The area (in square units) of the region bounded by $x^2=8y$,$x=4$ and the $X$-axis is:

If \(x_1\) be the size of the magnified image and \(x_2\) the size of the diminished image in Lens Displacement Method, then the size of the object is :

Two ions having masses in the ratio $1: 1$ and charges $1: 2$ are projected into a uniform magnetic field perpendicular to the field with speeds in the ratio $2: 3$. The ratio of the radii of circular paths along which the two particles move is :

The natural frequency of an $L-C$ circuit is $1,25,000 \text{ cycles/s}$. Then the capacitor $C$ is replaced by another capacitor with a dielectric medium of dielectric constant $K$. In this case,the frequency decreases by $25 \text{ kHz}$. The value of $K$ is

If the area of the auxiliary circle of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ is twice the area of the ellipse,then the eccentricity of the ellipse is

When bauxite powder is mixed with coke and heated with nitrogen at $2075 \ K$,carbon monoxide and $X$ are formed. Which of the following gases is produced when $X$ reacts with water?