JEE Advanced Problem 2

Calculus Level 5

Suppose f(x)f(x) and g(x)g(x) are 2 continuous functions defined for 0x1 0 \leq x \leq 1.

f(x)=01ex+t.f(t)dt\displaystyle f(x) = \int_0^1 e^{x + t} . f(t) dt and g(x)=01ex+t.g(t)dt+x\displaystyle g(x) = \int_0^1 e^{x + t} . g(t) dt + x

AA. The value of f(1)f(1) is

1) 00

2) 11

3) e1e^{-1}

4) ee

BB. The value of g(0)g(0) - f(0)f(0) is

5) 23e2\frac{2}{3 - e^{2}}

6) 2e22\frac{2}{ e^{2} - 2}

7) 2e21\frac{2}{ e^{2} - 1}

8) 00

CC. The value of g(o)g(2)\frac{g(o)}{g(2)} is

9) 00

10) 13\frac{1}{3}

11) 1e2\frac{1}{ e^{2}}

12) 2e2\frac{2}{ e^{2}}

Details for entering the answer:

If your answer comes as option 1 for A, option 7 for B and option 12 for C, then write your answer as 1712.

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