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),0lttlt1` 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)/(9sqrt (1+t^2)))`;

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) and vecf(t) , 0 < t < 1 are (a) parallel to each other (b) perpendicular to each other (c) inclined at cos^(-1)(2/(sqrt(7(1-t^2)))) (d) inclined at cos^(-1)((8+t)/(9*sqrt(1+t^2)))

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

If y = cos^(-1) ((1)/( sqrt(1+t^(2)))), x = sin^(-1) (sqrt((t^(2))/(1 + t^(2)))), "find " (dy)/(dx)

The value of the integral I=int_(1)^(2)t^([{t}]+t)(1+ln t)dt is equal to ( [.] and {.} denotes the greatest integer and fractional part function respectively)

If Vectors vec(A)= cos omega t(i)+ sin omega t(j) and vec(B)=cosomegat/(2)hat(i)+sinomegat/(2)hat(j) are functions of time. Then the value of t at which they are orthogonal to each other is

Find the Cartesian form the equation of the plane vec r=(s-2t) hat i+(3-t) hat j+(2s+t) hat kdot

If A=[1ta n x-t a n x1],t h e n detA^T A^(-1) is

A particle is moving with a position vector, vec(r)=[a_(0) sin (2pi t) hat(i)+a_(0) cos (2pi t) hat(j)] . Then -

If vec a=4 hat i+3 hat j+ hat k\ a n d\ vec b= hat i-2 hat k\ t h e n\ fin d \ |2 vec bxx vec a|dot

If S_n=sum_(r=1)^n r !(n >6),"t h e nf i n dt h ev a l u eo f" [cot^(-1)(cot(s_n-7[(S_n)/7]))],"w h e r e"[dot]"d e n o t e sg r e a t e s t" integer function