# JEE ADVANCED Problem 2

**Calculus**Level 5

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

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

\(A\). The value of \(f(1)\) is

1) \(0\)

2) \(1\)

3) \(e^{-1}\)

4) \(e\)

\(B\). The value of \(g(0)\) - \(f(0)\) is

5) \(\frac{2}{3 - e^{2}}\)

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

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

8) \(0\)

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

9) \(0\)

10) \(\frac{1}{3}\)

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

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

**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.