If y² + log_e (cos²x) = y,
$$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$, then :

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 :