In order to eliminate the first degree terms from the equation $2x^2 + 4xy + 5y^2 - 4x - 22y + 7 = 0$,the point to which the origin is to be shifted is:

Choose the correct statement$(s)$:
$(A)$ The position of the centre of mass of a system is dependent on the choice of the coordinate system.
$(B)$ Newton's second law of motion is applicable to the centre of mass of the system.
$(C)$ Internal forces cannot change the state of the centre of mass.
$(D)$ Internal forces can change the state of the centre of mass.

The potential energy of a body of mass $m$ is given by $U = ax + by$,where $x$ and $y$ are the position coordinates of the particle. The acceleration of the particle is

During the titration of $0.1 \, M$ $CH_3COOH$ and $0.1 \, M$ $KOH$ solution,what will be the $pH$ of the solution after $40\%$ neutralization of the acid? [$K_a$ for $CH_3COOH = 2 \times 10^{-5}$]

$A$ point moves with uniform acceleration and $v_1, v_2$ and $v_3$ denote the average velocities in the three successive intervals of time $t_1, t_2$ and $t_3$. Which of the following relations is correct?

The gravitational force acting on a particle,due to a solid sphere of uniform density and radius $R$,at a distance of $3 R$ from the centre of the sphere is $F_1$. $A$ spherical hole of radius $(R / 2)$ is now made in the sphere as shown in the figure. The sphere with hole now exerts a force $F_2$ on the same particle. Ratio of $F_1$ and $F_2$ is