@ 8% =_ = Pnr = # 5 # 8 = ` 80, O=` ~ = ` ~ " ` O^ " ` = = 2,00, ,000 = 2,80,000. L^ ~ }.32 " n+ u < ~ O = `500 K C# 5 O=` ~ = 7.5% =_ ~@ K C# ` H ~ = #O^ x K #. S^ O=` ~ = ` ~ " ` J` # O^ " ` = # H # Q # =. u < x K # " ` =, P = ` n = 5 2 = 60 < = =_ ~@, r = 7 % % x " ` = = Pn = J# =~ # x, N = = 24 # 60 # 6 = = ` 30,000 nn ^ + h < i " ~ ^ # P < U ` k < < K e OK =K #. ; E O=` ~ = 305 O=` ~ =

38 J^ = =_ = I = PNr " ` = = Pn = 500 # 305 # 2 # = `5, PNr = 30, ,78.75 = `35,78.75 L^ ~ }.33 q u < ~ O = # `200 K C# S^ O=` ~ = ` H ~ = #O^ x K #. 5 O=` ~ ` ~ " ` J` # `3,830 " ` = # O^ #. =_ ` = # H # Q # =? H = =k ` ~ " ` Ok# " ` =, A = ` 3,830, P = ` 200, n = 5 2 = 60 < H =, N = nn ^ + h 2 2 = ; E O=` ~ = # # = OII x K # " ` = = Pn = = 2, 000 H = =k " ` = = Pn + PNr 3830 = # 305 # r = 305 r = 305 r r = = 6%.6. J^ ^ ~ H # Q = i " ~ ^ K e O (Hire Purchase and Instalments) OH = i xk x~ } O (financial institutions) qx Q ^ ~ H ~ J^ ^ ~ H # Q, " ~ ^ K e O =O\ ^ H = # ik = K #q. J^ H # Q ^ H = (Hire purchase): J^ H # Q ^ H = ^ ~ = #= H # Q K " #= XH xs ` H = =~ = # O` = H * ^. xk x~ } O = #= H # Q K # = = H " ` = q =# O ~ = Q K e Oz# ^ P = = H " #= = # O` = JQ #. " ~ ^ H # Q q^ #= (Instalment): = = H ^ ~ ^ xh Q =_ = i ` k` ~ K s # H e = #= K e O = O_ #. J ` =_ " ` = # = # < ` yo = #. D q^ = Q =z # " ` = < " ~ ^ " ` = JO^ ~.

39 J# k# JOH Q } ` = g H iok < i " ~ ^ (Equated Monthly Instalment ( E.M.I. )) g H iok _ # < i " ~ ^ J# #k ` y O q^ # H = = # J# io _ " ~ ^ q^ #= Q #. = = H ^ ~ ^ xh Q =_ = i ` k` ~ Y~ # H e = #= K e OK = O_ #. J ` Jk H = Q " ` = # = # < ` yo = #. P q^ = Q =z # " ` = < g H iok _ # < i " ~ ^ (E. M. I.) J# _ #. E. M. I. = J { =_ = = OY q^ = # " ~ ^ K e OK ^ H = (Different schemes). 0% =_ ^ H = (0% interest scheme): 0% =_ ^ H = " H H K s # (processing charge) = K ^ ~. 4 H 5 " ~ ^ H = # = O^ Q < f H O^ ~. =_ ^. 2. % ^ = ^ H = (% Finance): =_ = i < i " ~ ^ ` (E.M.I.) H ^ " [ K s # (processing charges) H ey = O_ #. 3. ` _ J= H = (Discount Sale): " } [ \ = # J= H q = O^ ` _ K _ K # k. " ~ Ja =$k H Q # J^ ^ ~ H # Q O^ _ ` _ W= _ K # k. 4. ~ O #Q ^ K e O ^ H = (Initial Payment): = = " H O` ` = # = O^ Q < H \ OK H x q ye# Q = # =_ x K i < i " ~ ^ ^ ~ = = J= _ K # q. nx< ~ O #Q ^ K e O ^ H = JO^ ~. L^ ~ }.34 XH " + OQ q + < ^ ~ `8,940 W= _ #k. H Ok \ H " ~ ~ " ~ ^ ^ H = H # Q H q=~ = ~ = _ #q. U ^ H = i #^ H # Q # =. =~ OY (i) (ii) " ~ ~ ^ H = 75% }= % Jq # " ~ O K e O =_ ~@ 25 = # 8940 = `4, H K s 8,940 25% 2% % 8,940 Nil 6% 2% H = 24 < }= < 0% 24 (iii) 8,940 4 E. M. I Nil 2% }= < \ H ~ # ^ H = g H ioz# < i " ~ ^ (E. M. I) = i " ` = # H # Q # =. (i) 75% } H ~ ^ H = P = `8, 940, ~ O K e O 25% ~@, ~@ = 2%, " # = `8, 940 % = # 8940 = ` ` 89 ~ O K e O = `8,940 H 25% 24

40 J^ = } " ` = = 8,940 4,735 = `4,205 =_ = E. M. I. = 4205# 2# 2 = `3, = `3,409 = }= + =_ < OY = = ` - `734 K e OK = # " ` = = 4, , , = `22,538 (ii) % } H ~ ^ H = " # = `8940 2% =_ ~@ = 6% = 2 # 8940 = ` `379 =_ = 8940 # 6 #2 E. M. I. = = ` `6,06 }= + =_ < OY = = = `, = `,04.7 = `,042 K e OK = # " ` = = 6,06 + 8, = `25,380 (iii) 0% =_ ^ H = " # = `8940 2% E. M. I. = = 2 # 8940 = ` = ` 379 = }= + =_ < OY = = ` `789 = O^ Q K e Oz# E. M. I. = `789 4 = `3,56 K e OK = # " ` = = 8, , = `22,475 H = # 0% =_ ^ H = = Oz ^ H =.

41 37 37 J# k# JOH Q } ` = L^ ~ }.35 XH OQ }H O ` = " `20,000 nxx 36 < 0% =_ ~@ K C# =K #x H O h K #k. H # Q ^ ~ _ XH < H ZO` = < H # Q # =? OQ }H O ` = " = 20,000, =_ = 0%, H = = 36 < (3 O=` ~ = ) " ` = =_ = # 0 #3 = `6,000 " ` = K e O = 20, ,000 < i K e O = = `26,000 " ` = < OY = = ` `722 J =.6. ` `25,000 # ~ x M ` [= K #. 4% O=` ~ =_ ~@ ` 2 O=` ~ = [= K #k. = yo = = P" H ZO` " ` = a OK #? 2. xu< `75,000 # 3 O=` ~ = 5% O=` ~ =_ ~@ ` ~ x M ` [= K #. = yo = = ZO` a OK #? 3. W = < u < ~ O = `400 # ~O_ O=` ~ = 2% =_ ~@ K C# ` H ~ = #O^ J# =~ # x K #. ~O_ O=` ~ = ` ~ " ` J` # O^ " ` = # H # Q # =? 4. ~ " + H O` " ` = # u < ~ O = 6 O=` ~ = ` H ~ = 8% =_ ~@ K C# [= K #. 6 O=` ~ = ` ~ " ` J` x `7,904. " ` = Q a Oz#k. J~ # J` # < i ZO` _ H > #? 5. H $+ u < ~ O = ` 700 # 6 O=` ~ = OH #O^ J# =~ # x [= K #. 6 OII ` ~ " ` J` xh ` 64,97 O^ #. =_ ` = # H # Q # =? 6. XH " H X" < ^ ~ ` 6,000, 5 " ~ ^ K e OK = #x i ` z#k. H O h 0% O=` ~ =_ ~@ K C# Jq # < i P" ZO` _ = #? " ` = # H # Q # =? 7. XH j` bh ~ } = " `6,800, ~ Ol` ^ xx 0% =_ ^ H = H #^ K #. 3 < " ~ ^ = O^ Q K e OK #. 3% " K s _ ~ Ol` K e OK #. 24 < J` _ < i K e O = i " ` = H # Q # =. 8. XH [#= " * ^ ~ ` 8,400, " OH \ ^ xx 0 " ~ ^ K e OK ^ H = H < #. H O h ^ xx 5% O=` ~ =_ ~@ K C# H Oz# J` x < i " ~ ^ K e O = i " ` = # H # Q # =?

42 J^ =.7 q = # ` = H Ok ` ~ Q ` J# = #, q = # ` = # Q ioz K kq < ~. W C_ " \ x [ H ` K H O^ =. J# = # ` = U^ x ~O_ i= }= # f H x# " \ XH \ Z " ##, ^ ` y ##, ~O_ =k Z = Q # ^ ` Q #. Wk J# = # ` =. J# = # ` = # L^ ~ }=. ^ ~ =, H = J# #k J# = # ` = # XH L^ ~ }. ZO^ H #Q ^ ~ = Jk H = ~ # H = Jk H = Q #. 2. J = i =_ J# #k XH L^ ~ }. Z@ #Q J Jk H = ~ # ^ xh a OK =_ Jk = Q #. 3. = = H ^ xh " z OK ^ #= _ XH L^ ~ }. ZO^ H #Q = = Z " #K ^ xh JQ Y~ Jk H = Q #. q = # ` = ~O_ i= }= XH \ Z " ## H ` " ## =~ Q = ~ H \ ` Q # ^ Z =Q #. nx< q = # ` = JO^ ~. q = # ` = # L^ ~ }=. x" ~, H = J# #k K H x L^ ~ }, Z@ #Q x K = # + Z =~ # ^ H = ` Q #. 2. " Q =, H = XH L^ ~ }. " Q = Z =~ # ^ H = ` Q #. 3. [< ~ Q ^ = i " ih H = # P ~ = J# #k XH L^ ~ }. [< iy#k P ~ ` ` y = #. q = # ` = ~O_ H JO` H O> Z = J# ` = H ey O_ H # q = # ` = JO^ ~. H Ok \ H ~O_ J# ` = # H ey O_ qq^ J=H = W= _ #q

43 J# k# JOH Q } ` = J# ` = I J# ` = II J# = # J# = # q = # q = # J# = # q = # q = # J# = # H O^ W= _ # H ^ ~ q = # ` = Q ioz ` H #=K #. L^ ~ }.36 2 g. _ = Q XH Q _ # 20 = Ok = # + 6 ~ A Q ey#k, 25 = Ok 3 ~ A ZO` _ = Q Q _ # Q ~? ^ u : = = _ ~ H = # J# ` = Q =. Jx = # + OY, ~ A, Q _ _ =. = # + OY ~ A Q _ _ = x : = # + # Q _ _ = # f H # =. = # + OY 20 # O_ 25 Jk H = ~ #K Q _ _ = Jk H = Q #. JO^ = # ~ k J# = # ` =. JO^ = x+ u 20 : 25 :: 2 : x... () 2 : ~ A #, Q _ _ = ` =. ~ A 6 # O_ 3 ` y #K Q _ _ = ` Q #. JO^ = # Wk J# = # ` =. x+ u 6 : 3 :: 2 : x... (2) () = i (2), ^ ~ = #= W " =K #. 20 : 25 S2 : x 6: 3 JO` = = = ^ = = JO` = = ^ = JO` = 20 : 25 ::2 : x 6 :

