The eccentricity of the curve $ 2x^2 + y^2 - 8x - 2y + 1 = 0 $ is:
Let $ f(x) = \int \frac{x^2 \, dx}{(1 + x^2)(1 + \sqrt{1 + x^2})} $ and $ f(0) = 0 $, then the value of $ f(A) $ is:
A particle of mass $ M $ originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation $ F = F_0 \left[ 1 - \left( \frac{t - T}{T} \right)^2 \right] $ where $ F_0 $ and $ T $ are constants. The force acts only for the time interval $ 2T $. The velocity $ v $ of the particle after time $ 2T $ is:
If the system of equations $ x + y + z = 6 $, $ 2x + 5y + \alpha z = \beta $, $ x + 2y + 3z = 14 $ has infinitely many solutions, then $ \alpha + \beta $ is equal to: