If $z = x + iy$ is a complex number satisfying $\left| z + \frac{i}{2} \right|^2 = \left| z - \frac{i}{2} \right|^2$,then the locus of $z$ is

If $-\frac{\pi}{4} < x < \frac{\pi}{4}$,then the general solution of the differential equation $\cos ^2 x \cdot \frac{d y}{d x}-(\tan 2 x) y=\cos ^4 x$ is

$A$ mass $M$ attached to a horizontal spring executes $S.H.M.$ of amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio $\left(\frac{A_1}{A_2}\right)$ is

One die and a coin (both unbiased) are tossed simultaneously. The probability of getting $5$ on the top of the die and tail on the coin is

$A$ lens forms real and virtual images of an object,when the object is at $u_1$ and $u_2$ distances respectively. If the size of the virtual image is double that of the real image,then the focal length of the lens is (take the magnification of the real image as $m$):

Polysaccharides have .............. linkage?