JEE Advanced 2020 Full Test 6 Paper 1

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Answer questions Q. 47, Q. 48 and Q. 49 by appropriately matching the information given in the three columns of the following table.

Let f(x) be a double differentiable function with continuous second derivative defined on the interval [0,1] satisfying the equation
x
0
1(f(t))2dt
=
x
0
f(t)dtx[0,1]
and f(0)=f(0)=0. Column-1; contains the information about interval of value of f(x); Column-2; contains information about value of f(x); Column-3: contains information about interval of value of definite integral of f(x); Match the following Column(s)
   Column 1    Column 2    Column 3
 (I)  f(
1
2
)
<
1
8
 (i)  f(
1
3
)
=
1
18
 (P)  
1
0
f(x)dx
>1
 (II)  f(
1
2
)
>
1
8
 (ii)  f(
1
4
)
=
1
8
 (Q)  
1
0
f(x)dx
>
1
6
 (III)  f(
1
3
)
<
1
18
 (iii)  f(
1
6
)
=2sin2(
1
12
)
 (R)  
1
0
f(x)dx
<
1
6
 (IV)  f(
1
4
)
<
1
4
 (iv)  f(
1
8
)
=2sin2(
1
16
)
 (S)  
1
0
f(x)dx
<
1
4
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