The value of (2.¹P₀ – 3.²P₁ + 4.³P₂ .... up to 51^th term)
+ (1! – 2! + 3! – ..... up to 51^th term) is equal to :

The function, f(x) = (3x – 7)x^2/3, x $$ \in $$ R, is increasing for all x lying in :

The solution curve of the differential equation,

(1 + e^-x)(1 + y²)$${{dy} \over {dx}}$$ = y²,

which passes through the point (0, 1), is :

If $$\alpha $$ and $$\beta $$ are the roots of the equation
x² + px + 2 = 0 and $${1 \over \alpha }$$ and $${1 \over \beta }$$ are the
roots of the equation 2x² + 2qx + 1 = 0, then
$$\left( {\alpha - {1 \over \alpha }} \right)\left( {\beta - {1 \over \beta }} \right)\left( {\alpha + {1 \over \beta }} \right)\left( {\beta + {1 \over \alpha }} \right)$$ is equal to :