Solutions to all questions of Orange phase part 2.
Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, what is the third term in the GP? 92 -92 46 231 Choice A. 92 DETAILED SOLUTION Sum of first 12 terms is equal to sum of first 14 terms. Sum of first 14 terms = Sum of first 12 terms + 13th term + 14th term => 13th term + 14th term = 0 Let us assume 13th term = k, common ratio = r. 14th term will be kr. k + kr = 0 k (1 + r) = 0 => r = -1 as k cannot be zero Common ratio = -1. Now, if the first term of this GP is a, second term would be -a, third would be a and so on The GP would be a, -a, a, -a, a, -a,… Sum to even number of terms = 0 Sum to odd number of terms = a Sum to 17 terms is 92 => a = 92 Third term = a = 92 Correct Answer: 92 ARITHMETIC PROGRESSION If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of the A.P., what is 13 times the 13th term of this A.P.? 7 times the 13th term 0 13 times the 7th term 4 times the 4th term + 9 times the 9th term DETAILED SOLUTION 4t 4 = 9t 9 , we need to find t13 4(a + 3d) = 9(a + 8d) 4a + 12d = 9a + 72d => 5a + 60d = 0 => a + 12d = 0 => t13 = 0 => 13 × t”13″ = 0 As a simple rule, if n × tn = m × tm, then tm+n = 0. See, if you can prove this. Answer choice (b). Correct Answer: 0 Q.2: Find the smallest number that leaves a remainder of 4 on division by 5, 5 on division by 6, 6 on division by 7, 7 on division by 8 and 8 on division by 9 2519 5039 1079 979 DETAILED SOLUTION LCM (5, 6, 7, 8, 9) – 1 = 2519. Choice (A) Correct Answer: 2519 .7: The sum of two non co–prime numbers added to their HCF gives us 91. How many such pairs are possible? 2 4 3 6 DETAILED SOLUTION Let HCF of the numbers be h. The numbers can be taken as ha + hb, where a, b are coprime. h + ha + hb = 91 h(1 + a + b) = 91 h ≠ 1 h = 7 => 1 + a + b = 13 a + b = 12 h = 13 => 1 + a + b = 7 => a + b = 6 Case 1: h = 7, a + b = 12 (1, 11), (5, 7) => Only 2 pairs are possible as a, b have to be coprime. Case 2: h = 13, a + b = 6 (1, 5) only one pair is possible as a, b have to be coprime. Overall, 3 pairs of numbers are possible – (7, 77) (35, 49) and (13, 65) Answer choice (c)