[" The largest real value for "x" such that "sum_(k=0)^(4)((5^(4-k))/(4-k!))((x^(k))/(k!))=(8)/(3)" is : "],[2sqrt(2)-5],[2sqrt(2)+5],[-2sqrt(5)-5],[-2sqrt(2)-5]

Number of integral values of k so that int_(1)^(k)(1+(3x-2)/(2sqrt(x)))dx<4sqrt(k)

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

If (6)/(2sqrt(3)-sqrt(5))=(12sqrt(3)+6sqrt(5))/(k), then 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 .]

Simplify each of the following : (i) 3/(5-sqrt(3))+2/(5+sqrt(3)) (ii) (4+sqrt(5))/(4-sqrt(5))+(4-sqrt(5))/(4+sqrt(5)) (iii) (sqrt(5)-2)/(sqrt(5)+2)-(sqrt(5)+2)/(sqrt(5)-2)

Simplify each of the following : (3)/(5-sqrt(3))+(2)/(5+sqrt(3)) (ii) (4+sqrt(5))/(4-sqrt(5))+(4-sqrt(5))/(4+sqrt(5))(sqrt(5)-2)/(sqrt(5)+2)-(sqrt(5)+2)/(sqrt(5)-2)

If one root x^(2)-x-k=0 is square of the other,then k=a.2+-sqrt(5)b.2+-sqrt(3)c3+-sqrt(2)d.5+-sqrt(2)

The sum sqrt((5)/(4)+sqrt((3)/(2)))+sqrt((5)/(4)-sqrt((3)/(2))) is equal to

Given x, y in R , x^(2) + y^(2) gt 0 . Then the range of (x^(2) + y^(2))/(x^(2) + xy + 4y^(2)) is (a) ((10 - 4 sqrt(5))/(3),(10 + 4 sqrt(5))/(3)) (b) ((10 - 4 sqrt(5))/(15),(10 + 4 sqrt(5))/(15)) (c) ((5- 4 sqrt(5))/(15),(5 + 4 sqrt(5))/(15)) (d) ((20- 4 sqrt(5))/(15),(20 + 4 sqrt(5))/(15))