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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)dt‌∀x∈[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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