$A$ particle of mass $m_1$ is moving with a velocity $v_1$ and another particle of mass $m_2$ is moving with a velocity $v_2$. Both of them have the same momentum,but their kinetic energies are $E_1$ and $E_2$ respectively. If $m_1 > m_2$,then:

$x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right), y=\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right) \Rightarrow \frac{d y}{d x}$ is equal to

If the length of the tangent from $(h, k)$ to the circle $x^2+y^2=16$ is twice the length of the tangent from the same point to the circle $x^2+y^2+2x+2y=0$,then

When an object is placed at a distance of $25 \, cm$ from a concave mirror,the magnification is $m_1$. The object is moved $15 \, cm$ farther away with respect to the earlier position,and the magnification becomes $m_2$. If $m_1/m_2 = 4$,the focal length of the mirror is $..... \, cm$ (Assume image is real,$m_1$ and $m_2$ are numerical values).

The sugar present in milk is

In the circuit shown,the heat produced in $5 \Omega$ resistance due to current $I_2$ is $50 \text{ J/s}$. Then,the heat generated per second in $2 \Omega$ resistance is (in $\text{ J/s}$)