Answer by appropriately matching the lists based on the information given in the paragraph.
Let $$f(x) = \sin (\pi \cos x)$$ and $$g(x) = \cos (2\pi \sin x)$$ be two functions defined for $$x > 0.$$ Define the following sets whose elements are written in the increasing order:
$$X = {x : f(x) = 0}, Y = {x : f'(x) = 0},$$
$$Z = {x : g(x) = 0}, W = {x : g'(x) = 0}.$$
List —I contains the sets X, Y, Z and W. List —II contains some information regarding these sets.
Which of the following is the only CORRECT combination?
Which ofthe following is the only CORRECT combination?
Answer by appropriately matching the lists based on the information given in the paragraph.
Let the circles $$C_1 : x^2 + y^2 = 9$$ and $$C_2 : (x - 3)^2 + (y - 4)^2 = 16,$$ intersect at the points X and Y. Suppose that another circle $$C_3 : (x - h)^2 + (y - k)^2 = r^2$$ satisfies the following conditions:
(i) centre of $$C_3$$ is collinear with the centres of $$C_1$$ and $$C_2$$,
(ii) $$C_1$$ and $$C_2$$ both lie inside $$C_3$$ and
(iii) $$C_3$$ touches $$C_1$$ at M and $$C_2$$ at N.
Let the line through X and Y intersect $$C_3$$ at Z and W, and let a common tangent of $$C_1$$ and $$C_3$$ be a tangent to the parabola $$x^2 = 8 \alpha y.$$
There are some expressions given in the List-I whose values are given in List-II below:
Which of the following is the only CORRECT combination?
Which of the following is the only INCORRECT combination?