The acrosome of a sperm contains:

If $u(x, y)=y \log x+x \log y$, then $u_x u_y-u_x \log x-u_y \log y+\log x \log y$ is equal to :

At a given instant of time,the position vector of a particle moving in a circle with a velocity $\vec{v} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ is $\vec{r} = \hat{i} + 9 \hat{j} - 8 \hat{k}$. Its angular velocity $\vec{\omega}$ at that time is:

The major product of the following reaction is:

The derivative of $f(\tan x)$ with respect to $g(\sec x)$ at $x=\frac{\pi}{4}$,where $f^{\prime}(1)=2$ and $g^{\prime}(\sqrt{2})=4$ is

The stress versus strain graphs for wires of two materials $A$ and $B$ are as shown in the figure. If ${Y_A}$ and ${Y_B}$ are the Young's moduli of the materials,then