RMO Sample Paper 1 Solutions :
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1 RMO Sample Paper Slutis :. The umber f arragemets withut ay restricti = 9! 3!3!3! The umber f arragemets with ly e set f the csecutive 3 letters = The umber f arragemets with ly tw sets f the csecutive 3 letters = The umber f arragemets with all three sets f the csecutive 3 letters = 3! Hece the required umber f arragemets is 9! 7! 5! ! 3!3!3! 3!3! 3!. d a, a, 0, 0 As is dd d, 4 0, 4 0 Agai usig the fact that is dd we get d, 80, 8 Hece all pssible values f d d ! 3 3!3! 5! 3 3! by iclusi exclusi priciple. (Marks 0) are factrs f 8 ad they are atteded whe is the factr f 8 as deted i fllwig table. S there are ly 4 pssible psitive values f viz, 4, 3, 4 0 givig crrespdig values f d 3, 9, 7, 8 (Marks ) 3. Clearly amg 8 itegers there are exactly divisible by 9. If the sum f the digits f ay e f them is 9 the we are thrugh. Nw the maximum sum f digits f a three digit umber ca be s S we csider the case whe factrs f 9 are givig digital sums 8 ad 7 Clearly the ther umber has t be 990 givig s ad (Marks 0) 4. Let A F E x ad F E P Therefre y 4 givig Als Nw 3 8 y The F E P F B P FP E P C B P C P B 3 y Hece A F E A F C AF F E B F B C F B 3 y 4 4 ad x x y 3 y 4 3 x 3 9 x y 3 x 3 as PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
2 x x Hece 3. x 3 3. A F P E square uits (Marks 5) 5. Sice B D C B A C, D B C A D C, A is the circumcetre f B C D. Hece A C A D A C D A D C A C D A D C A D E A D C B D C A E B C A D A D B 9 5 My sluti : Sice m arc D C 60 B D C B A C, D B C A D C, A is the circumceter f B C D. Hece Let P be the ther ed f the diameter thrugh C, the m arc P A B P A B 30 Hece A E B m arc D C m arc P A B 9 5 (Marks 5) 6. 7! 36! ! !36! m d Hece m d 7 3 (Marks 9) 7. The umber f rdered pairs such that A B x A B r x A, x B r x A B is give by 3 (As every x i S has 3 chice The umber f rdered pairs such that A = B is give by. (As every x is S has chice x A r x A ). Hece the umber f rdered pairs required is 3 (Marks 9) RMO Sample Paper Slutis :. We use the fllwig ccept, Fr travlig frm (a, b) t (c, d), he has t take ttally c a d b steps. c a steps X-axis ad d b steps Y-axis. S umber f paths are c a d b c a d b c a d b Ttal umber f OA paths OA Ttal umber f OPA paths O P A Ttal umber f OQA paths OQA Ttal umber f OPQA paths O P Q A P P P P PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
3 By Iclusi Exclusi priciple the ttal umber f OA paths t passig thrugh ay f pits P r Q will be P P P P 0 5. O A O P A O Q A O P Q A NOTE: Studets are t expected t fid the values f terms f r. r. It s perfectly k t keep fial aswer i. T... S m! m! T m! The pit f ccurrecy f agle bisectrs is iceter be I Let us dete m A a, m B b ad ad m C c. The C A D B A D a, C B E A B E b ad D C I E C I c a b c 8 0 ad a b c 9 0 Nw C D A is a extrerir agle f A B D C D A a b Hece