Let the circles

C₁ : x² + y² = 9 and C₂ : (x $$-$$ 3)² + (y $$-$$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $$-$$ h)² + (y $$-$$ k)² = r² satisfies the following conditions :

(i) Centre of C₃ is collinear with the centres of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃ and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$$\alpha $$y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only INCORRECT combination?

Let the circle C₁ : x² + y² = 9 and C₂ : (x $$-$$ 3)² + (y $$-$$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $$-$$ h)² + (y $$-$$ k)² = r² satisfies the following conditions :

(i) centre of C₃ is collinear with the centers of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃, and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$$\alpha $$y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only CORRECT combination?

An electric dipole with dipole moment $${{{p_0}} \over {\sqrt 2 }}(\widehat i + \widehat j)$$ is held fixed at the origin O in the presence of a uniform electric field of magnitude E₀.



If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are, ($$ \in $$₀ is the permittivity of the free space, R >> dipole size)

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure.



When the distance of the piston from closed end is L = L₀, the particle speed is v = v₀. The piston is moved inward at a very low speed V such that $$V < < {{dL} \over L}{v_0}$$, where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?