SSC MAINS MATHS 68 (SOLUTION)

Transcription

1 . ().6 yards SS MINS MTHS 68 (SOLUTION). yards. yards rea of trapezium Height (Sum of parallel sides). (.6 +.).. sq yards sq metre ost of ploughing the land ` 600. () (8 ) (8 ) 0 ( )(8 ) 0. (), ( ) ( ) 0 ( )(0 ) 0, 0 So, common root ube and square Required difference E O is the incentre F. () O 8 O 6 and OE O Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR s, O 8... (i) OE and 6... (ii) From (i) and (ii) } and } and So, and similarly from equation (i) : : : : O OF [cosec cot ][cosec cot cosec cot ] [cosec cot ] [cosec cot ][cosec cot cosec cot ] [cosec cot ] cot cosec cot cosec + cot + cosec cot cot cosec cot cosec cot. () sin 8 + cos 8 s, sin + cos Squaring on both sides. (sin + cos ) () sin + cos + sin cos sin + cos sin cos gain squaring on both sides sin 8 + cos 8 + sin cos + sin cos sin cos + sin cos + sin cos sin cos sin cos sin cos sin cos 6. () a(a + b + c) ; b(a + b + c) and c(a + b + c) 0 0 a ; b ; c a b c a b c a b c Thus, ( a b c) ( a b c) So, (a + b + c) Thus, a ; b and c

2 (a + b + c ) ( + + ) ( ) 8. () sec sec + sec sec + sec + sec sec ( + sec ) + sec sec so, 0º cosec 6 cosec + 8cosec + (cosec 0) 6 (cosec 0) + 8(cosec 0) () Time needed by the minutes hand to show pm (60 t) minutes t 60 t t + t 0 t + 8t t 0 (t + 8)(t ) 0 So, t 9. () Raghav pays half of `,0,000 ` 60,000 in cash and balance ` 60,000 in annual instalment of ` 000 each. mount of first instalment ` ` ` 00 mount of second instalment ` ` ` 6600 ` 600 mount of third instalment ` ` ` 6000 ` 000 Thus, it forms an P where a 00, common difference 600 Sum n [ a ( n ) d ] [ 00 ( 600)] 6[ ],06,800 So, Total cost of shop ` 60,000 + `,60,800 `,66, () Given that O 0º O 0º E [tangent property] O, Let O In O O sin sin 0º So, O 0º 60º and 0º ngle made on minor segment 80º 80º 0º 0º. () istance covered by in 8 second 8 m istance covered by in 6 second + 8 m istance covered by in second m istance covered by in second m istance covered by in 0 second m istance covered by in 8 second m Remaining distance (00 ) 88 m To cover 88 m distance with speed 0 m/s sec 0 Total time 6.8 sec In 6.8 sec, cycles m So, should have a 0.8 m lead to result in a dead heat. Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

3 . () istance is equal. Let original speed s s( s ) s( s 0) 0 0 s s 0 s 6 km/hr 6 0 istance 90 km. () P Profit for n articles 0n n% of 0 n 00 Profit for rest articles (00 n)% of 0 (80 n) 0 (80 n)(00 n) 00 Minimum possible value of Total profit 0 at what value of n 0n 0 (80 n)(00 n) n + (80 n)(00 n) 0 n 80n n 90n n 90n (n ) So, at n, the minimum profit is made. % of 0 + % of () I for years ` 6 I for years ` 880 Let principal P, rate of interest r% I for years P( + r) P 6 P(r + r + r)... (i) I for years P( + r) P 880 P(r + r)... (ii) ivide (i) by (ii) r r r 6 r r r 880 r 6 r 880 r r 8 r 0 0r + 0r + 0 8r + 6 0r + 9r 6 0 0r + 680r 88r 6 0 0r(r + ) 8(r + ) 0 (0r 8)(r + ) 0 8 r, r 0 So, r 00 0% r I for years P 00 P P P 880 P P 80 P ` 000. () MRP ` SP after discount of 60% and 0% iscount Net discount Net P for man Spending on repairing + 9 ` 80 SP for article SP for article is ` () Length of pool 6 m readth of pool 0 m Height or rise in level h m Volume 6 0 h Volume used by 80 men m TQ, h. () h h 00 y y 0. cm 0 y 8 + y y y 9 6y y y y ( y) ( y) Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