44 J^ = ~O_ = ^ u 20 # 6 # x = 25 # 3 # 2 x = # 6 # # = 70 = # + ~ A Q _ _ = x : = # + OY Jk H = ~ #K Q _ _ = Jk H = Q #. = # + 20 # O_ 25 Jk H = ~ #K Q _ _ = Jk H = Q #. JO^ = # Wk J# = # ` =. Q }H = 25 = 20 2 : W C_ ~ A, Q _ _ = # f H # =. ~ A OY 6 # O_ 3 ` y #K Q _ _ = ` Q #. JO^ = # Wk J# = # ` = Q }H = = 3. 6 x = # # = 70 L^ ~ }.37 XH xx 6 = Ok ~ A 0 Q O@ xk # 24 ~ A i K ^ ~. J^ xx 9 = Ok ~ A 8 Q O@ x K # Zx ~ A i K Q ~? ^ u : D = 3 ~ H = # J# ` = H =. Jq ) = # + 2) xk H = 3) ~ A = # + Q O@ ~ A x : = # +, ~ A # ijeoz#, = # + 6 # O_ 9H iy# ~ A OY ` Q #. Wk q = # ` =. x+ u 9 : 6 :: 24 : x... () 2 : x K H = ~ A # ijeoz# x K Q O@ 0 # O_ 8 H ` y #, ~ A Jk H = Q #. H \ Wk q = # ` =. x+ u 8 : 0 :: 24 : x... (2) () = i (2), # H e #K 9: 6 :: 24 : x 8: 0

45 JO` = = = = ^ = = JO` = = ^ = JO` = 9 : 6 : : 24 : x 8 : x = x = 9 # 8 # # = 20 ~ A 4 4 J# k# JOH Q } ` = Q = xh :. J# = # ` = (.) }= Q ~ H O^ Q i OK = 2. q = # ` = (-) }= Q ~ H Q i OK = 3. Q }H = # } Q ~ J^ ~ OQ " = #. } Q ~ # ` Q = OY # == Q # ` H Q = OY # ~ = Q # f H # =. ~O_ = ^ u H Ok W= _ # K # Q ~ OK H # =. ^ u 2 : ( } Q ~ L yoz) = # + x K Q O@ ~ A OY x : = # + #, ~ A ` =. = #+ OY 6 # O_ 9 H iy# ~ A ` Q #. Wk q = # ` = Q }H = = : x K Q O@ # ~ A ` =. Q O@ 0 # O_ 8H ` y # ~ A Jk H = Q #. JO^ = # ~ k q = # ` =. Q }H = = 0 8 x = 6 # 0 # 24 = 20 ~ A 9 8 J =.7. 2 = Ok = _ OQ XH H xx ~ A 0 Q O@ xk 8 ~ A = yozi. J^ xx 5 = Ok ~ A 6 Q O@ xk # Zx ~ A Q #? O ` 6 Q O@ X H ~ H " # 4800 # # ` ~ K #K Q O\ O ` = Zx # # ` ~ K Q ^? J^ q^ = Q 25 O ` 5 Q O@ Zx # # ` ~ K Q =? 3. 4 = Ok = ^ } Jk H ~ 5 Q O@ 70 r H = # = koz#k, r H = # 9 Q O@ = ZO` = Ok H = #? 4. 2 = Ok 0 ~ A 2400 K ^ ~ < # K ^ # K #K, 5400 K.g. < # 8 ~ A K ^ # ZO` = Ok H = #? 5. u 8 ~ A ~ A 4 Q O@ xk 5 p~ ZO ~ _ s K #. 6 Q O@ xk 0 p~ ZO ~ _ s Zx ~ A Q #? 6. XH O J ` < ` K C# =_ ~ z #K ` < ZO` =_ =K #?

46 J^ =.8 H = = i x XH xx, ^ xx K = #+ = O^, XH _ XH ~ A ZO` x K Q _ H J= ~ =. ZO^ H #Q x, H = ~O_ q = # ` = = O_ #. Z@ #Q Z = = Ok H xk #K P xx H = ` Q #. D = # i+ H Ok q+ = # Q ~ OK H #= #.. XH = x+ XH xx n ~ A = yoz#k J` # K # x " ` = n =O` JQ #. L^ ~ } XH _ XH xx '4 ~ A = yoz#k J C_ J` x XH ~ A x = 4 2. XH = x+ XH ~ A K Q " ` = x i= }= ~ z #K J` # " ` = xx H = = /(XH ~ A x). L^ ~ } XH _ XH ~ A 0 =O` xk #K J` # P x H =. = = # 0 = 0 ~ A. ` 0 j L^ ~ }.38 A J# #` _ 20 ~ A #, B J# #` _ 30 ~ A # XH xx i K Q ey#k W^ ~ H P xx ZO` H = i K Q ~? XH ~ A A K x = 20, XH ~ A B K x = 30 A = i B ~ ^ ~ H XH ~ A K x = = 5 60 = = = =O` x W^ ~ H P xx H = = = 2 ~ A L^ ~ }.39 2 A = i B J# ~ ~ = ~ H XH xx 8 ~ A i K Q ~. JO^ A XH _ 2 ~ A i K #. B XH _ H = ZO`? A = i B XH ~ A K x = = =O` x XH ~ A A K x = 2 = =O` x XH ~ A B K x = 8 - = = 24 J^ xx B H = = = 24 ~ A

47 L^ ~ }.40 A = i B W^ ~ H e x K ^ ~. P xx A, 2 ~ A #, B, 20 ~ A # i K Q ~. = i ~ ^ ~ H 3 ~ A xk # ` ~ ` A P xx =^ e " Q q ye# xx B ZO` H = i K Q _? XH ~ A A K x = 2 XH ~ A B K x = XH ~ A W^ ~ H K x = 2 = _ ~ A W^ ~ H e K x = 5 2 = q ye# x = q ye# x H = = J# k# JOH Q } ` = = 8 = # 3= = = 3 5 # 20 = 2 ~ A L^ ~ }.4 XH xx A, B J# W^ ~ 2 ~ A, B = i C W^ ~ H 5 ~ A, C = i A W^ ~ H 20 ~ A i K Q ~. " ~ JO^ ~ H = i q_ q_ H = # H # Q # =? XH ~ A A, B W^ ~ H K x = 2 XH ~ A B, C W^ ~ H K x = 5 XH ~ A C, A W^ ~ H K x = 20 XH ~ A (A+B)+(B+C)+(C+A) H K x = 2 5 XH ~ A (2A + 2B + 2C) H K x = XH ~ A 2(A + B + C) H K x = 2 60 XH ~ A A, B, C = Q ~ H K # x = A, B, C J# = Q ~ q_ q_ Q =~ Q 20, 5, 4 ~ A xx i K Q ~. J~ #, " ~ = Q ~ H xx Zx ~ A i K Q ~? # 2 =

48 J^ = ` A,B,C = Q ~ H e 0 ~ A i K ^ ~. XH ~ A A K # x (i.e.) [(A + B + C) K # x (B + C)K # x ] = 0 ` A P xx 30 ~ A i K #. XH ~ A B K # x (i.e.) [(A + B + C)K # x (C + A)K # x ] = ` B P xx 20 ~ A i K #. XH ~ A C K # x (i.e.) [(A + B + C)K # x (A + B)K # x ] = = = - = = - = = ` C P xx 60 ~ A i K #. L^ ~ }.42 XH xx A J# #` _ 0 ~ A #, B J# #` _ 5 ~ A # i K Q ~. " i^ ~ H x = yoz O koz# " ` = ` 500 J~ # " i^ ih XH H ih ZO` a OK #. XH ~ A A K x = 0 XH ~ A B K x = 5 " i^ i x x+ u = : 0 " ` = Q = = 500 A H Q = = = 3 : 2 3 # 5 = ` # 5 0 = ` 600 B H Q = = 50 L^ ~ }.43 XH h\ ` > ~O_ ~ q_ q_ H = 30,40 xq + =. P ` \ h~ " eh " ~ H ~ H = 24 xq + = J~ #K = _ O\ x X H i ` ~ z#k M m ` > H " O`? " ^ \ ~ ^ ~ XH xq + = xo_ h\ H ` = ~O_ = ~ ^ ~ XH xq + = xo_ h\ H ` = 40 = _ = ~ ^ ~ XH xq + = " eh = h\ H ` =

49 J# k# JOH Q } ` = 3 H ~ X H i ` ~ z#k ` > xo_ h\ H ` = = = 2 = = ` H = = = 60 xq + = 60 = Q O@ J =.8. XH xx i XH ~ + H = 4 ~ A, J^ xx i H 2 ~ A \ #K " i^ ~ H Zx ~ A i K Q ~? 2. W^ ~ ~ H XH xx 0 ~ A i K Q ~. " ^ \ J ~ XH _ 5 ~ A i K Q _. ~O_ = J ~ XH _ H = ZO`? 3. A, B = i C J# = Q ~ XH xx =~ Q 8, 2 = i 6 ~ A i K Q ~. g\ " ^ \ A, B W~ = ~ H 3 ~ A x K # ` ~ ` B =ke " Q C, A ` H x K #. " i^ ih P xx H = ZO`? 4. XH ` > h~ A ~ H 0 xq + =, B ~ H 20 xq + #. ` > h~ " eh C ~ H 5 xq + #. = _ O\ x X H i ` ~ z#k M m ` > H = ZO`? 5. XH xx A J# #` _ 20 ~ A, B J# #` _ 30 ~ A i K Q ~. P xh " ih a Oz# " ` = `600. " i^ ih XH H ih ZO` a OK #? 6. XH xx = A, B = i C J# = Q H = =~ Q 2,24 = i 8 ~ A. " ~ = Q ~ H XH ~ A x K # ` ~ ` C =ke " Q q ye# x A = i B H = ZO`? 7. XH ` > " ^ \ H ~ H 5 xq + =, ~O_ = ~ H 20 xq + #. ~O_ ~ ` ~ z= OK Q M m ` H " O`? OH = : C.P. = H # ", S.P. = Jq # ^ ~, M.P. = Q i O ^ ~, R = ~@, P = J n = H =, A = " ` =, C.I = K H =_

50 J^ = ` = J# #k # \ H H " #. a # = H ~ = J~ # ^ xx ` = JO^ ~. = =z # _, = = Jq #" H # " ; ` = = = H # " X Jq #" = { % X H.".; H # " = XJq # " { % #+ = =z # _, #+ = = H # " Jq #".; #+ ` = = #+ = H # " X Jq #" = #+ % XH.".; H # " = XJq # " #+ % Q i O " O^ ` y O _ #k ` _ JQ #. Q i O " O^ W= _ # ` _ ` y Oz# ` ~ " ` Q " # J= H " JO^ ~. ` y O = Q i O ^ ~ Jq #". - ` y O % Q i O ^ ~ = J." ; Jq #" = - ` y O % Q.^ ~ - ` y O % H # " = { % Q.^ ~ ; Q i O ^ ~ = { % - ` y O % H.". ` y O ` y O ` = = Q i O ^ ~ =_ J# #k (i) O=` ~ K H =_, A = P r n + (ii) J~ O=` ~ K H =_, A = (iii) = O=` ~ K H =_, A = ~ Q ^, A = P r n + ` j r 2n P8 + 2 ` jb r 4n P8 + ` 4 jb n j r 2 P` j ` j ; ` ~ Q ^, A = P- r ` ~O_ O=` ~ K H =_, ~ =_ H = ^ = ` = = A J# #` x XH ~ A x = A f H # ~ A x ~ A = yok x = XH ~ A x x.

