If `a_1, a_2, a_3,54,a_6,a_7, a_8, a_9` are in H.P., and `D=|[a_1,a_2,a_3],[5, 4,a_6],[a_7,a_8,a_9]|` , then the value of `[D]i sw h e r e[dot]` represents the greatest integer function

Evaluate: int_1^(e^6)[(logx)/3]dx ,w h e r e[dot] denotes the greatest integer function.

Evaluate: int_(-5)^5x^2[x+1/2]dx(w h e r e[dot] denotes the greatest integer function).

Find the value of : int_0^(10)e^(2x-[2x])d(x-[x])w h e r e[dot] denotes the greatest integer function).

The integral int_0^(1. 5)[x^2]dx ,w h e r e[dot] denotoes the greatest integer function, equals ...........

Discuss the differentiability of f(x)=[x]+|1-x|, x in (-1,3),w h e r e[dot] represents greatest integer function.

If a_r=(cos2rpi+i sin 2rpi)^(19) , then prove that |[a_1,a_2,a_3],[a_4,a_5,a_6],[a_7,a_8,a_9]|=0 .

The number of roots of x^2-2=[sinx],w h e r e[dot] stands for the greatest integer function is 0 (b) 1 (c) 2 (d) 3.

Suppose a_1, a_2, are real numbers, with a_1!=0. If a_1, a_2,a_3, are in A.P., then (a) A=[(a_1,a_2,a_3),(a_4,a_5,a_6),(a_5,a_6,a_7)] is singular (where i=sqrt(-1) ) (b)The system of equations a_1x+a_2y+a_3z=0,a_4x+a_5y+a_6z=0,a_7x+a_8y+a_9z=0 has infinite number of solutions. (c) B=[(a_1,i a_2),(ia_2,a_1)] is non-singular (d)none of these

The population of a country increases by 2% every year. If it increases k times in a century, then prove that [k]=7,w h e r e[dot] represents the greatest integer function.

IF A={1,2,3,4,5} and U={0,1,2,3,4,5,6,7,8,9}, then find A^c .