The value of $\lambda$ such that $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines is:

The angle between the lines $\vec{r}=(2 \hat{i}-3 \hat{j}+\hat{k})+\lambda(\hat{i}+4 \hat{j}+3 \hat{k})$ and $\vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}+2 \hat{j}-3 \hat{k})$ is

On a new scale of temperature (which is linear) called the $W$ scale,the freezing and boiling points of water are $39\,^{\circ}W$ and $239\,^{\circ}W$ respectively. What will be the temperature on the new scale,corresponding to a temperature of $39\,^{\circ}C$ on the Celsius scale? ............. $^{\circ}W$

The ratio of the total energy of the $2^{\text{nd}}$ orbit electron for the hydrogen atom $1H^1$ to that of the helium ion $(He^{+})$ $2He^4$ is:

If one side of a triangle is double the other and the angles opposite to these sides differ by $60^{\circ}$,then the triangle is

The value of $\cot \left(\sum_{n=1}^{23} \cot ^{-1}\left(1+\sum_{k=1}^n 2 k\right)\right)$ is