Show that a six-dight number abcdef is divisible by 11, if and only if ab+cd+ef is divisible by 11. Hence or otherwise find one set of values of two-dight numbers x,y and z, so that the value of the determinant `|{:(x,23,42),(13,37,y),(19,z,34):}|` is divisible by 99 (without expanding the determinant ).

If [] denotes the greatest integer less than or equal to the real number under consideration and -1lt=xlt0,0lt=ylt1,1lt=zlt2, the value of the determinant |{:([x]+1,[y],[z]),([x],[y]+1,[z]),([x],[y],[z]+1):}| is

For positive numbers x ,\ y\ a n d\ z the numerical value of the determinant |1(log)_x y(log)_x z(log)_y x1(log)_y z(log)_z x(log)_z y1| is- a. 0 b. logx y z c. "log"(x+y+z) d. logx\ logy\ logz

In the given figure, AB||CD and CD||EF Also EAbotAB . If angleBEF=55^(@) , find the values of x, y and z.

Determine the maximum value of z = 11x + 7y subject to the constraints : 2x+y le 6, x le 2, x ge 0, y ge 0

If the determinant |[a+p, l+x, u+f], [b+q, x+y, v+g], [c+r, n+z, w+h]| splits into exactly K determinants of order 3, each element of which contains only one term, then the values of K is

In the given figure three lines bar(AB) , bar(CD) and bar(EF) intersecting at O. Find the values of x, y and z it is being given that x : y : z = 2 : 3 : 5

If the determinant |{:(x+a,p+u,l+f),(y+b,q+v,m+g),(z+c,r+w,n+h):}| splits into exactly K determinants of order 3, each elements of which contains only one term, then the value of K is 8

A radioactive nucleus X decay to a nucleus Y with a decay with a decay Concept lambda _(x) = 0.1s^(-1) , gamma further decay to a stable nucleus Z with a decay constant lambda_(y) = 1//30 s^(-1) initialy, there are only X nuclei and their number is N_(0) = 10^(20) . Set up the rate equations for the population of X , Y and Z The population of Y nucleus as a function of time is given by N_(y) (1) = N_(0) lambda_(x)l(lambda_(x) - lambda_(y))( (exp(- lambda_(y)t)) Find the time at which N_(y) is maximum and determine the populations X and Z at that instant.