51 ~M Q } ` = 2 2. ik = 2.2 u [= J# + H = 2.3 ^ Q ~ ^ O` = 2.4 =$` =. 2. ik = ~M Q } ` = H. 0 O=` ~ = ~ " Dl K Ja =$k ~ K _ #k. w J= ~ " # ^ q H H $`, ` =K < ^ ~ ~M Q } ` = # ` i H ^ ` ^ OuH x~ } WK ~. ~M Q } ` = = # rq` O XH = Y ` + OK K # k. = Y = Q u [= # J< H q^ = Q L yok K < =. [= Q Q q " = # u [= Q q loz ` H #=K #. Jx u [= " ` = " = P l H " = Q #. u [= O H } u [= #K " = H # Q K = Q #. J@ H xk u [ P^ ~ [= # O_ Z^ \ j~ = # O = wz " = # H # Q #= #. D ~ Q = #O^ u [= J# + H = " \ ^ ~ = # Q ioz ` H < ^ =. ^ Q ~ ( H ) Q } ` " ` O^ i # ~x H Q # D # XH w Q } ` A _. O H } u [= # O Ok Oz# q+ = P # k@. O H } u [ O O^ = D # ~ ` < = _ K # q. 47

52 J^ = u [= J# + H = (Concurrency in triangles) XH ` = #O^ = _ H JO` H O> Z =Q ~Y # w =. Zx q^ = Q w Q =? H Ok q^ = Q w Q =. A C E (a) (b) (c) (a) #O^ AB,CD, EF = O` ~ ~Y, H = # Jq YO_ OK H (b) #O^ AB = i CD P =^, AB = i EF Q =^ YO_ OK H # K # q. H = # P, Q ~O_ YO_ # ao^ (c) #O^ P, Q, R = _ YO_ # ao^ =. J~ (d) #O^ P XH > YO_ # ao^ =. WH _ AB,CD,EF X H ao^ = 'P ^ ~ = K # q. D ~Y # J# + H ~Y (concurrent lines) JO^ ~. 'P' ao^ = # ''J# + H ao^ = JO^ ~. XH u [= #O^ H x = Y " # J# + H ao^ = H =. " \ x u [ Q ~ ` HO ^ =, u [ O HO ^ =, u [ JO` ~ HO ^ = = i u [ i=$` HO ^ = Jx JO\ ~. W _ = #= D u [ ao^ = # Z@ O^ Q " K ^ = H = K < = u [ Q ~ ` HO ^ = #O^, ABC XH u [=. BC = ^ ao^ = D. AD x H =. WH _ 3ABC #O^ AD XH = ^ Q ` ~Y. XH u [ j~ = # O_ ^ x Z^ \ [= H = ^ ao^ = w _ # ~M YO_ = # ''= ^ Q ` ~Y JO^ ~. # Q = xok =. 3 ABC AD, BE, CF = _ = ^ Q ` ~Y. Jq G =^ YO_ OK H # #. D ao^ = < Q ~ ` HO ^ = JO\ ~. XH u [= x = _ = ^ Q ` ~Y J# + H = Q #. " \ J# + H ao^ = # Q ~ ` HO ^ = JO\ ~. nxx 'G Jx Q i ~. Q = xh : B D F E C Q B P A F (i) Q ~ ` HO ^ = u = ^ Q ` ~Y# 2 : x+ u q lok #. (ii) Q ~ ` HO ^ = = H ~ } ao^ = JQ #. D E Q B C A P R D A F C A E F P D B B2 O.g D2 O.g C A F G E B D C

53 ~M Q } ` = u [ O HO ^ = #O^, ABC XH u [=. A # O_, BC H XH O = # w =. BC H AD XH O =. + ADB = + ADC = 90 WH _ 0 D = ^ ao^ = H = # J= ~ O ^. WH _ AD O > ` JQ #. F A E XH u [= #O^ j~ = # O_, ^ x H^ ~ Q Q [= # H wz# O = # ' O > ` JO\ ABC #O^ B AD,BE,CF = _ O > `. Jq H =^ YO_ OK H # K # q. D ao^ = < O HO ^ = JO\ ~. D C XH u [= x = _ O > ` J# + H =. " \ J# + H ao^ = # ' O HO ^ = JO\ ~. O HO ^ = H qq^ #= (Different positions of orthocentre) A A A B D C B (a) (b) (c) O^ ~ = (i) (a) #O^ ABC XH J H } u [=. nx O HO ^ = u [ JO` ~ Q = #O^ LO_ #. O^ ~ = (ii) (b) #O^ ABC XH O H } u [=. nx O HO ^ = u [ O H } j~ = =^ LO_ #. O^ ~ = (iii) (c) #O^ ABC XH Jk H H } u [=. nx O HO ^ = u [ Q = #O^ LO_ #. C D B C u [ JO` ~ HO ^ = #O^ ABC XH u [=. + A # AD ~O_ = Q = Q = k YO_ # K K # k. + BAD = + DAC. WH _ AD x H } = k YO_ # ~Y JO\ ~

54 J^ = 2 XH u [= x H }= # = k YO_ # K ~M YO_ = # H } = k YO_ # ~Y JO\ ~. W _ 3ABC x AD, BE, CF = _ H } = k YO_ # ~Y. Jq I =^ YO_ OK H # K # q. D ao^ = < '' u [ JO` ~ HO ^ = '' JO\ ~. XH u [= x = _ H } = k YO_ # ~Y J# + H =, " \ J# + H ao^ = # ''JO` ~ HO ^ = '' JO\ ~ u [ i=$` HO ^ = H Ok ` ~ Q u g ~ O = k YO_ # ~Y# Q i < ~ H x < ~. u [= #O^ O = k YO_ # ~Y J#Q < q? H # ijeok =. (a) (b) (a) #O^ A # O_ BC H w _ # XH O = AD J~ ` Jk BC x = k YO_ # (b) #O^ BC x AD = k YO_ # K K # k. H = # BD=DC = i BC H AD O = Q L# (c) #O^ BC H w _ # O ~Y DX = i BC x DX = k YO_ # K K # k. BD=DC J~ ` XD j~ = A ^ ~ ^. XH u [= x XH [= # O = Q LO_ P [= # = k YO_ # K A ~Y# ' O = k YO_ # ~Y JO^ ~. M P S R O N B C # Q = xok =

55 ~M Q } ` = WH _ O =^ YO_ OK H # K # PQ, RS = i MN J# #q BC, AC = i AB H = _ O = k YO_ # ~Y. O J# #k i=$` HO ^ =. XH u [= H O = k YO_ # ~Y J# + H =. " \ J# + H ao^ = # '' i=$` HO ^ = JO\ ~. Q = xh : (i) U^ < XH u [= ABC #O^, i=$` HO ^ = (O), Q ~ ` HO ^ = (G) = i O HO ^ = (H) X H ~Y L# K ^ xx ~ ~Y (Euler Line) JO\ ~. = i OG : GH = : 2. (ii) ` H = Q = [ u [= #O^ i=$` HO ^ = (O), JO` ~ HO ^ = (I) O HO ^ = (H), Q ~ ` HO ^ = (G), X H ao^ = =^ LO_ #. 2.3 ^ Q ~ ^ O` = 2.3. ^ Q ~ ^ O` = XH O H } u [= #O^ H ~ = g k =~ = #, q ye# ~O_ [= =~ = " ` = # = #=. A D ABC #O^ + C = 90 o J# H # =. BC = a, CA = b = i AB = c ^ Q ~ ^ O` = H ~ =, a + b = c. Wk J< H = Ok Q } ` " ` K qq^ ^ ` ^ ~ x~ OK _ #k. = #= XH ^ u ^ Q ~ ^ O` = H x~ } K ^ =. ^ O` = : = #= K #@ Q [= (a+ b) ` XH K ` ~ = # wk ^ =. x~ } : a XH K ` ~ = H " = ^ x [= S H =~ = Jx = # `. K ` ~ [= (a + b) H =~ = (a + b) 2. # O_ (a + b) [= Q Q K ` ~ " = I =(a + b) 2 = I, II, III = i IV u [= " = + PQRS K ` ~ " =. IV B c H ~ = q a b R a c c Euler ( ) Switzerland A B M c H c M M a c c a P b E H H a III II M b H b C Z` C b b Q a 5 5

56 J^ = 2 i.e., (a b) 2 + = 4 ( O H } u [= " = ) + (PQRS K ` ~ " = ) (a + b) 2 = 4 ` a + b + 2ab = 2ab+ c ` a + b = c 2 ^ Q ~ ^ O` = x~ OK _ #k. ^ Q ~ ^ O` = # a# bj+ c 2 o + C = 90, AB = 5 O.g., AC = 4 O.g., BC = 3 O.g.LO_ #@ XH O H } u [= ABC # xi OK =. P u [ [= = _ K ` ~ = # xi OK =. P K ` ~ = # K. O.g " = LO_ #@ Q z# z# K ` ~ = Q q lok =. z# K ` ~ = # H Oz, ^ Q ~ ^ O` = # x~ OK =K #. ABPQ #O^ K ` ~ = = 25 BCRS #O^ K ` ~ = = 9 ACMN #O^ K ` ~ = = 6 `ABPQ #O^ K ` ~ = = BCRS #O^ K ` ~ = OY + ACMN #O^ K ` ~ = OY ^ Q ~ ^ O` = # x~ OK Q OY # '' ^ Q ~ u k JO\ ~. L^ ~ } 2. DABC +B=90 0, AB = 8 O.g., BC = 24 O.g. J~ # AC _ = # H # Q # =. ^ Q ~ ^ O` = H ~ = AC = AB + BC 2 2 = = = 900 ` AC = 900 = 30 O.g. L^ ~ } 2.2 XH K ` ~ = H ` 40 O.g. " \ H ~ = _ = " ` = ZO`? 52 52

57 53 53 ~M Q } ` = K ` ~ [= _ = a J# H # =. AC XH H ~ =. ABCD K ` ~ O H ` = 4 a x@ 4a = 40 O.g. [^ ` O = ] a = 40 = 0 4 O.g. XH K ` ~ = x uh }= O H }= = i H ~ = = #= Jx = # ` #. D ABC#O^ AC = AB BC = = + = 200 ` AC = 200 = 2 # = 0 2 = 0.44 # = 4.4 O.g. H ~ = AC = H ~ = BD ` H ~ = " ` = = = O.g. L^ ~ } 2.3 u [= PQR #O^ PT XH O > `, PQ = 25 O.g., PR = 7 O.g. = i PT = 5 O.g. J~ # QR = x O.g. x # H # Q # # O_ QR = QT + TR. H # Q # = #k : QT = i TR. PTQ O H } u [= #O^, Q PT XH O > ` + PTQ = 90 o ^ Q ~ ^ O` = H ~ =,PQ = PT + QT 2 PQ x O.g 2 PT 25 O.g ` - = QT 2 ` QT = 25-5 = = 400 QT = 400 = 20 O.g....() J^ q^ = Q O H } u [= PTR #O^ ^ Q ~ ^ O` = H ~ = PR = PT + TR ` TR = PR - PT 2 2 = 7-5 = = 64 TR = 64 = 8 O.g.... (2) () = i (2) # O_ QR = QT + TR = = 28 O.g. 5 O.g P T 7 O.g R

58 J^ = 2 L^ ~ } 2.4 XH n~ K ` ~ H ~ = H ` 40 g X 30 g Q L# q. P = A # O_ C H H ~ = g ^ Q #_ z " o #K ZO` ^ ~ = ` Q < H # Q # =. ^ ` O = :ABCD XH n~ K ` ~ = _ = = 40g, " _ = 30g, + B = 90 o O H } u [= ABC #O^ ^ Q ~ ^ O` = H ~ =, AC = AB + BC 2 2 = = = 2500 AC ` = 2500 = 50 g, A # O_ C H B ^ ~ " = # ^ ~ = = = 70 g, ` y # ^ ~ = = = 20 g. J = 2.. ~ # [" # Q i OK = : (i) XH u [= x = ^ Q ` ~Y J# + H ao^ =... Z) JO` ~ HO ^ = a) i=$` HO ^ = ) O HO ^ = _ ) Q ~ ` HO ^ = (ii) XH u [= x O > ` J# + H ao^ =... Z) JO` ~ HO ^ = a) i=$` HO ^ = ) O HO ^ = _ ) Q ~ ` HO ^ = (iii) XH u [= x H } = k YO_ # ~Y J# + H ao^ =... Z) JO` ~ HO ^ = a) i=$` HO ^ = ) O HO ^ = _ ) Q ~ ` HO ^ = (vi) XH u [= x O = k YO_ # ~Y J# + H ao^ =... Z) JO` ~ HO ^ = a) i=$` HO ^ = ) O HO ^ = _ ) Q ~ ` HO ^ = 2. XH = k [ u [= #O^ AB = AC = i + B = 65c J~ # ` = _ = Q [= Uk? 3. P =^ O H }= Q ey# u [= PQR. PQ=0 O.g., PR=24 O.g. QR # H # Q # =. 4. Wz # [= O H } u [ [= JQ <? ik _ =. AB = 25 O.g., BC = 24 O.g., AC = 7 O.g. 5. PQR u [= #O^ Q=25, R=65 J~ # DPQR XH O H } u [= JQ <? PQ = 4 O.g.,PR = 3 O.g. J~ # QR # H # Q # = g. _ = Q ey# XH xk ## 2 g. Z` Q H \ H =^ " _ #k. Jk Q _ # O_ q x g ^ ~ O LOK _ #k. x q = H # Q # = O.g. [= Q = [ u [= H Z` # H # Q # =

