Let y = y(x) be the solution of the differential equation

$${{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)$$, 0 < x < 2.1, with y(2) = 0. Then the value of $${{dy} \over {dx}}$$ at x = 1 is equal to :

Let the system of linear equations

4x + $$\lambda$$y + 2z = 0

2x $$-$$ y + z = 0

$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R.

has a non-trivial solution. Then which of the following is true?

The area bounded by the curve 4y² = x²(4 $$-$$ x)(x $$-$$ 2) is equal to :

If 15sin⁴$$\alpha$$ + 10cos⁴$$\alpha$$ = 6, for some $$\alpha$$$$\in$$R, then the value of

27sec⁶$$\alpha$$ + 8cosec⁶$$\alpha$$ is equal to :