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Question

In a triangle ABC, D,E,F are taken on the sides BC,CA,AB respectively. If BDDC=CEEA=AFFB=n and  Area (DEF)=f(n)× Area (ABC), then the correct statements is/are
  1. f(n) is monotonic function.
  2. f(n) is increasing in [1,)
  3. 10f(n) dn=523ln2
  4. 10f(n) dn=12+3ln2


Solution

The correct options are
B f(n) is increasing in [1,)
C 10f(n) dn=523ln2
Assuming A to be origin and the position vectors of B and C be b and c respectively.

Position vectors of
D:nc+bn+1E:cn+1       F:nbn+1
Now,
FD=ADAF=nc+(1n)bn+1FE=AEAF=cnbn+1
We know that,
Area (ABC)=12|b×c|
So,
Area (DEF)=12|FD×FE|=12(n+1)2(nc+(1n)b)×(cnb)=12(n+1)2n2(b×c)+(n1)c×b=n2n+1(n+1)2×12(b×c)
Therefore, 
f(n)=n2n+1(n+1)2f(n)=13n(n+1)2f(n)=3[(n+1)22n(n+1)(n+1)4]f(n)=3(n1)(n+1)3
So, f(n) is increasing when n1
Now let
I=10f(n) dnI=1013n(n+1)2 dnI=1310[n(n+1)2] dnI=1310[1(n+1)1(n+1)2] dnI=13[ln(n+1)+1(n+1)]10I=13ln2+3210f(n) dn=523ln2
 

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