59 ~M Q } ` = 8. 2, 5 = i 3 OY ^ Q ~ u k JQ <? 9. WO\ H ~ OQ " " _ xk ##, 6 J_ Q Z` # =Ou H ~O_ = JO` x H \ H =^ ` #@ Q LOK #. P xk # H Ok Q = WO\ # O_ 2 J_ Q ^ ~ = L# k. J` _ ~ OQ H H # = =, XH H P xk ## WO\ # O_ 2 J_ Q ^ ~ = Q < \ " #. W _ xk # WO\ H Zx J_ Q Z` =~ K ~#? 2.4 =$` = H Ok = = h Q ik ". H Ok = = U PH ~ = L# " h= K Q "? (a) H K H = (b) * f z = x J H K H = (c) ~ K O ^ _ Yz ` = Q g [" =$` ". XH ` = #O^ XH ~ " # ao^ = # O_ = # ^ ~ = P J# ao^ = uiy#k U~ _ #k =$` = Jx g `. =$` = x~ K #= XH ` = #O^ XH ~ ao^ = # O_ = # ^ ~ O Q ao^ = = ^ = < '=$` = JO\ ~. ~ ao^ = # '=$` HO ^ = JO\ ~. ~ ^ ~ = # '=$` " ~ = JO\ #O^ 'O # HO ^ = Jx OA, OB, OC # '" ~ = Jx JO\ ~. WH _, OA = OB = OC = r Q = xh : =$` = x " ~ = x = #". * (Chord) : =$` = U" x ~O_ ao^ = # H ~M YO_ = # '* JO\ #O^ CD, AB = i EF * JQ #. WO^ AB =$` HO ^ = ^ ~ = K # XH ` H *

60 J^ = 2 " = =$` HO ^ = ^ ~ = * # '" = JO\ ~. " = Jx O\ H O> _ " # * JQ #O^ AOB =$` " =. AB H = ^ ao^ = 'O = i OA= OB = =$` " ~ =. H = # " = = 2 " ~ = ( H ) " ~ = = " = 2 Q = xh : (i) u " = H = ^ ao^ = =$` HO ^ ". (ii) =$` " = x J# + H =. " \ J# + H ao^ = =$` HO ^ =. =$` YO_ # ~Y XH =$` = ^ ~ = K, =$` = # ~O_ ao^ = =^ YO_ OK ~Y# =$` = H YO_ # ~Y JO\ ~. Wz #O^ AB XH YO_ # ~Y Wk =$` = # A,B ao^ = =^ YO_ OK K # k. W _ YO_ # ~Y AB x H Ok " Q [~ =. W _ A,B, =^ #, ` ~ ` A 2 B 2 =^ LO_ #. =$` YO_ # ~Y AB H OkH [~ Q H k A,B XH ^ xh XH \ ^ Q ~ Q #. A H = # A, B = ^ ^ ~ = x^ #= Q ` Q K B LO_ #. A B A 2 B 2 XH u YO_ # ~Y AB =$` = # X H XH ao^ = L N =^ ` #. D u LM ~Y# ' ~ ~Y JO\ ~. L M Wk =$` = # X H XH ao^ = =^ ` #. ~ ~Y =$` = # X H XH ao^ = =^ ` ~Y# ' ~ ~Y JO\ ~. P ao^ = # ' ~ ao^ = JO\ ~. =$` K #O^ AB XH *. AB * =$` = # ~O_ Q = Q q lok K # k. = H Q = ALB = i AMB # K = JO\ ~. K = #! Q ~ ` Q i OK ^ ~.! z# K = ALB x J K = Jx, 56 56

61 ~M Q } ` =! ^ K = AMB x Jk H K = Jx JO\ ~. A =$` YO_ = XH * =$` = # ~O_ Q = Q q lok #. XH H Q = # ''=$` YO_ = JO\ ~. J K = # H ey # Q = # =$` YO_ = JO\ ~. M Jk H K = # H ey # Q = # n~ =$` YO_ = Jx JO\ ~. =$` H } Q = XH =$` K =, ~O_ " ~ = ` = _ LO_ =$` Q = # =$` H } Q = JO\ ~. z# =$` H } Q = OALB x J =$` H } Q = (minor sector) JO\ ~. ^ =$` H } Q = OAMB x Jk H =$` H } Q = (major sector) JO\ ~. J = 2.2 L =$` YO_ = n~ =$` YO_ = B. ~ # [" # zok = : (i) =$` HO ^ = #, =$` ik H = ^ Q ^ ~ = #... JO^ ~. (A) =$` H } Q = (B) =$` YO_ = (C) " = (D) " ~ = (ii) =$` " ~ =, " = = ^ Q O O^ =... (A) " ~ = = 2 " = (B) " ~ = = " = + 2 (C) " = = " ~ = + 2 (D) " = = 2 (" ~ = ) (iii) =$` = Ju ^ * (A) " ~ = (B) =$` YO_ # ~Y (C) " = (D) ~ ~Y 2. ~O_ " = " ` = 200 q.g. =$` " ~ = # O.g. H # Q # =. 3. =$` YO_ = = i =$` H } Q = # x~ zok =. 4. =$` K = # x~ zok =. 5. =$` YO_ # ~Y = i ~ ~Y # x~ zok =

62 J^ = 2 Ä Q ~ ` HO ^ = : = _ = ^ Q ` ~Y J# + H ao^ =. Ä O HO ^ = : = _ O > ` J# + H ao^ =. Ä JO` ~ HO ^ = : = _ H } = k YO_ # ~Y J# + H ao^ =. Ä i=$` HO ^ = : = _ [= O = k YO_ # ~Y J# + H ao^ =. Ä =$` = : XH ~ ao^ = # O_, = # ^ ~ O Q ao^ = = ^ " =$` =. Ä * : JO` ao^ = =$` = Q XH ~M YO_ = * Ä " = : =$` HO ^ = ^ ~ = * < " = JO^ ~. Ä =$` = Q O_ = K =$` = # ~O_ ao^ = =^ YO_ OK ~Y# =$` YO_ # ~Y JO\ ~. Ä =$` = # X H XH ao^ = =^ ` K = ~Y# ' ~ ~Y Jx, P ao^ = # ' ~ ao^ = Jx JO\ ~. Ä =$` YO_ = : =$` = H * =$` = # ~O_ Q = Q q lok #. Ä =$` H } Q = : K =, ~O_ " ~ = = ^ Q =$` Q = # =$` H } Q = JO\ ~. Q } ` O H $` = ^ Q ~ u k m 2 + n 2, m 2 - n 2, 2mn WH _ m > n; n! N J# x = = # L yoz = #= J< H " # ^ Q ~ u ^ # H # Q #Q =. m = 2, n = J~ # m 2 + n 2 = = 5, m 2 - n 2 = = 3, 2mn = 2 x 2 x = 4. 5, 3, 4 XH ^ Q ~ u k. ^ Q ~ u k H Q } [= _ ^ Q ~ u ^ JQ #. L^ ~ } (5, 3, 4) H Q } [= : (0, 6, 8), (5, 9, 2), (20, 2, 6)... _ ^ Q ~ u ^. WOH # H x ^ Q ~ u ^ # H # Q #Q ~? 58 58

63 ^ ` O = # L 3 3. ik = 3.2 #: # \ H # ` ~ #i = ~ 3.3 =s H $` ^ ` O = # Q " = i #: # l # 3.4 ^ ~ } =$` # xi 3.5 HO n # H ` 3. ik = u k#= = #= " ~ uh ^ ~ #, W` ~ = ^ = ^ ~ # qq^ ~ H = = K ~ = # OY ~ = K < =. = # ^ O x P ~ L` u, OK [<, qq^ ^ = kq = ` = i ZQ = `, XH ~ + O ~ xez ~ # q^ ~ OY, = ^ O K x ~ # " i q=~ = " ^ Q = K ~ = WO^ LO_ =K #. D = K ~ = x O\ # = #= OY # L y < =. D OY < ^ ` O = JO\ ~. D ^ ` O = x~ f H L Q _ #. u ~ x rq` O # D ^ ` O = XH = Y ` + < ~. H = # J= ~ " # = i Yz ` " # = K ~ = #, ^ ` O = # O_ Z\ ` H K = Y =. H Oz# ^ ` O = K ^ J~ O K H = i J# = Q LO_ H =K #. H \ ^ ` O = # qq^ ^ ` H ~ * Q ` Q H iok = #. XH ^ ~ } = H _ ^ ` O = # J~ O K H x, ^ x # O_ H O` = K ~ = O^ Q Q #@ Q LO_ = #. P~.. + ~ (7 =i ) ^ + ^ O` = W+ = ` '' + ~ OY H = # i k # H < #. W` _ = ^ H = Q } `, uh = L ^ _ Q xk #. i ~ H # ^ $z H `, q $u # W` _ L yoz# q^ #= < < _ OK = L y < ~. D ## ''P^ xh OY H ` Jx JO\ ~. 59

64 J^ = #: # \ H # ` ~ Q ~ K H = #= 7= ` ~ Q u #: # \ H # ` ~ K _ O < ~ H x < =. Q ~ K O^ =! 3.2. = _ ^ ` O = # (ungrouped data) #: # \ H L^ ~ } 3. H O^ W= _ # ^ ` O = # #: # \ H # ` ~ K =. 5, 7, 7, 20, 5, 8, 6, 25, 6, 5, 6, 8, 20, 28, 30, 27, 8, 8, 20, 25, 6, 6, 20, 28, 5, 8, 20, 20, 20, 25. #: # \ H H O^ W= _ #k : OY Q }# Q ~ (x) #: # = (f) " ` = =s H $` ^ ` O = # (grouped data) #: # \ H ` ~ L^ ~ } 3.2 XH Q } ` sh 50 = Ok q^ ~ = ~ Q # " ~ ` K H # = ~ H O^ W= _ #q

65 ^ ` O = # L 43, 88, 25, 93, 68, 8, 29, 4, 45, 87, 34, 50, 6, 75, 5, 96, 20, 3, 8, 35, 25, 77, 62, 98, 47, 36, 5, 40, 9, 25, 39, 60, 37, 50, 9, 86, 42, 29, 32, 6, 45, 68, 4, 87, 6, 44, 67, 30, 54, 8. ^ ` O = # ` ~ Q u JO` ~ = # L yoz #: # \ H # ` ~ K =. " ` O q = OY = 50 " = Q i+ q = H x+ q = = 98 8 = 90 W _ ^ ` O = # 0 ` ~ Q ` Q q lok ^ =. " ` ` ~ Q u JO` ~ = H i= }= = ` ~ Q u JO` ~ = OY = Q } ` sh 50 = Ok q^ ~ Ok# = ~ #: # \ H W LOk : 90 0 = 9 ` ~ Q u JO` ~ = Q }# Q ~ #: # = (f) " ` =

