If `(x)=|alpha+xtheta+xlambda+xbeta+xvarphi+xmu+xgamma+xpsi+x v+x|` show that `"Delta^"(x)=0a n d"Delta"(0)+S x ,w h e r eS` denotes he sum of all the cofactors of all elements in `"Delta"(0)a n d` dash denotes the derivative with respect of `xdot`

Evaluate: int_0^(2pi)[sinx]dx ,w h e r e[x] denotes the greatest integer function.

Prove that int_0^oo[cot^(-1)x]dx ,w h e r e[dot] denotes the greatest integer function.

int_(-1)^2[([x])/(1+x^2)]dx ,w h e r e[dot] denotes the greater integer function, is equal to -2 (b) -1 z e ro (d) none of these

("lim")_(xvec1)(xsin(x-[x]))/(x-1),w h e r e[dot] denotes the greatest integer function is equal to (a) 0 (b) -1 (c) not exist (d) none of these

int_0^x(2^t)/(2^([t]))dt ,w h e r e[dot] denotes the greatest integer function and x in R^+ , is equal to

Find the value of int_(-1)^1[x^2+{x}]dx ,w h e r e[dot]a n d{dot} denote the greatest function and fractional parts of x , respectively.

Given that alpha , gamma are roots of the equation Ax^(2)-4x+1=0 and beta , delta are roots of the equation Bx^(2)-6x+1=0 . If alpha , beta , gamma and delta are in H.P. , then

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval

Solve 2[x]=x+{x},w h r e[]a n d{} denote the greatest integer function and the fractional part function, respectively.

Consider the points O(0,0), A(0,1), and B(1,1) in the x-y plane. Suppose that points C(x,1) and D(1,y) are chosen such that 0ltxlt1. And such that O,C, and D are collinear. Let the sum of the area of triangles OAC and BCD be denoted by S. Then which of the following is/are correct?.