If e₁ and e₂ are the eccentricities of the ellipse, $${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$$ and the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ respectively and (e₁, e₂) is a point on the ellipse, 15x² + 3y² = k, then k is equal to :

Let the observations x_i (1 $$ \le $$ i $$ \le $$ 10) satisfy the
equations, $$\sum\limits_{i = 1}^{10} {\left( {{x_1} - 5} \right)} $$ = 10 and $$\sum\limits_{i = 1}^{10} {{{\left( {{x_1} - 5} \right)}^2}} $$ = 40.
If $$\mu $$ and $$\lambda $$ are the mean and the variance of the
observations, x₁ – 3, x₂ – 3, ...., x₁₀ – 3, then
the ordered pair ($$\mu $$, $$\lambda $$) is equal to :

The number of real roots of the equation,
e^4x + e^3x – 4e^2x + e^x + 1 = 0 is :

In a fluorescent lamp choke (a small transformer) 100 V of reverse voltage is produced when the choke current changes uniformly from 0.25 A to 0 in a duration of 0.025 ms. The self-inductance of the choke (in mH) is estimated to be ________.