Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$$. If $$\theta$$ $$\in$$ (0, $$\pi$$) is the angle between $$\widehat a$$ and $$\widehat b$$, then among the statements :

(S1) : $$2|\widehat a \times \widehat b| = |\widehat a - \widehat b|$$

(S2) : The projection of $$\widehat a$$ on ($$\widehat a$$ + $$\widehat b$$) is $${1 \over 2}$$

If $$y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} < {x^3} < {{3\pi } \over 2}$$, then

Consider the following statements:

A : Rishi is a judge.

B : Rishi is honest.

C : Rishi is not arrogant.

The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by $${{{x^2}} \over {xy - {x^2}{y^2} - 1}}$$. If the curve passes through the point (1, 1), then e . y(e) is equal to