If $(1+x)^{15}=a_0+a_1 x+\ldots+a_{15} x^{15}$,then $\sum_{r=1}^{15} r \frac{a_r}{a_{r-1}}$ is equal to

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $|\overrightarrow{a}|=3, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=\sqrt{37}$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is:

$A$ capacitor of capacity $0.1 \mu F$ connected in series to a resistor of $10 M \Omega$ is charged to a certain potential and then made to discharge through the resistor. The time in which the potential will take to fall to half its original value is (Given,$\log _{10} 2=0.3010$ ) (in $~s$)

$A$ screen is placed $50 \, cm$ from a single slit, which is illuminated with $6000 \, \mathring{A}$ light. If the distance between the first and third minima in the diffraction pattern is $3 \, mm$, the width of the slit is.....$mm$

$A$ tiny air bubble in a glass slab $(\mu = 1.5)$ appears from one side to be $6\, cm$ from the glass surface and from the other side,$4\, cm$. The thickness of the glass slab is......$cm$

If pairs of straight lines $x^2-2 p x y-y^2=0$ and $x^2-2 q x y-y^2=0$ be such that each pair bisects the angle between the other pair,then