The value of

$${\sec ^{ - 1}}\left( \matrix{ {1 \over 4}\sum\limits_{k = 0}^{10} {\sec \left( {{{7\pi } \over {12}} + {{k\pi } \over 2}} \right)} \sec \left( {{{7\pi } \over {12}} + {{(k + 1)\pi } \over 2}} \right) \hfill \cr} \right)$$

in the interval $$\left[ { - {\pi \over 4},\,{{3\pi } \over 4}} \right]$$ equals ..........

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?

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 combinations is correct?

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?