Q Let a b c d be four integers such that abcd 4m1 where m is a positive integer

# Q let a b c d be four integers such that abcd 4m1

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Q.Let a, b, c, d be four integers such that a+b+c+d = 4m+1 where m is a positive integer. Given m, which one of the following is necessarily true? 1. The minimum possible value of a2+ b2+ c2+ d2is 4m2–2m+12. The minimum possible value of a2+ b2+ c2+ d2is 4m2+2m+13. The maximum possible value of a2+ b2+ c2+ d2is 4m2–2m+14. The maximum possible value of a2+ b2+ c2+ d2is 4m2+2m+1Soln. (2) — (a + b + c + d)2= (4m + 1)2Thus, a2+ b2+ c2+ d2+ 2(ab + ac + ad + bc + bd + cd) = 16m2+ 8m + 1a2+ b2+ c2+ d2will have the minimum value if (ab + ac + ad + bc + bd + cd) is the maximum.This is possible if a = b = c = d = (m + 0.25) ……….since a + b + c + d = 4m + 1In that case 2((ab + ac + ad + bc + bd + cd) = 12(m + 0.25)2= 12m2+ 6m + 0.75 Thus, the minimum value of a2+ b2+ c2+ d2= (16m2+ 8m + 1) 2(ab + ac + ad + bc + bd + cd)= (16m2+ 8m + 1) – (12m2+ 6m + 0.75)= 4m2+ 2m + 0.25Since it is an integer, the actual minimum value = 4m2+ 2m + 1Q.In the figure below, ABCDEF is a regular hexagon and 90AOF. FO is parallel to ED. What is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF? A B C D E F O 11/16 All rights reserved by CraZZyUstad.com 18 1 + DIRECTIONS for next three questions: Answer the questions on the basis of the information given below. A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from  • • • 