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Q.

Let f be a function satisfying  f(x)=axax+a=ga(x) (a>0) On the basis of above information, answer the following questionsLet fx=g9x, then the value of ∑r=11995f(r1996) is (where [.] denotes the greatest integer function) Let f(x) = g4(x), the ∑r=11996f(r1997) isThe value of g5(x)+g5(1−x) is

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a

995

b

996

c

997

d

998

e

zero

f

even

g

odd

h

None of these

i

1

j

5

k

10

l

None of these

answer is , , .

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Detailed Solution

fx=g9x=9x9x+9=9x9x+3therefore fx+f1-x=1∑r=11995f(r1996)=f(11996)+f(21996)+f(31996)+…+f(19931996)+f(19941996)+f(19951996)=f(11996)+f(19951996)+f(21996)+f(19941996)+f(31996)+f(19931996)+…+f(9971996)+f(9991996)+f(9981996) =1+1+1…+1+1⏟997times+f(12) =997+12 =997.5∑r=11996f(r1997)=f(11997)+f(21997)+f(31997)+…+f(19941997)+f(19951997)+f(19961997)=f(11997)+f(19961997)+f(21997)+f(19951997)+f(31996)+f(19941996)+…=1+1+1+... 998 times or = 998 (even)g5(x)+g5(1−x)=f(x)+f(1−x)=1
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