66 J^ = 3 ` ~ Q u JO` ~ = #: # = D q^ OQ ^ ` O = # =s H ioz, \ H O^ O^ ~ K =K # =s H ioz# ^ ` O = # (Grouped Data) # z ` = = i #: # l # Wz # ^ ` O = # ~v z ` = Q Q h, Q Q x Q i OK =K #. ^ x< ~M z ` = (graphs) JO\ ~. ^ ` O = # ~M z ` = Q #, K ^ P H H eyok #, ` = = = = Q J~ = K H Q =. g\ x J< H ~ H = Q ~M z ` = K =K #. D J^ = #O^ = #= ~O_ ^ ` # ` H = K < =. Jq (i) # z ` = (Histogram) (ii) #: # l (Frequency Polygon) 3.3. # z ` = XH JO` " O^ x #: # q [## ~M z ` = ^ ~ < '' # z ` = JO^ ~. # z ` = #O^, n~ K ` ~ = (Bars) # =~ Q XH ^ x H # XH \ w = #. J#Q, # z ` = #O^, n~ K ` ~ =, ` ~ Q u JO` ~ = x _ LO_ #. n~ K ` ~ = " = #: # = J# ` = LO_ # (a) JO` " O^ x #: # q [# (continuous frequency distribution) # z ` = ^ u : ^ ` O = JO` " O^ ~ O LO>, ^ xx JO` " O^ x ~ O " = #. 2 : X-JH = ` ~ Q u JO` ~ = # ~ # = }= f H x Q i OK = 3 : #: # = # Y-JH O ~ # = }= f H x Q i OK =. 4 : ` ~ Q u JO` ~ = # P^ ~ = Q #, #: # = # Z` Q # f H x n~ K ` ~ = # xi OK =. H Ok L^ ~ } # z ` = # Z@ w ^ ~ q=iok _ #k. L^ ~ } : 3.3 XH sh = Ok q^ ~ Ok# = ~ H Ok \ H W= _ #q. nxh # z ` = # w =

67 ^ ` O = # L = ~ q^ ~ OY ` ~ Q u JO` ~ = 0 = ~ = # JO` ~ = H ey L# q. g\ x X-JH = Q i OK =. q^ ~ OY # Y-JH = ~ # L yoz Q i OK =. # z ` = H O^ w _ 3. Q = xh : \ H n~ K ` ~ = (bars) Jx =~ Q w _ #q. n~ K ` ~ = _ =, " \ #: # = J# ` = L# q. ` ~ Q u JO` ~ = = #= Q L# O^ #, n~ K ` ~ = " = " \ #: # = J# ` = L# q (b) ` ~ Q u JO` ~ = =~ Q # _ # z ` = L^ ~ } 3.4 XH J_ q x Z` W W= _ #. Wz # ^ ` O = # # z ` = # w =. Z` ) OY

68 J^ = 3 D =, W= _ # ` ~ Q u JO` ~ = =~ Q =. nxh Wz # q^ OQ < # z ` = # wz#k ` ~ Q u JO` ~ = = ^ M m U~ _ #. J~ ` # z ` = #O^ n~ K ` ~ = =~ Q LO_ = #. H = # ` ~ Q u JO` ~ = # =~ Q " = #. nxh = # XH J= ih H ~ H = (adjustment Factor) J= ~ =. J= ih H ~ H = = [(` ~ Q u JO` ~ = kq = ^ ) 2 (^ xh = O^ ` ~ Q u JO` ~ = ZQ = ^ )] = 2 (2 20) = 0.5 ` ~ Q u JO` ~ = #O^, = #= u kq = ^ # O_ 0.5 # f ", u ZQ = ^ #O^ 0.5 _ = #. H = # Wz # \ H # = #O W = i " = #. Z` ) OY W _ \ H JO` " O^ x #: # q [#Q = i#k. # z ` = H O^ w _

69 ^ ` O = # L Q = xh : # z ` = 3.2) X-JH = #O^ " ^ \ q = 5.5 # O_ ~ O = Q K # k. H = # = ao^ = =^ XH = H ~Y (kink)w _ #k. nxh ~M z ` = = ao^ = # O_ H O_, 5.5 # O_ ~ O = Q K # k Jx J~ =. JO` ~ = # lq * Q ~Y` zok ^ ~ #: # l #: # q [## ~M z ` = zok = ~ H ^ u #: # l. W= _ # JO` " O^ x ^ ` O = # # z ` = # w =. Q = H H # L# n~ K ` ~ = = ^ ao^ = # Q i OK =. D = ^ ao^ = x O\ x =~ Q ~M YO_ = K H e #, l U~ _ #. D lx #: # l JO\ ~. l H ~O_ z=~ # ` x iq #: # = H kq = ^, ZQ = ^ =~ w = #. #: # lx ~O_ q^ = Q w =K # (i) # z ` = # L yoz (ii) # z ` = O_ (a) # z ` = # L yoz #: # l w ^ u : Wz # ^ ` O = # O_ #: # q [# K, # z ` = # w =. 2 : # z ` = x n~ K ` ~ = Q = ^ ao^ = # ~M YO_ = K H =. 3 : ~O_ ` ~ Q u JO` ~ = #, " \ #: # = # (0) Q # T Oz, " \ XH \ " ^ \ n~ K ` ~ = # Z_ = H #, ~O_ =k z=i n~ K ` ~ = # _ H # Q i OK =. D ` ~ Q u JO` ~ = # T Oz# ` ~ Q u JO` ~ = JO\ ~. 4 : T Oz# ` ~ Q u JO` ~ = ` " ^ \ = i z=i ` ~ Q u JO` ~ = # H e #: # lx i K =. L^ ~ } 3.5 H O^ W= _ # ^ ` O = # # z ` = #: # lx w =. ` ~ Q u JO` ~ = #: # = :

70 J^ = 3.3 K #@ i # ` ~ Q u JO` ~ = # X-JH = #, #: # = # Y-JH = # f H # =. Wz # ^ ` O = # # z ` = w =. =~ Q L# n~ K ` ~ = Q = = ^ ao^ = # Q i OK =. T Oz# ` ~ Q u JO` ~ = 0 0 = i 90 _ = ^ ao^ = # Q i OK =. = ^ ao^ = # H =. l H ~O_ z=~ 0 0 = i 90 = ^ ao^ = ` H _ #q. W _, = # H = # #: # l U~ _ 3.3# ijeok =. 3.3 L^ ~ } 3.6 # z ` = # L yoz H Ok ^ ` O = # #: # l w =. ` ~ Q u JO` ~ = #: # = K #@ Q ~ # ` X-JH = ` ~ Q u JO` ~ = #, Y-JH = #: # = # Q i OK =

71 ^ ` O = # L Wz # ^ ` O = # # z ` = # w =. W _ n~ K ` ~ = Q = = ^ ao^ = # Q i OK =. T Oz# ` ~ Q u JO` ~ = (-0) - 0 = i f H # =. = ^ ao^ = # H =. l z=~ # T Oz# ` ~ Q u JO` ~ = (-0) - 0 = i = ^ ao^ = ` H =. W _ = # #: # l U~ _ #k. 3.4 K _ = 3.4 Q = xh : H x O^ ~ = T Oz# ` ~ Q u JO` ~ = LO_ =. L^ ~ } q^ ~ XH sh Ok# = ~, D q+ O # H O> z# ` ~ Q u, J` k H = ~ H O> ^ ` ~ Q ux T OK =. W O\ O^ ~ = ~M YO_ = z=~ #, " ^ \, z=i n~ K ` ~ = Z` = ^ ao^ = =^ w = #. L^ ~ } 3.7 Q = xh # L yoz H Ok L^ ~ } #: # lx w ^ =. ` ~ Q u JO` ~ = #: # =

72 J^ = 3 X-JH = ` ~ Q u JO` ~ = #, Y-.JH = #: # = # Q i OK =. ^ ` O = # # z ` = # w =. n~ K ` ~ = Q = x = ^ ao^ = # Q i Oz, " \ x H =. #: # l z=i JOK, " ^ \ = i z=i n~ K ` ~ = Z` = ^ ao^ = ` H (b) # z ` = # L yok O_ #: # lx xi w ^ u : #: # q [# \ H #O^, ` ~ Q u JO` ~ = = ^ ao^ = # H OK =. 2 : ` ~ Q u JO` ~ = # X-JH = #, #: # = # Y-JH = # Q i OK =. 3 : #: # = # ^ xh O Ok Oz# ` ~ Q u JO` ~ = = ^ ao^ = =^ Q i OK =. 4 : ~M YO_ = ` D = ^ ao^ = #, =~ Q H =. 5 : lx i ^ x z=~ # X-JH = H Ok ` ~ Q u, ` ~ Q u = ^ ao^ = ` H =

73 ^ ` O = # L #: # = ^ # : X- JH = ` ~ Q u JO` ~ = #, Y- JH = #: # = # Q i OK =. T Oz# ` ~ Q u JO` ~ = 0 0 = i 90 # #: # = '0 LO_ #@ f H # =. ^ ` O = # \ H ~ K =. H # # \ H # L yoz, ` ~ Q u JO` ~ = = ^ ao^ = #: # = A (5, 0), B (5, 4), C (25, 6), D (35, 8), E (45, 0), F (55, 2), G (65, 4), H (75, 7), I (85, 5) = i J (95, 0). ao^ = # Q i OK =. = #= AB, BC, CD, DE, EF, FG, GH, HI, IJ H e # = # #: # l ABCDEFGHIJ U~ 3.6 K _ H = # Q } ` G= L^ ~ } 3.8 H Ok ^ ` O = # #: # lx w =. ` ~ Q u JO` ~ =

74 J^ = 3 J = 3.. H Ok ^ ` O = # # z ` = w =. ` ~ Q u JO` ~ = #: # = H Ok ^ ` O = O` # z ` = w =. XH ZH ~ kq _ (H O\ ) = OY XH H H\ P@ x H OY ^ ` O = W= _ #k. nxh # z ` = w =. = ( OII) H OY XH Q = = x K H ~ " k Q q=~ = H iok _ #q. = ( OII) " k Q OY ^ ` O = # # z ` = O` #: # lx w =. 5. H Ok ^ ` O = #, # z ` = #, #: # lx xi OK =. ` ~ Q u JO` ~ = #: # = = Ok q^ ~ x# k ` (Intelligence test) sh " i u \ H ~ K _ #k. nxh # z ` = #, #: # lx w =. * zh q^ ~ OY H Ok ^ ` O = # # z ` = # L yoz, #: # lx w =. ` ~ Q u JO` ~ = q^ ~ OY

75 ^ ` O = # L 8. H Ok ^ ` O = # # z ` = O_ # # lx xi OK =. = OII = OY H Ok ^ ` O = # # z ` = O_ #: # lx xi OK =. ` ~ Q u JO` ~ = #: # = = Ok q^ ~ XH POQ sh Ok# = ~ (50 = ~ ) H O^ W= _ #q. D ^ ` O = # # z ` = = i #: # lx w =. 29, 45, 23, 40, 3,, 48,, 30, 24, 25, 29, 25, 32, 3, 22, 9, 49, 9, 3, 32, 39, 25, 43, 27, 4, 2, 3, 32, 44, 27, 43, 5, 35, 40, 23, 2, 48, 49, ^ ~ } =$` # xi (Construction of Simple Pie 3.7 = i 3.8 K #@ Q =$` H ^ ` O = # W= _ O, g ~ C_ < K ~? XH ~ q^ i XH ~ A (24 Q OII ) qq^ \.q. K # K H. Q _ = = kiq w _ # =$` JO\ ~. XH =$` XH " ` = #, ^ x Q = = ^ Q O O^ = # zok #. WH _ =$` = #, =$` YO_ = Q q lok ^ ~. u =$` YO_ =, Jk zok = K ~ = # J# ` = LO_ #. =$` YO_ = pie x = H ~ = # nxx pie chart JO\ ~. 7 7 Pie J# #k XH J" ih < =O@H =

