If `p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot]` denotes the greatest integer function, then the value of `p(1-f)` is equal to a.`1` b. `2` c. `2^n` d. `2^(2n)`

Let R=(5sqrt(5)+11)^(2n+1)a n df=R-[R]w h e r e[] denotes the greatest integer function, prove that Rf=4^(2n+1)

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

Evaluate: int_1^(e^6)[(logx)/3]dx ,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 ...........

Evaluate lim_(n->oo) [sum_(r=1)^n(1/2) ^r] , where [.] denotes the greatest integer function.

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

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

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.

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

Let vecf(t)=[t] hat i+(t-[t]) hat j+[t+1] hat k , w h e r e[dot] denotes the greatest integer function. Then the vectors vecf(5/4)a n df(t),0lttlti are(a) parallel to each other(b) perpendicular(c) inclined at cos^(-1)2 (sqrt(7(1-t^2))) (d)inclined at cos^(-1)((8+t)/sqrt (1+t^2)) ;