Time: 120 mins                                                                               Max Marks: 100

1.(a) Find the range of f(x)=cot^-1(2x-x²).                                                [5]

(b) Evaluate                                                                 [5]

2. Prove that                                                                                                                                   [10]

3.Prove that if three prime numbers exceeding the number 3 are in Arithmetic Progression, then the common difference of the progression is divisible by 6.                                                                                        [10]

4. Let S₁ be a circle passing through through A(0,1),B(-2,2) and S₂ is a circle of radius  units such that AB is common chord of S₁ and S₂. Find the equation of S₂.                                                                                        [10]

5. Let f(x)=x2-2x, x R and g(x)=f((f(x)-1)+f(5-f(x)). Show that  Also find the critical point of g(x) if any.                                     [10]

6.Let A and B be two the objects lying on same side of a straight road on a horizontal plane. The line AB subtends two maximum angles a and b at two points on the road which are at ‘c’ distance apart from each other. Show that the distance between the two objects is  given by

7. Evaluate 

                                                                       [10]

8. Solve the differential equation

, given m+n=1.                                            [10]

9.If f(x) is a real valued function not identically equal to zero such that f(x+yⁿ)=f(x)+f(y)ⁿ, y“R and n is a natural number > 1 and f’(0)], then find the value of f(5) and f’(10).                                                             [10]

10. Find the area of the region enclosed by the curve , the line y=x and the +ve X axis.