4 y y y y 9. () sec + tan sec tan (sec + tan )(sec tan ) 8. () O In O, + r E (sec tan ) From (i) and (ii), tan tan 8 tan tan tan... (ii) () Syrup: Water I : II : III : Syrup in the first bottle 0 litre Water in first bottle 0 6 litre Syrup in the second bottle a OE In OE, O OE + E r ( ) r. () 9 litre Water in second bottle 9 litre Let n quantity of bottle be taken out n n 9 6 n 0 0th min th min r km r So, diameter of the circle cm P º 60º Speed of ballon 8 km/hr t 0th min, km Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

5 . () km t th min, s, P 9 km In P, tan 60º P P P 9 9 S 60 t 9 km/hr cm rea of triangle ase height cm. () Raman marks 6 Sita marks Total marks TQ, Minimum passing marks So, Raman scored (6 ) 8 marks more than minimum passing marks.. () Let the speed of boat in still water km/hr Let the speed of current y km/hr Let upstream speed U Let downstream speed 0U (i) 0U +... (ii) O E 6 cm cm and after putting this value U So, speed of current ( U) ( ) Radius cm E (6 8) 8 cm O cm O and O O ~ () O O 6 km/h. () P of milk ` 6. per litre P of miture Milk Water : Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

6 Required ratio : 0 6. () y, y 6 & y 9 a b cos( y)... (iii) ivide equation (i) by (ii) sin siny a cos cos y b tan y tan y ( y) a tan b a y b cos( ) cos( y) a b a b a b a a b b a b a a b b a b b a b b a b b a b b 8. () Let tower and height h cm (9 9) Height 9 (0) rea of O M 9 sq units. () sin + siny a... (i) cos + cosy b... (ii) dding after squaring them sin + sin y + sin siny + cos + cos y + cos cosy a + b + (sin siny + cos cosy a + b + cos( y) a + b In, In, h tan º 6 h tan º 9 From (i) and (ii), tan º tan º h 6 9 h 6 m 9. () º h (i)... (ii) Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

7 In, sin º 0 0 h 0 + h 0 h 0( ) cm 0. () y cos z cos y cos z cos y z y z k k, y k, z k y + yz + z k( k) + ( k)( k) + ( k)(k) k + k k 0. () Use options sin cos sin cos ivide by cos tan sec tan sec sec tan (sec tan ) tan sec (sec tan )[ sec tan ] [ tan sec ] sec + tan sin cos. () Let age of years Let age of ( ) years Let age of ( + ) years fter years, 6 + years So, sum of present ages + + years. () verage age of 80 boys years Sum Sum of boys 6 0 Sum of another boys (00 90) 60. () verage yrs M Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR : : 8000 : : : 0000 : : 0 : 's share of profit. () 6. () ; ivide and N by () () 9 so, required answer.... (i) 6... (ii) Multiply (i) and (ii), 6 6. () 6M 8W TQ, M W 0 M + W + 0 M + 6M + 8M

8 Remaining work (6M 8M ) 6 6 units Men required to do in one day 6 6 men 8. () Let the required quantity of coal consumed metric tonnes Let 6 engine of former type consume unit in hour and 8 engines of latter type consume unit in hour Engines : 6 Working hours 8 : :: 0 : Rate of consumption : () can do 8 0. () can do can do metric tonnes days 8 days 0 days Remaining work 60 [ ] Required percentage % Usual time taken to fill hrs 8 minutes + 9 minutes 9 hours Work done by leak in one hour Required time hours. () Let no. of students Each student got TQ, So, no of students 0. () iagonal of square P (a + b) rea of square P (a + b) rea of new square (a + b) Side (a + b) So, diagonal of new square ( a b). () Total upstairs covered by kbar ( + 9) 0 Time taken by him 0 00 seconds Total downstairs covered by kbar (9 + +) 6 Time taken by him 6 89 seconds Required time ( ) 89 seconds or 6 minutes 9 seconds. () Let total mothly salary 00 House rent 0 hildren's education Spending of % 00 8 Total monthly salary Total yearly salary () Let price of a chair Price of a table + 0 TQ, ( + 0) Price of chair 60, Price of table 80 Required % % hrs 8 minutes 000 ut due to leakage 6. () Error (gm) 0 00 Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

9 Reqd weight gm () Rahul's score 9 8 Manish score 68 Sum of Rahul, Manish & Suresh 89 Suresh's score 89 (68 + 8) Sum of Manish and Suresh's score gm. () ost price of horse and dog together ( ) SP of (horse + dog) ost price of dog (0 000) 0 ost price of horse () SI given by to Mr. X after years SI gained by from and Etra amount paid by to Mr. X () Let length of bus L m Speed of man 8 L m/s Speed of bus L m/s L Required ratio L 9 : 8 0. () Let side of square a cm a 96 a cm radius of larger circle 8 cm radius of smaller circle 8 cm ircumference of smaller circle cm. ().; y + y + y () verage score of Rahul, Manish and Suresh 6 jay's score Mº M Length of r 60º M 60º M 60º r 8 M rea of sector 60º r 8 r M and 60º M 0º r 8 M 0º So, 0 r. () Height is same for both traingles &. E So, ratio of their areas are in the Ratio of their bases. rea of rea of rea of rea of So, area of rea of 60 Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

