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What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

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Let f(x) be a polynomial of degree four having extreme values at $x=1$and $x=2$. If $(lim)_{x0}[1+x_{2}f(x) ]=3$, then f(2) is equal to :

The coagulating power of electrolytes having ions $Na_{+},Al_{3+}andBa_{2+}$ for arsenic sulphide sol increases in the order:

If the lines $1x−2 =1y−3 =−kx−4 $ and $kx−1 =2y−4 =1x−5 $are coplanar, then k can have

If z is a complex number of unit modulus and argument q, then $arg(1+z1+z )$equal

The sum of coefficients of integral powers of x in the binomial expansion of $(1−2x )_{50}$ is:

A particle of mass m moving in the x direction with speed 2v is hit by another particle of mass 2m moving in the y direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is T, density of liquid is $ρ$ and L is its latent heat of vaporization.

The I - V characteristic of an LED is