76 J^ = 3 L^ ~ } =$` ''3.7 x ^ = Q O@ # zok x ^ = Q O@ 3 =$` YO_ = H J# ` = = XH i ~ A 8 Q O@ = 24 Q O@ = 3 H = #, D =$` YO_ =, =$` = 3 = =O` Q w _ #k. ~ Q _ # Q O@ # zok ~ Q _ # Q O@ =$` YO_ = H J# ` = 3 = XH i ~ A 6 Q O@ = 24 Q O@ = 4 H = #, D =$` YO_ =, =$` = 4 = =O` Q w _ #k. P@ _ = = # zok =$` P@ _ Q O@ YO_ = H J# ` = 3 = XH i ~ A 3 Q O@ = 24 Q O@ = 8 H = #, D =$` YO_ =, =$` = 8 = =O` Q w _ #k. WO\ x K = = # zok WO\ x K Q O@ =$` YO_ = H J# ` = 3= XH i ~ A 3 Q O@ = 24 Q O@ = 8 H = #, D =$` YO_ =, =$` = 8 = =O` Q w _ #k. WO\ x K = = # zok W` ~ = L yok Q O@ =$` YO_ = H J# ` = 3= XH i ~ A 4 Q O@ = 24 Q O@ = 6 H = #, D =$` YO_ =, =$` = 6 = =O` Q w _ #k. # # Jx a # = # _ =, " ` = = 3 = = =. 24 Jx a # = " ` = '' H = #=. WH _ XH q^ i XH ~ A Q _ = = # =$` = = ` K _ #k. i =$` " = # ' Q f H =_ O [iy#k. q^ i H qq^ H ~ H = = # J# ` = H Oz =$` YO_ = K _ = [iy#k. D J# ` Q = # H } H ` ^ ~ _ H OK =K #. =$` HO ^ = =^ Jx H }= " ` = 360 H = # = #= u =$` YO_ = # H } H ` ^ ~ K =K #.

77 ^ ` O = # L H Ok L^ ~ }, H } H ` # L yoz =$` (pie chart) xi Z\ q=iok _ #k. L^ ~ } 3.9 XH q^ i XH ~ A ` # = = # qq^ # Z qx yok K < _ H O^ W= _ #k. H } H ` # L yoz =$` # (pie chart) xi OK =. H ~ H = = x ^ ~ P@ WO\ x W` ~ = x Q O@ XH ~ A x 24 Q OII K qq^ # Jx O\ x 360.Q = ~ H # =. x ^ = = = 8 Q OII, H = # nxx 8 # =20 24 Q zok =. H = #, x ^ = = = # zok =$` YO_ HO ^ H }= 20 Q LO_ = #. J^ q^ OQ, ~, P@, WO\ x, = i W` ~ = qx yok = = # W^ q^ OQ _ w H Oz =$` YO_ HO ^ H }= Q = H Ok \ H W= _ #k. x x Q O@ HO ^ H }= x ^ 8 ~ # = 20 0 # = P@ 3 # 360 = WO\ x # = W` ~ = " ` = 24 =$` 24 # = h J# " # " ~ = ` =$` = w =. P =$` = #O^, U^ XH " ~ = =^ ~ Oa Oz HO ^ = =^ 20 Q i Oz ~O_ = [= w =. D =$` YO_ = x ^ = = = # zok #.

78 J^ = 3 D [= # O_, # wz# q^ OQ < =$` YO_ = 90 LO_ #@ = ~ H \ w =. Wk ~ Q _ # = = # zok #. W^ q^ OQ Jx JO = =$` YO_ = # wk ^ =.. D =$` YO_ = qq^ ~ OQ = # JO = = ^ ` _ K =K #. i K _ # =$` K _ 3.9 Q = xh : =$` qq^ JO = # =$` YO_ = K ` # = i JO = " ` = # =$` = K ` # zok ^ ~. HO ^ = =^ H } " ` = 360 # JO = q = J# Q }OQ q lok = #. XH JO = H HO ^ H }= JO = q = = " ` = q = # H x O^ ~ = JO = # ` = ` e ^ ~. J@ =O\ = = HO ^ JO = H ` = H }= = # Wz # ^ ` O = # =$` xi OK q^ =. # K # ` = # L yoz, u JO = # HO ^ H }= # H OK =. 2. J# " # " ~ = ` XH =$` = # w =. 3. P =$` = J_ = Q XH " ~ = # w =. 4. P " ~ = # O_ " ^ \ JO = H HO ^ H ` ` =$` YO_ = # w =. nx# O_ ~O_ = JO = # HO ^ H ` Q i Oz " ~ = wz =$` YO_ = # w =. D q^ OQ Jx Q = =$` YO_ = w =.. 5. Jx Q = " ~ ~ Q ~ OQ " =. 6. zh " =. 7. JO = # =. D q^ OQ Wz # ^ ` O = # =$` w Q = 74 74

79 L^ ~ } 3.0 H Ok \ H O < i Y~ q=~ = W= _ #q. ^ ` O = # L q=~ = P ~ = WO\ K ^ = ^ W` ~ = Y~ (~ ) ^ ` O = # =$` w =. H Ok q^ OQ u JO = # HO ^ H }= # H OK = : q=~ = Y~ (~ ) HO ^ H }= P ~ = # 360 = WO\ J^ # 360 = # 360 = K ^ = # 360 = ^ 0 0 # 360 = W` ~ = # = 42 " ` = H Ok q^ OQ =$` wk ^ =. O < i

80 J^ = 3 L^ ~ } 3. XH ~ ^ = ` ~ Q u e` = W L# q. e` = ^ = } Lf ~ k f } Lf ~ q^ ~ Lf ~ ` ` = = K ~ = # =$` w =. JO = H ` = JO = # HO ^ H }= = qq^ JO = HO ^ H }= # H OK ^ = : ` $f } Lf ~ Lf ~ H x" ~ 25% 35% 30% 0% # 360 = 0 G e` = q^ ~ ` = HO ^ H }= ^ = } Lf ~ 25% 25 k f } Lf ~ 35% 360 # = # = 26 0 ` $f } Lf ~ 30% # = 08 0 Lf ~ H x" ~ 0% # = 36 " ` = % = # H = # =$` H Ok q^ OQ wk ^ = ^ = ` ~ Q u sh e`

81 77 77 ^ ` O = # L J = 3.2. QO ^ < H Q u ` = (progress card ) = ~ W L# q. ~ O = ` q = POQ = Q } ` = = # = O H = = ~ J` _ qq^ ~ O = Ok# = ~ =$` w =. 2. 8= ` ~ Q u 36 = Ok q^ ~ H ~. " ~ qq^ O = Q L< ~. O = Q } ` = Z<... *.P~.. \ q^ ~ OY ^ ` O = # =$` K =. 3. XH = u Q $ = #O^ qq^ + = \ _ q^ ~ OY H O^ W= _ #q. ~ O = ` q = ` Q = O H # _ POQ = W` ~ = q^ ~ OY D ^ ` O = # =$` K =. 4. XH ~ x 8= ` ~ Q u qq^ Ja ~ K (hobbies) # " i OY W= _ #k. =$` w =. Ja ~ z Ow` = = = = O = q^ ~ OY XH q = = H Ok = # H ey # k. ^ ` O = # =$` K =. = OQ ~ O = " O_ ~ y lo ~ = XH H i JOQ _ XH ~ A qq^ ~ H = ^ ~ = J= H O (~ ) D q^ OQ [iy#k. ^ ` O = # =$` K =. ~ H = ^ ~ } ~ > ~ > HH a \ W` ~ = " (~ ) XH H = # JK XH K ~ } H ~ K # Y~ W W= _ #k. ~ H = H y` = = ^ } O_ OQ K ~ = ~ b Y~ ^ ` O = # =$` K =. 8. XH = = ^ ~ _ = = = K # Y~ q=~ = W= _ #q. ~ H = ^ H Z~ = q` #= = O^ h\ ~ ^ " ` = ~ II ^ ` O = # =$` K =.

82 J^ = 3 9. XH [O` ^ ~ # 900 } L# q. " \ q=~ = W= _ #q. =# = $Q ~ [O` = q x= OK [O` = h\ [O` = _ } } OY D ^ ` O = # =$` K = 0. XH H O h XH OII ` ~ K qq^ ~ H = " #= OY W= _ #k. D ^ ` O = # =$` K =. " " \ ~ H H ~ r " # OY XH P ~ = L# + H q = ` = H Ok \ H W= _ #k. D ^ ` O = # K =$` # w =. + H = \ < H = O_ ^ ~ = q@q # q #~ ` = 30% 0% 40% 5% 5% 2. qq^ ~ K (flavours) S H " # W+ _ XH ~ q^ ~ ` = H Ok \ H W= _ #k. nxh =$` # w =. ~ K (Flavours) K H \ " < i W` ~ = q^ ~ ` = 40% 30% 20% 0% 3. XH ~ q^ ~ ~ qx yok } = q=~ = H O^ W= _ #q: } q^ = H #_ H ~ q^ ~ ` = 40% 30% 5% 0% 5% ^ ` O = # =$` # w =. 4. ~ [< ` # P^ = 20% WO\ J^, 30% P ~ = #, 0% K ^ =, 25% ^ K #. q ye#k W` ~ Y~ L yok #. D = K ~ = ## =$` K =. 5. XH ~ + = x qq^ =~ = x" i ` = H Ok \ H W= _ #k. x" i =s H ~ } = = ^ ~ = = b H i " } [ x" ~ W` ~ ` = 40% 25% 2.5% 0% 2.5% = K ~ = # =$` K =.