10 . () sq units y y and Let and y So, In + + y 80º + y 80º 0º + y 60º + y 0º 80º ( + y) 80º 0º 80º 0º 0º 6. () Tangents drawn to a circle from the same points are equal in length. 8. () y property (MP + OQ ) MO (MP + OQ ) (0) MP + OQ 00 MP + OQ 00 ' O E O' E' is an iscosceles trapezium. E E' lso, O' ' 08 ' cm nd O '. cm So, 9. cm E E E F G H F y G I k E I H z Now, + + k + z + k + + y + z + y (k + + y + z) 8 0 So, k + + y + z 0 z + k + y 8 So, + y 8. () MO 0 cm Q N M P O (9.) (6.) 90 cm Thus, diameter of circle 90 cm 9. () O O s, O O radius So, O O y O y O O 80º y So, + y + 80º y 80º y 0 y 60. () Total volume Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

11 Volume of new cube 9 a 9 a 9 cm So, side of new cube 9 cm iagonal of new cube 9 cm 6. () Ist year interest 0 ` 0 00 So, Total amount after st year ` 6 So, fter Ist installment, remaining amount 6 ` 00 nd year interest 00 ` 00 Total amount 00 + ` 6. () beats by 0 seconds and beats by seconds. So beats by (0 + ) seconds So distance travelled by in seconds 80 m Speed of 80 m/second So, time taken by to travel km sec. 6. () a b 9 c 9 a b c? a b c 0 or a + ( b) + ( c ) 0 So, (a ) + ( b ) + ( c ) a ( b )( c ) 0 or, a 9 b 9 c 9 a b c 0 6. () istance cover km. with speed of p km/hr. ue to slippery ground Speed (p q) km/hr r ( p q) (p q) r 6. () Total quantity of milk 8 litre 66.() rd of milk is replaced by water 8 litre Process is followed by times Then quantity of pure milk Quantity of water 8 6 litre Ratio of milk and water 6 : : Total distance covered by the man in hrs. is 9 km. ue to stop, etra time ( ) minute 60 hrs. Total time hrs. verage speed km./hrs. 6. () st nd day day rd day Let the speed of train be km./hrs. Remaining distance 8 ( ) 6 km. Total time days 8 hrs. 6 hrs. ccording to question, verage speed, 8 6 km. 6 istance in two day ( ) 08 km. In rd day speed of train 6 6 km./hrs. 8 ifference (6 6) km./hrs. 68. ()a ( + y + z) y, a y ( + y + z) z, a z ( + y + z) Multiply a, a y and a z. We get, a.a y.a z (+y+z) y. (+y+z) z. (+y+z) a +y+z ( + y + z) + y + z + y + z a So, + y + z a 69. () a + y, b y, c + y (Given) a + b + c ab bc ca a(a b) + b(b c) + c(c a) Put the value of a, b and c ( + y)( + y + y) + ( y)( y y) + ( + y)( + y y) ( + y)(y) + ( y)( y) + ( + y)(y) y + y + ( y) + y + y + y + y + y y + y + y + y y 0.() Let etra men needed y learly, 90 8 / y 60 0 / 90 ( + y) 0 + y y 0 Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

12 . () Let, or Putting the value of or, 6. () Height of cone 0 cm TQ, Volume of first cone Volume of second cone Height of first cone Height of second cone Height of new cone Height of new cone cm Section is made ( 0 cm) above the base. () rea of base of prism 0 cm Height of prism 0 cm Surface area of prism 0 ( + + ) sq cm. () Volume of sphere r Volume of cone (r) r TQ, Volume of new sphere Volume of cone R r + + r r R () () Required ratio Surface area of smaller sphere : surface area of larger sphere r R : r : r () : (). () ab + bc + ca 0 bc ab + ac ac ab + bc ab bc + ca a bc b ca c ab a ab ac b ab bc c bc ca a( a b c ) b( a b c) c( a b c) ab bc ca abc( a b c) 0 6. () E E : :. () n() n() + n() n() () Minimum distance HF of 6 cm, 0 cm and cm () From figure, O P O + O + º + O ( + ) E º + O P O 0º º O 60º º º tan P + cot O tan 0º + cot º Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

