The area of the region bounded by the curve $y=x^{2}$ and the line $y=16$ is

Two sources $A$ and $B$ are sending notes of frequency $680 \, Hz$. A listener moves from $A$ towards $B$ with a constant velocity $u$. If the speed of sound in air is $340 \, ms^{-1}$, what must be the value of $u$ so that the listener hears $10$ beats per second (in $ \, ms^{-1}$)?

If the circle $x^2+y^2+8x-4y+c=0$ touches the circle $x^2+y^2+2x+4y-11=0$ externally and cuts the circle $x^2+y^2-6x+8y+k=0$ orthogonally,then $k$ is equal to

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\overline{a} \times(\overline{a} \times \overline{c})+\overline{b}=\overline{0}$,then the acute angle between $\overline{a}$ and $\overline{c}$ is

$A$ ray of light refracts from medium $1$ into a thin layer of medium $2$,crosses the layer,and is incident at the critical angle on the interface between medium $2$ and $3$ as shown in the figure. If the angle of incidence of the ray is $\theta$,the value of $\theta$ is

If $\left(\frac{2}{3}, 0\right)$ is the centroid of the triangle formed by the lines $4x^2-y^2=0$ and $lx+my+n=0$,then $l+m+n=$