83 3.5 HO n # H ` ^ ` O = # L Wz # = _ ^ ` O = # \ H ~ ^ ~, = # P q [# H ^ ~ } ~ = = ` " ` e #. WOH + = Q ` H " O> P ^ ` O = # Zx H K _ # XH OY Q K = #. P ^ ` O = H = Y ` = # XH ` H q = ^ ~ K Oz# WOH L H = Q LO_ #. D q^ OQ " ` = ^ ` O = # iq } OK H ` < HO n # H ` JO\ ~. Jq (i) JOH = ^ = (ii) = ^ Q ` = = i (iii) H = 3.5. JOH = ^ = (Arithmetic Mean (A.M)) JOH = ^ = J# #k ~ " ` = #, ~ OY = ^ Q x+ u (a) =s H iok _ x ^ ` O = # JOH = ^ = x = x, x, x,, 2 3 g x n K ~ ~ t x n zh JOH = ^ = # x Jx zok ^ ~. nxx x x 2 x 3 g x n n Jx " ~. Q } ` = #O^ " ` = # w JH ~ = R (Sigma) # L yok ^ ~. D O H` = ^ ~ n x, x 2, x 3, g, x n # / x i Jx ( H ) x = Rx i i = n Jx zok ^ =. Q = xh : JOH = ^ = # ~ i Jx _ JO\ ~. L^ ~ } 3.2 XH sh 0 = Ok q^ ~ Ok# = ~ W L# q. 5, 75, 33, 67, 76, 54, 39, 2, 78, JOH = ^ = # H # Q # =. WH _, ~ OY, n = 0 JOH = ^ = = x = x = 460 = R Q ~ # Q ioz WOH O` / k = = 6 k = 6 / n = = 8 n = 3 4 / 2n = = 8 = 3 3 / 5 = / 5 k 0 k = k = = n 2 = = 5 / ^k - h = (2 ) + (3 ) + (4 ) = 6 = 2 K

84 J^ = 3 L^ ~ } 3.3 9, 6, 7, 8, 5 = i x ~ i 8 J~ # x q = H # Q # =. Wz # q = 9, 6, 7, 8, 5 = i x, WH _ n = 6. ` = H ~ = JOH = ^ = = x = x 6 ^ ` O = H ~ = JOH = ^ =, x = 8 H = #, 35 + x = = 35 + x 6 6 i.e x = 48 x = = 3. L^ ~ } 3.4 XH ` ~ Q u x 0 = Ok q^ ~ ~ i Z` 66 O.g. Jx H OK _ #k. J~ ` ik z# _ XH q = 50 O.g. ^ Q 60 O.g. f H _ #k Jx ` e #k. ~ # ~ i Z` # H # Q # =. WH _, x = 66 O.g = i n= 0 = # x = n x i.e. 66 = R = x 0 Rx 0 R ( H ) Rx = 660 ` # Rx = 660 ~ # Rx = ` # Rx ` CQ " _ # q = + ~ # q = = = 650 H = #, ~ # JOH = ^ = = 650 = 65 0 O.g (b) =s H $` ^ ` O = (grouped data) # JOH = ^ = =s H $` ^ ` O = # ~O_ q^ = Q JOH = ^ = # H # Q # =K #. Jq, (i) ` H ^ u = i (ii) T Oz# JOH = ^ = ^ u (i) JOH = ^ = # H ( ` H ^ u) = # H Ok #: # q [# W= _ #k J# O^ =. K ~ ~ t x x 2 x 3 g x n #: # = f f 2 f 3 g f n D \ H # H Ok q^ OQ q=iok ^ = nx q = x, f = ~ =z #k. x 2, f 2 = ~ =z #k. x 3, f 3 = ~ =z #k. ggggg ggggg x n, f n = ~ =z #k.

85 WH _ x, x 2, x 3, g, x n J# #q x H " ~ ~ q =. WH _, ~ OY # N K z = n = N ( H ) (i.e.,) f f 2 f 3 g f n ^ ` O = # L / f i = N = i = ijeok = # q = " ` = = (x + x +x f ~ ) + (x 2 + x 2 +x f 2 ~ ) (x n + x n +x n f n ~ ) = f # x + f # x + g + f # x = fx 2 2 n n R i i H = #, x = ^ ~ }OQ, nxx W " =. x = L^ ~ } 3.5 ij # x q = " ` = ijeoz# q = OY R fx Rf R fx =, N = Rf. N H Ok ^ ` O = # JOH = ^ = # ` H ^ u H # Q # =. x f x f f x = Rfx i i Rf i (ii) JOH = ^ =, x = " ` = N = 25 R fx = 390 Rfx N = 390 = L* ~ O ^ u JOH = ^ = # H # Q (Assumed Mean Method) L^ ~ }, OY z# qq # Q }H ~ = = Q L# k. J~ ` OY ^ qq L# _ " \ Q }H ~ = n~ = Q LO_ P H O_ ~ ` C ^ i f =K #. 8 8

86 J^ = 3 D H + = O_ " ~ H ^ u H # Q # _ #k. D ^ u Wz # q = XH q =# JOH = ^ = (A) Q J# H O^ ~. nxx L* ~ O JOH = ^ = JO\ ~. ` ~ `, Jx q = x, x 2, x 3, g, x n L* ~ O JOH = ^ = (A) # O_ qk #= d, d2, d3,, g, dn # H # Q O^ ~. i.e. d = x- A, d2= x2- A, d3= x3- A,g,dn= xn-a W _ d, d2, d3,, g, d # =~ Q n f, f 2, f 3, g, f n K Q } Oz, P q = # _ # R fd =K #. W _ JOH = ^ = x = x = Rfd A + Rf Rfd A N R ) + (WH _ A T Oz# JOH = ^ = N= f W _ = #= L^ ~ } 3.5 # T Oz# JOH = ^ = ^ u JOH = ^ = # H # Q O^ =. T Oz# JOH = ^ = A = 5 Q f H # =. x f d = x A f d " ` = N = 25 R fd = 5 A + Rfd N JOH = ^ = = x = = 5+ 5 = 5+ 3 = 25 5 = = ~ ` JOH = ^ = (Weighted Arithmetic Mean (W.A.M.)) qq^ ~ = ` _ O_ #. J@ =O\ = = H # Q # JOH = ^ = # ~ ` JOH = ^ = JO\ ~. L^ ~ } XH ~ tx ~ = w, ` #, ~ t x 2 ~ = w 2 ` # z=iq x n ~ t, w n ~ = ` # _ L# q. J _ W. A. M. = wx wx wx wx g + n w + w + w + g + w n n = Rwx Rw

87 83 83 ^ ` O = # L L^ ~ } 3.6 H Ok ^ ` O = # ~ ` JOH = ^ = # H # Q # = : P ~ = i= }= ^ ~ (H. Q. ) ~ II (H. Q. ) w i x i a = K H ~ 2 30 # < 8 70 WH _ x-j# #k q = ^ ~ #, w i= }= # (H. Q.) # K K # k. ~ ` JOH = ^ = (W.A.M) = wx wx wx wx g n n w+ w2+ w3 + g + wn = 25 # # # 70 = = ` = ^ Q ` = HO n # H ` = ~ H H ` ''= ^ Q ` = (a) = _ ^ ` O = # = ^ Q ` = = ^ Q ` = # H Ok q^ OQ H OK ^ ~. (i) ij # OY (~ OY ) OY (odd number) Q L# _, " \ x P~ } H J=~ } H = = " = #. " \ = ^ ^ = < = ^ Q ` = JO\ ~. L^ ~ }, S^ ~ # 33, 35, 39, 40, 43 # ijeok =. g\ = ^ L# q = 39 H = # Wk = ^ Q ` =.. (ii) ij # OY ( H ) ~ OY ' i OY Q LO> " \ x P~ } ( H ) J=~ } H = = " =. " \ = ^ x ' ~O_ ~ ~ i< '= ^ Q ` = JO\ ~. L^ ~ }, 33, 35, 39, 40, 43, 48 = i 48 = ^ Q ` = JQ #. 2 Q = xh : = ^ Q ` = J#Q = O^, " # H = # OY ~ # H ey # XH ~ t. L^ ~ } 3.7 7, 5, 9, 3, 2, 7, 32. = ^ Q ` = H # Q # =. q = # P~ } H = = W " =. 7, 9, 3, 5, 7, 2, = i 32, WH _, n = 7 ( OY ) H = #, = ^ Q ` = = = ^ q = = n + 2 ` j = ^ = = H = #, = ^ Q ` = 5. 7 ` + 2 j q = = 4= ^ = = 39.5

88 J^ = 3 L^ ~ } 3.8 XH H H\ P@Q _ K # ~ Q 3, 28, 6, 70, 4,, 33, 0, 7, 92. = ^ Q ` = H # Q # =. ~ Q # P~ } H = = " Q 0, 4,, 3, 28, 33, 6, 70, 7, 92. WH _, n = 0 ( i OY ) WH _, ~O_ = ^ q = 28 = i 33 H = = ^ Q ` = = ~O_ = ^ q = ~ i = = 6 = (b) =s H $` ^ ` O = # = ^ Q ` = Oz` #: # = XH ` ~ Q u H Oz` #: # = J#Q P ` ~ Q u =~ Q #: # = " ` =. L^ ~ } = Ok q^ ~ = ~ = ^ Q ` = H # Q # =. = ~ q^ ~ OY = ~ (x) q^ ~ OY (f) Oz` #: # = (2 + 4 = ) (6 + 6 = ) 2 43 ( + 2 = ) ( = ) ( = ) ( =) 50 WH _ #: # = " ` =, N = R f = 50 N = 50 = = ^ Q ` = ( N ) 2 = ^ = = 25= ^ =. W _, 25= q =, Oz` #: # = 35, LO_ #, ^ xh O Ok Oz# = ~ 58. H = #, = ^ Q ` = = 58

89 ^ ` O = # L H = HO n # H ` '' H = _ XH \ JQ #. H = # H Ok q^ = Q H # Q #=K # (a) = _ ^ ` O = # H = : H x q = = ^ = # Wz # _, " \ Z = ~ =z # q =# H = JO^ ~. L^ ~ } , 4, 5, 2,, 2, 3, 4, 4, 6, 2 H = H # Q # =. D L^ ~ } '2 J# OY Z = ~ J#Q < Q ~ =z #k. H = # H = = 2. L^ ~ } , 25, 2, 22, 29, 25, 34, 37, 30, 22, 29, 25 H = H # Q # =. WH _ ''22 3 ~ = i 25 _ 3 ~ =z #q. ` 22 = i 25 D ^ ` O = # H = JQ #. Wz # ^ ` O = # ~O_ H = L# q Jx Q = xok Q =. L^ ~ } , 25, 35, 45, 55, 65, u q = X H XH i = ` " =z #k. H = # D ^ ` O = # H = ^ (b) =s H $` ^ ` O = # ( #: # q [# ) H = ^ ` O = #: # \ H ~ = LO>, #: # = Z =Q Q ` ~ Q ux H ` ~ Q u JO\ ~. P ` ~ Q u H q = # H = JO\ ~. L^ ~ } 3.23 H Ok #: # \ H H = H # Q # =. " ` #= x" i OY

90 J^ = 3 " ` #= x" i OY \ H # Q = xoz#, JO^ Ju ^ #: # = 6. D Ju ^ #: # = # K Ok# q = 350 W^ D ^ ` O = H H =. UH H = k H = u H = H = Wz # =~ X H Wz # =~ Wz # =~ Wz # =~ XH H = ~O_ H = = _ = _ H O> Z = LO> ^ xx UH LO> ^ xx k H = H = H = H = LO> ^ xx u LO> ^ xx JO\ ~. JO\ ~. H = H = L^ ~ } : 0, 5, 20, 25, 5, 8, 2, 5. WH _, H = 5. L^ ~ } : 20, 25, 30, 30, 5, 0, 25. WH _ 25, 30 k H =. J = 3.3 JO\ ~. L^ ~ } : 60, 40, 85, 30, 85, 45, 80, 80, 55, 50, 60. WH _ 60, 80, 85 u H =. I. JOH = ^ = =. 2, 4, 6, 8, 0, 2, 4, 6 JOH = ^ = H # Q # =. JO\ ~. L^ ~ } :, 2, 3, 8, 5, 4, 5, 3, 4, 2, 3,, 3, 5, 2, 7, 4,. WH _, 2, 3, 4, 5 H =. 2. 8, 4, x, 36, 3, 24, 37, 35, 27, 36, ~ i 3, x q =# H # Q # = = Ok q^ ~ Q ` ~ Q u 5 = Ok 76 = ~ 7 = Ok 77 = ~, = i 8 = Ok 78 = ~ k OK ~. P ` ~ Q u JOH = ^ = # H # Q # =.. 4. XH ` ~ Q u x 20 = Ok q^ ~ ~ i Z` 60 O.g. ijeoz# K # _ XH q = 52 O.g. ^ 32 O.g. Q " _ #k. J~ # ~ # JOH = ^ Z` H # Q # =