13 80. () Milk : Water Milk: Water ontainer : ) : ontainer : ) 0 : 8 Required ratio : 8. () Total ependiture ` ( ) ` 900 Total saving ` verage salary ` ` 8 8.() It is easy to solve this question by using option. y option(d), Total number of apples at the starting Number of apples sold to first customer + 8 Remaining apples 8 Number of apples sold to second customer 6 Remaining apples 6 6 Number of apples sold to third customer 6 Remaining apples 6 Number of apples sold to fourth customer 6 Remaining apples 6 i.e. condition satisfied 8.()Required remainder () [ 9 8 +, so 9 is replaced by ] 8.() Let etra hours per day are. y M H W M H W (6 ) (6 6 ) 8 6.9% (appro.) 86.() Let the fied charges ` and the additional charges ` y / km ccording to the question, + y 0 (i) + 0y 800 (ii) On solving Eqs. (i) and (ii), we get 00, y 0 harge for a distance of 0 km + y ` 90 8.() Let the price of shirt ` 00 0.T.Q, Hence, the price of shirt ` () Let 's share ` then 's share ` (9 ) 0.T.Q, 00 0 (9 ) 00 (9 ) 6 (9 ) (9 ) ` 9 Hence is 'share ` 9 89.() Let the height of circular cylinder H total volume of the solid volume of the cone () Let marked price ` 00 and selling price ` 80 If the loss is 0%, the cost price of article ` Required profit percentage Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR O' O h H Total volume of the solid Volume of circular cone r H r h r h

14 r H r h H h r h r H r h 90. () radius of pipe cm, (given) 0 height of pipe 000 cm radius of vessel 0cm and height cm Volume of water that flows in one minute through cylindrical pipe cm lso, volume of conical vessel (0) 00 cm Time taken by pipe to fill the vessel 00 or min s 9.() Given area of right-angled triangle cm h 6cm cm 0cm Now, area of triangle 0 h h 0.8cm 9.() Let the radius of solid spheres be r and r respectively..t.q Surface area of 00% of surface area of surface area of r 6r r r or : Now, Volume of 00 k Volume of 00 r 00 k 00 r r r 00 k k 00 k () Perimeter of circular sheet r 0 cm The perimeter of base of conical surface cm 60% 0% Radius of base of conical surface 6cm & the height of conical surface 0 6 8cm So, The required ratio 6 : 8 or : 9.() Given Now, () ( +) ( +) ( +)( 8 +)( 6 +) ( ) (+) ( +)( +)( 8 +) ( 6 +) ( ) ( +) ( +)( 8 +) ( 6 +) ( ) ( +) ( 8 +)( 6 +) ( 8 ) ( 8 +) ( 6 +) ( 6 ) ( 6 +) ( ) 96.() Number of workers in scale V % of Number of working male in scale V % of Number of working female in scale V () In scale VII. Total number of workers 8% of 00 0 Number of male workers 0% of Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

15 Number of female workers Required ratio 80: 0 : 98.() Number of females in scale I 0-98 Number of females in scale VI 0-8 Number of females are same in scale I and VI. 99.(d) verage of working females in all scales Required no. of scales (II, IV, V, VII) 00.() Number of females in scale VII 8% of 600-0% of Required percentage % + 0% (increase) Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR

16 SS MINS MTHS - 68 (NSWER KEY). (). (). (). (). (). () 6. (). () 8. () 9. (). (). (). (). (). (). () 6. (). () 8. () 9. (). (). (). (). (). (). () 6. (). () 8. () 9. (). (). (). (). (). (). () 6. (). () 8. () 9. (). (). (). (). (). (). () 6. (). () 8. () 9. () 6. () 6. () 6. () 6. () 6. () 6. () 66. () 6. () 86. () 96. (). (). (). (). (). (). () 6. (). () 8. () 9. () 8. () 8. () 8. () 8. () 8. () 8. () 68. () 8. () 88. () 98. () 9. () 9. () 9. () 9. () 9. () 9. () 69. () 9. () 89. () 99. () 0. () 0. () 0. () 0. () 0. () 60. () 0. () 80. () 90. () 00. () Note:-. For any issue related to RESULT Processing, kindly contact us on If your opinion differs regarding any answer, please message the mock test and question number to For any complaint/suggestion, all/msg. or Whatsapp to Ph: 0-608, (M) entres at: MUKHERJEE NGR MUNIRK UTTM NGR ILSH GREN ROHINI RPUR JIPUR GURGON NOI LXMI NGR GHZI MEERUT VRNSI ROHTK PNIPT SONEPT PTN GR HNIGRH LUKNOW LLH KNPUR HISR