SSC MAINS NEON CLASSES, JAIPUR /34

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2 Web.: What is the value of ? dk eku D;k gsa (a) (b) 101 (c) (d) (SSC Mains 017, 17 Feb. 018) ( ) 3 [( ) - ( ) ]. a 5 + b 5 = a + b a 4 a 3 b + a b ab 3 + b 4 = a 5 a 4 b + a 3 b a b 3 + ab 4 + a 4 b a 3 b + a b 3 ab 4 + b 5 = a 5 + b 5 3. a 7 + b 7 = a + b a 6 a 5 b + a 4 b a 3 b 3 + ab4 ab5+b6 = a 7 a 6 b + a 5 b a 4 b 3 + a 3 b 4 a b 5 + ab 6 + a 6 b a 5 b + a 4 b 3 a 3 b 4 + a b 5 Formula: n 3 = n n + 1 ab 6 + b 7 = a 7 + b 7 General Form : [(10) (1) ] = 8 (13959) = Neon Approach : a n + b n = (a + b) a n 1 a n b + an 3b +bn 1, if n is odd. There are n terms in bracket. Extension of this formula : a n + b n + c n is divisible by (a + b + c), ;g la[;k = 198 ls fohkkftr gksxha 198 dk digit sum 9 gs] vr% answer dk digit sum Hkh 9 gh gksxka,slk dsoy,d gh option (d) gsa Concept : (a n + b n ) is divisible by (a + b), if n is odd (a n + b n ) dks (a + b) fohkkftr djsxk ;fn n,d fo"ke la[;k gksa Explanation: 1. a 3 + b 3 = a + b a ab + b = a 3 a b + ab + a b ab + b 3 = a 3 + b 3 if n is odd and a, b and c are in arithmetic progression (A.P) and similarly, a n + b n + cn+dn is divisible by (a + b + c + d). a n + b n + c n dks (a + b + c) fohkkftr djsxk ;fn n,d fo"ke la[;k gks rfkk a, b, c lekurj Js<+h esa gksa blh izdkj a n + b n + c n + d n dks (a + b + c + d) fohkkftr djsxka Hence, the above mentioned situation can be extended for any number of terms. The condition is that, the terms should b in AP. And n is odd. (a n + b n + c n x n ) is divisible by (a + b + c + x) Download our app : NEON CLASSES Page

3 Example: a, b, c are in arithmetic progression and n is odd. What is the remainder obtained when is divided by 9? It can be seen that 17, 1, 5 and 9 are in Arithmetic progression and power n is odd. Further, Denominator = = 9 Hence it will be divisible by 9. Therefore remainder obtained = 0. 17, 1, 5 rfkk 9 lekurj Js<h es a gs] rfkk budk ;ksx 9 gs vksj n,d fo"ke la[;k gsa vr% 'ks"kqy 0 vk;sxka CAT Questions Q. 1 The remainder, when is divided by 19, is? (1) 4 () 15 (3) 0 (4) 18 (CAT 004, November) n is odd, hence it is divisible by = 38, and therefore, it will be divisible by 19 also. All the terms except the last one contain 19 and the last terms get cancelled out. Hence, the remainder obtained on dividing by 19 will be Zero. bl expression ds lhkh inksa esa 19 ekstwn jgsxk rfkk dsoy vafre in es a 19 dh txg 4 3 vk;sxk tks fd,d ckj /kukred rfkk,d ckj _.kkred gksxka vr% ;s nksuksa eku vkil es a cancelled out gks tk;saxsa vr% 19 ls Hkkx nsus ij 'ks"kqy 0 vk;sxka Binomial Theorem: (x + y) = x + xy + y (x + y) 3 = x 3 + 3x y + 3xy + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x y + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y + 10x y 3 + 5xy 4 + y 5 For a binomial involving subtraction, the theorem can be applied by using the form (x y) n = (x + (-y)) n. This has the effect of changing the sign of every other term in the expansion. (x - y) = x - xy + y (x - y) 3 = x 3-3x y + 3xy - y 3 (x - y) 4 = x 4-4x 3 y + 6x y - 4xy 3 + y 4 (x - y) 5 = x 5-5x 4 y + 10x 3 y - 10x y 3 + 5xy 4 - y 5 Hence, remainder = 0. Hence, answer is option (3) Alternatively: The expression becomes Q. If x = , then x is divided by 70 leaves a remainder of? (1) 0 () 1 (3) 69 (4) 35 (CAT 005) Download our app: NEON CLASSES Page 3 Like our FB

4 Web.: n is odd, and these terms are in arithmetic progression with common difference, d = 1. Hence it is divisible by = 70. And therefore, remainder is Zero, Alternatively, This problem can be done by theorem method also. x = = ( ) = = ( ) = 35( ) The expression in the bracket is divisible by, because it will be an even number. dks"bd esa nh xbz la[;k,d le la[;k gs] vr% ;g ls fohkkftr gksxha x is divisible by both 35 and. x is also divisible by 35 = 70. Remainder = 0 Hence, answer is option (1) The sum of the squares of first n natural n n+1 (n+1) numbers is = 3 n n+1 (n+1) izfke n izkd`r la[;kvksa ds oxksza dk ;ksx = 3 K = = ( ) K = K = = 96. Neon approach, To find the last digit of answer, add the last digit in each term Last digits, = 46 Hence, last digit is 6. Only one option contains 6 as last digit Hence, answer is option (a) 96. Q. 78. What is the fourth proportional to 189, 73 and 153? 189, 73 vksj 153 dk pksfkk vuqikfrd D;k gsa (a) 117 (b) 99 (c) 1 (d) 187 (SSC Mains 017, 17 Feb. 018) Q. The sum of =? (a) 96 (b) 3017 (c) 315 (d) 3311 (SSC Mains 01) Solution: a : b : : c : d a d = b d Product of extremes = Product of means 189 : 73 : : 153 : x Download our app : NEON CLASSES Page 4

5 189 x = x = 1 Neon Approach Last Digit 9 x = 3 3 x = 1 Hence, the last digit of x is 1. The correct answer is option (3) 1. Range : 189 : 73 :: 153 : x 180 : 70 :: 150 : x : 3 :: 150 : 5 (Approx.) Download our app: NEON CLASSES Page 5 Like our FB

2009 (Odd) xzqi&a ds lhkh 20 ç'uksa ds mùkj nsaa ¼izR;sd ds 1 vad gsaa½

2009 (Odd) xzqi&a ds lhkh 20 ç'uksa ds mùkj nsaa ¼izR;sd ds 1 vad gsaa½ 009 (Odd) Time : Hrs. Full Marks : 80 Pass Marks : 6 SemI-G Engg. Math.-I GROUP-A.(A) Write down the most correct answer for the following question from four given alternatives : = Answer all 0 questions