91 ^ ` O = # L 5. H Ok ^ ` O = # JOH = ^ = # H # Q # =. x f H Ok ^ ` O = #O^, XH ` ~ Q u x 40 = Ok ~ = W= _ #k. P ~ = JOH = ^ = H # Q # =. = q^ ~ OY H Ok ^ ` O = # JOH = ^ = H # Q # =. = ~ q^ ~ OY H Ok \ H, XH H ~ Q ~ O xk 2 = Ok H i ~ = W= _ #q. H i ~ = JOH = ^ = H # Q # =. ~ = (H. Q = ) H i OY O = #, XH < H = # ~, " \ ~ = W= _ #q. ~ ` JOH = ^ = H # Q # =. ~ ~ = (H. Q. ) ^ ~ (~. ) a = Q ^ = 5 20 C^ < 4 60 ~ Q 8 25 # < H Ok ^ ` O = # ~ ` JOH = ^ = H # Q # =. = = OY ^ ~ _ ~ 2 ` 45 4 ` 2 H = 5 ` 5 ih ~ = > 4 ` II. = ^ Q ` = =. H Ok q = = ^ Q ` = H # Q # = : (i) 83, 66, 86, 30, 8. (ii) 45, 49, 46, 44, 38, 37, 55, 5. (iii) 70, 7, 70, 68, 67, 69, 70. (iv) 5, 55, 46, 47, 53, 55, 5,

92 J^ = 3 2. H Ok ^ ` O = # = ^ Q ` = H # Q # =. x f XH ` ~ Q u x 50 = Ok q^ ~ Z` ( O.g ) W= _ #k. = ^ Q ` = H # Q # =. Z` ( O.g ) q^ ~ OY = Ok ~ Q Q O_ # X-H ~ }= ^ ~ sh Oz, ij # H Ok \ H W= _ #q. = ^ Q ` = H # Q # =. Q O_ " = (q.g. ) ~ Q OY = Ok L^ Q r` = H Ok \ H W= _ #q. = ^ Q ` = # H # Q # =. r` = (~ II) ,000 7,000 25,000 L^ Q OY III. H = =. H Ok ^ ` O = H = H # Q # =. i) 74, 8, 62, 58, 77, 74. iii) 43, 36, 27, 25, 36, 66, 20, 25. ii) 55, 5, 62, 7, 50, 32. vi) 24, 20, 27, 32, 20, 28, H Ok #: # \ H H = H # Q # =. x f H Ok \ H H = H # Q # =. L Q ` ( O\ Q_ ) ~ A OY XH ~ qq^ H ` Q K H Q _ = O_ H O^ W= _ #k. H = H # Q # =. H ` K H OY IV. JOH = ^ =, = ^ Q ` = = i H = =. H Ok #: # \ H JOH = ^ =, = ^ Q ` = = i H = # H # Q # =. x f

93 ^ ` O = # L 2. XH H O h x L^ Q = H Ok \ H W= _ #q. JOH = ^ =, = ^ Q ` =, H = # H OK =. = ( OII ) = OY H Ok \ H 20 = Ok q^ ~ ~ = W= _ #q. JOH = ^ =, = ^ Q ` =, H = # H OK =. ~ = H. Q q^ ~ OY XH # # q [## Q ^ ~ K ~O_ ^ ` # z ` = = i #: # l. # z ` = = i #: # l # Q K # _, ` ~ Q u JO` ~ = # X-JH = #, " \ #: # = # Y-JH = # f H = #. # z ` = #O^ n~ K ` ~ = # =~ Q H H # J#Q = ^ M m O_ J= ~ = #. # z ` = H n~ K ` ~ = Q = # = ^ ao^ = # Jx O\ x, H O^ \ ` ~ Q u JO` ~ = (succeeding c. i) = i ` ~ u ` ~ Q u JO` ~ = (preceding c. i) = ^ ao^ = # H e #K U~ _ l< #: # l JO\ ~. JO = H ` = XH JO = H HO ^ H }= = = # G ~ " ` = #, ~ OY Q x+ u < JOH = ^ = JO\ ~. JOH = ^ = H # Q ` = (i) x = (iii) x = A + ~ ` JOH = ^ = = R x (ii) x = n R fd R f R wx i i R w R fx R f A L* ~ O JOH = ^ = d = x A. i. = O^ " # H = # OY ~ # H ey O_ ~ t< = ^ Q ` = JO\ ~. XH q [# O^ Z = ~ =K q =# H = JO\ ~.

94 4 Q ` H ~M Q } ` = 4. ik = 4.2 UH HO n =$` = Q [ H ] U_ = ` = # K Ok# ~ ` Q } ` " `, ~ [ < " J~ # Q _ Q } ` =, YQ = J< H H = " _. L[ ~ x x YQ i ^ < HO ^ = # Jk uq = = iok _. k K Ok# ^ O` = J# Q O^ = # " _. r - 0 = i n [ OY =~ = " ` = = n( n+ )( 2n + ) 6 J# q = # H x > #. 4. ik = ~ H = Dl ` = ~M Q } ` = H Q ` H < } = # " i ^ H x~ }= #, i= }= # H # L yoz < ~. " i < Q ih ` _ Q ` H ~M Q } ` = # L yoz < ~. J< H Q x~ }= # " ~ H ^ = i =$` dx = ` xi Oz< ~. Q } ` = H ` e Y XH \ ~M Q } ` =. ~M Q } ` =, ^ O` ~M Q } ` = = i Q ` H ~M Q } ` = Q =s H io _ #. ~M Q } ` ^ O` = # K K Q w _ H x~ } H = = ` ^ O` ~M Q } ` = q=iok _ #. ~M Q } ` L H ~ }= # L yoz i # wz " \ x~ } H = = ` Q ` H ~M Q } ` = q=iok _ #. H x = ` ~M z ` = x~ }=, ^ ~ = = i " \ " = # H # Q # ` = # H Ok ` ~ Q ` < ~ H x < =. D Q = #O^ UH HO ^ =$` = # xi < ~ H O^ =. 90

95 Q ` H ~M Q } ` = 4.2 UH HO ^ =$` = D Q = #O^, UH HO ^ =$` = # Q i < ~ H < ^ =. =$` = # Q i = O^ < ~ H x < = ~ x " # h\ XH ~ ~ x " # C_, z# z# =$` H ~ J U~ Q = xok =K #. D =$` = HO ^ = LO_ #? ~ ~ ZH _ " " J^ JQ <? J= #. J^ JQ #. UH HO n =$` = XH = ` = #O^ X H HO ^ = ` " ~ ~ " ~ = ` w _ # =$` = # UH HO n =$` = O^ 4. = i 4.2 K _ 4.2 H O^ W= _ # # ijeok 4.3 #O^ ~O_ UH HO n =$` = L# 4.4 #O^ ~O_ UH HO n =$` = = ^ Q " = ~ OQ ` K K _ #. ~ OQ ` K K _ # " = # =$` = = JO^ ~. 9 9

96 J^ = 4 =$` = = - q=~ 4.5 #O^ C = i C 2 J# #q O # HO ^ = Q = i r, r 2 # a # " ~ = # H ey # =$` = Q #. C = i C 2 J# #q UH HO n =$` = Q #. P ~O_ =$` = = ^ Q " = # =$` = = JO^ ~. =$` = = H " _ = OB OA = r r 2 > r ) W= _ # " ~ = ` UH HO n =$` x~ }=. L^ ~ } 4. 3 O.g. = i 5 O.g. " ~ = ` UH HO n =$` = # xi Oz ^ =$` = = # K " =. =$` = " _ # H # Q # =. W= _ #k: " ~ = 3 O.g. = i 5 O.g. UH HO n =$` = x~

97 Q ` H ~M Q } ` = x~ } H = = ^ : # wz Wz # H ` # Q i OK =. ^ 2 : U^ # XH ao^ = O # f H x =$` HO ^ = Q Q i OK =. ^ 3 : O HO ^ = Q f H x OA = 3 O.g. " ~ = ` =$` = # w =. ^ 4 : O HO ^ = Q f H x OB = 5 O.g. " ~ = ` =$` = # w =. D q^ = Q UH HO n =$` = C = i C 2 w _ #q. =$` = = H " _ = OB OA J = 4. = 5 3 = 2 O.g.. H O^ W= _ # " ~ = H ` UH HO n =$` = # w =. =$` = = H " _ # H # Q # =. (i) 4 O.g = i 6 O.g. (ii) 3.5 O.g = i 5.5 O.g. (iii) 4.2 O.g = i 6.8 O.g. (iv) 5 O.g = i 6.5 O.g. (v) 6.2 O.g = i 8. O.g. (vi) 5.3 O.g = i 7 O.g. XH = ` = #O^ X H HO ^ = " ~ " ~ " ~ = ` w _ # =$` = # UH HO n =$` = JO^ ~. ~O_ UH HO ^ =$` = = ^ Q " = # =$` = = JO^ ~. =$` = = " _ = r 2 r (r 2 > r ) 93 93

98 [" J^ = [" J =.. i) D ii) C iii) B iv) B v) A 2. i) 200 b@~ ii) 20,000 H.g. iii) `,550 iv) 50 xq + = v) ` `40, i) 90% ii) 94% iii) 98% iv) 88% v) 95% vi) 93% ` 9, `, , H. Q.. i) 26, ii) 5, %, ` 8, P" Q } ` = 20% Jk H = Q k OK #. 4. ` 6, % J =.2. i) ` 7490 ii) ` 500 iii) ` 9,000 iv) ` 2,246 v) ` 6,57, i) = `64, ` = = 20% ii) = `200, ` = = 8% iii) #+ = `9, #+ ` = = 5% iv) Jq # " = `38, #+ ` = = 5% v) Jq # " = `5,500, ` = = 0%. 3. i) ` ii) `,260 iii) `2, `, % 6. 25% 7. `22,80, `34,40, % 0. " ` O # = `3.68 J =.3. i) A ii) D iii) B iv) B v) C 2. ` ` 8, ` 49, `8, ` 4, ` 3, % 9. `9, %. XH K H ^ ~ `, i) ` _ " ` = = `460; J= H ^ ~ = `,840 ii) ` _ " ` = = `35; ` _ ` O = 25% iii) Q i O ^ ~ = `20,000; ` _ " ` = = `4,000 iv) ` _ ` O = 5%; ` _ " ` = = `725 v) ` _ " ` = = `403; J= H ^ ~ = `2,

99 [" J =.4. i) A = `,57.63, =_ = `57.63 ii) A = `4,840, =_ = `840 iii) A = `22,869, =_ = `4, `2,25 3. i) `88,200 ii) `4,40 4. " ` = = `27,783, H # " = `3, `9, H # " = `,95 7. `20, `36, i) `92,400 ii) `92,60, ` _ = `20 0. `6. `25 2. `2, [ K e Oz# Jk H =_ ` J = `,25, OII 6. 0% 7. 0% J =.5. 2, `2,55,50 3. `46, ,3, ,45 6. `20, ,000 J =.6. `27, `86, `0, ` % 6. `, `9, < i " ~ ^ = `875, " ` = = `8,750 J = ~ A 2. 0; = Ok 4. 5 = Ok x" ~ A 6. `92 J =.8. 3 ~ A ~ A 3. 2 ~ A 4.2 xq 5. A = `360, B = ` ~ A 7. Q O@ 95 95

100 [" J^ = 2 J = 2.. i) D ii) C iii) A iv) B 2. Ju ` = _ = Q [= BC. 3. QR = 26 O.g 4. XH O H } u [= U~ _ #. 5. QR = 5 O.g 6. x = 9 g. 7. Z`, x = 5 3 O.g 8. J= # J_ Q J = 2.2. i) D ii) D iii) C 2. " ~ = = 5 O.g. J^ = 3 J = 3.3 I. JOH = ^ = = x = O. g `33 0. `2 II. = ^ Q ` = =. i) 8 ii) 45.5 iii) 70 iv) `0,000 III. H = =. i) 74 ii) H = ^ iii) 25 = i 36 iv) C IV.. JOH = ^ = 28; = ^ Q ` = 25; H = JOH = ^ = 25; = ^ Q ` = 25; H = JOH = ^ = 53.05; = ^ Q ` = 53; H =

101