Let `K` be a positive real number and `A=[2k-1 2sqrt(k)2sqrt(k)2sqrt(k)1-2k-2sqrt(k)2k-1]a n dB=[0 2k-1sqrt(k)1-2k0 2-sqrt(k)-2sqrt(k)0]` . If det `(a d jA)+det(a d jB)=10^6,t h e n[k]` is equal to. [Note: `a d jM` denotes the adjoint of a square matix `M` and `[k]` denotes the largest integer less than or equal to `K` ].

Let K be a positive real number and A=[(2k-1,2sqrt(k),2sqrt(k)),(2sqrt(k),1,-2k),(-2sqrt(k),2k,-1)] and B=[(0,2k-1,sqrt(k)),(1-2k,0,2),(-sqrt(k),-2sqrt(k),0)] . If det (adj A) + det (adj B) =10^(6) , then [k] is equal to ______ . [Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k .]

Let K be a positive real number and A=[(2k-1,2sqrt(k),2sqrt(k)),(2sqrt(k),1,-2k),(-2sqrt(k),2k,-1)] and B=[(0,2k-1,sqrt(k)),(1-2k,0,2),(-sqrt(k),-2sqrt(k),0)] . If det (adj A) + det (adj B) =10^(6) , then [k] is equal to ______ . [Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k .]

Let K be a positive real number and A=[(2k-1,2sqrt(k),2sqrt(k)),(2sqrt(k),1,-2k),(-2sqrt(k),2k,-1)] and B=[(0,2k-1,sqrt(k)),(1-2k,0,2sqrt(k)),(-sqrt(k),-2sqrt(k),0)] . If det (adj A) + det (adj B) =10^(6) , then [k] is equal to ______ . [Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k .]

Let K be a positive real number and A=[(2k-1, 2sqrtk, 2sqrtk),(2sqrtk, 1,-2k),(-2sqrtk,2k,-1)] and [(0,2k-1, sqrtk),(1-2k, 0,2),(-sqrtk, -2sqrtk,0)] If det (adjA)+det(adjB)= 10^6 , then [k] is equal to

Let k be a positive real number and let |A|=(2 k+1)^3 and B=[[0,2k,-sqrt k ],[-2k,0,2sqrt k],[sqrt k, -2sqrt k,0]] . If |adj A|+|adj B|=10^6 then [k]= ([k]= the greatest integer less than or equal to k)

sum_(k=1)^(oo)(1)/(k sqrt(k+2)+(k+2)sqrt(k))=(2+sqrt(2))/(a) then

If log (sqrt(2)+sqrt(3))=cosh^(-1)k then k=

If sum_(k = 1)^(oo) (1)/((k + 2)sqrt(k) + ksqrt(k + 2)) = (sqrt(a) + sqrt(b))/(sqrt(c)) , where a, b, c in N and a,b,c in [1, 15] , then a + b + c is equal to