i IC D, IC D c C D A a b C ID 80 a b c a b c a b c a c 0 0 Hece i IC Q, IC Q a c 9 0 a c c b c a c b Nw C P IQ is cyclic givig Q P I IC Q b E B A Hece P Q A B 4. If is eve, the k fr sme k Z k k k The is cmpsite. If is dd, the, ab with bth a ad b are bth dd itegers. Recllect the prperty that fr dd m m a m, a b a b Hece 5. Let us assume the cutarary. By Pigehle priciple atleast five f the lateral edges must have the same clur say black. Withut lss f gerality we assume these are segmets jiig vertex V t the base vertices B, B,..., B 5 which eed t be all adjacet. Hece B B, B B ad B Bare diagals s they must be clured. I ay e f B B, B B ad B Bis clured black say B 5 B 3 the we get V B B as a black triagle. 3 If e f the edges B B, B B ad B Bis clured black the we get V B B as a white triagle Hece we get a mchrmatic triagle. 6. Let M ad N be midpits f sides BC ad AC, respectively. Ntice that M O C B O C E A B, O M C 9 0 A E B, s triagles OMC ad AEB are similar ad we get O M A E O C. A B Fr triagles ON A ad BDA we als have O N O A B D B A 3 PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
4 The O N B D O M r O M.B D O N.A E A E S O B D B D.O M A E.O N S O A E. S, we see that Aalgusly, S O C D S O A F ad S O C E S O B F IPM MEGA FINAL SAMPLE PAPER- Aswer Key. (a). (c) 3. (a) 4. (d) 5. (b) 6. (a) 7. (c) 8. (b) 9. (b) 0. (a). (b). (d) 3. (c) 4. (c) 5. (b) IPM MEGA FINAL SAMPLE PAPER- Aswer Key. (a). (c) 3. (b) 4. (d) 5. (c) 6. (b) 7. (d) 8. (b) 9. (a) 0. (a). (d). (d) 3. (c) 4. (b) 5. (c) 4 PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
5 NTSE LEVEL- SAMPLE PAPER - ANSWER KEY MAT SAT Ques. N. As. Ques. N. As. Ques. N. As. Ques. N. As. b 6 a b 6 c b 7 d d 7 c 3 c 8 c 3 c 8 c 4 c 9 b 4 c 9 c 5 c 30 d 5 d 30 a 6 a 3 b 6 b 3 d 7 c 3 a 7 a 3 b 8 b 33 c 8 c 33 b 9 c 34 b 9 b 34 a 0 c 35 b 0 d 35 d a 36 d d 36 c c 37 d b 37 b 3 b 38 c 3 d 38 c 4 c 39 b 4 b 39 a 5 b 40 a 5 a 40 c 6 d 4 a 6 c 4 d 7 c 4 a 7 d 4 b 8 b 43 d 8 a 43 d 9 a 44 a 9 c 44 a 0 d 45 d 0 d 45 d a 46 b a 46 c c 47 b a 47 d 3 b 48 c 3 c 48 a 4 c 49 a 4 a 49 b 5 a 50 d 5 a 50 c 5 PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
6 NTSE LEVEL- SAMPLE PAPER - ANSWER KEY SAT LANGUAGE Ques. N. As. Ques. N. As. Ques. N. As. Ques. N. As. 5 b 76 d b 6 c 5 b 77 a d 7 b 53 a 78 c 3 b 8 c 54 d 79 d 4 d 9 a 55 d 80 c 5 a 30 a 56 d 8 a 6 d 3 d 57 d 8 b 7 c 3 c 58 c 83 c 8 a 33 a 59 d 84 d 9 c 34 b 60 c 85 b 0 b 35 b 6 c 86 a c 36 b 6 c 87 b d 37 d 63 a 88 a 3 c 38 d 64 c 89 a 4 b 39 d 65 c 90 c 5 d 40 a 66 c 9 a 6 a 4 d 67 b 9 b 7 c 4 d 68 c 93 b 8 b 43 d 69 c 94 d 9 d 44 d 70 d 95 a 0 c 45 c 7 a 96 d a 46 d 7 a 97 c d 47 a 73 a 98 c 3 c 48 b 74 b 99 c 4 b 49 a 75 c 00 a 5 b 50 b 6 PACE IIT & MEDICAL: Mumbai /Delhi & NCR / Luckw / Ga / Akla / Klkata / Pata / Nashik / Pue / Nagpur / Kapur / Bkar / Dubai
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