Let f : R $$ \to $$ R be defined as f(x) = e^$$-$$xsinx. If F : [0, 1] $$ \to $$ R is a differentiable function with that F(x) = $$\int_0^x {f(t)dt} $$, then the value of $$\int_0^1 {(F'(x) + f(x)){e^x}dx} $$ lies in the interval

Let S₁, S₂ and S₃ be three sets defined as

S₁ = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}

S₂ = {z$$\in$$C : Re((1 $$-$$ i)z) $$ \ge $$ 1}

S₃ = {z$$\in$$C : Im(z) $$ \le $$ 1}

Then the set S₁ $$\cap$$ S₂ $$\cap$$ S₃ :

Let the tangent to the circle x² + y² = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r² is equal to :

The number of solutions of the equation

$${\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}$$, for x$$\in$$[$$-$$1, 1], and [x] denotes the greatest integer less than or equal to x, is :