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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower, Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M) 09652351152, 09652661152 IAS-MATHEMATICS (Opt.) - 2018 PAPER - I : SOLUTIONS SOLU TION : UP SC-IAS-M AINS-2018-MA THEM A TICS -OPTIONA L-PAPER-1 PDF created with pdfFactory Pro trial version www.pdffactory.com

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Page 1: IAS-MATHEMATICS (Opt.) - 2018 · 2019-08-05 · IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA (16) S O L U TI O N: U P S C-I A S-M A I N S-2 01 8-MA T H E MA T I C S-O P T I ON A L-P

HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower, Mukherjee Nagar, Delhi-9.BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M) 09652351152, 09652661152

IAS-MATHEMATICS (Opt.) - 2018PAPER - I : SOLUTIONS

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: UPSC

-IAS-M

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

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IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA

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PDF created with pdfFactory Pro trial version www.pdffactory.com

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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA

(48)

SOLU

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PDF created with pdfFactory Pro trial version www.pdffactory.com

http://www.pdffactory.com
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HEAD OFFICE:25/8, Old Rajinder Nagar Market, Delhi-60. (M) 9999197625, 011-45629987. BRANCH OFFICE(DELHI): 105-106, Top Floor, Mukherjee Tower,Mukherjee Nagar, Delhi-9. BRANCH OFFICE(HYDERBAD): H.No. 1-10-237, 2nd Floor, Room No. 202 R.K’S-Kancham’s Blue Sapphire Ashok Nagar Hyd-20. (M)

09652351152, 09652661152

IAS/IFoS MATHEMATICS (Opt.) BY K. VENKANNA

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PDF created with pdfFactory Pro trial version www.pdffactory.com

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