Let `H_n:x^2/(1+n)-y^2/(3+n)=1, n in NN`.Let `k` be the smallest even value of n such that the eccentricity of `H_k` is a rational number. If `l`is the length of the latus rectum of `H_k` then `21l` is equal to

Let A = ((1,1,3),(5,2,6),(-2,-1,-3)) and let n be the smallest value of n in N such that A^(n) = O_(3) then det (I_(3)+A+A^(2)+ cdots+ A^(n-1)) is equal to _____ .

Let l_(1) be the length of the latus rectum of the parabola 9x^(2)-6x+36y+19=0. Let l_(2), be the length of the latus rectum of the hyperbola 9x2-16y^(2)-18x-32y-151=0. Let n be the unit digit in the product of l_(1) and l_(2) The number of terms in the binomial expansion of (1+x)^(n) is

Lines x=n, y=n^(3) intersect at the point A_(n).L_(n) is a line through A_(n) parallel to x+y=0 and make an intercept of length l_(n) on the axis of x, then the value of sum_(n=1)^(5)l_(n) is equal to K^(4)-16 , where K =

Let A={n in N : n is a 3- digit number } B={ 9k +2 : k in N} and C= { 9k +l : K in N } for some l (0 lt l lt 9) if the sum of all the elements of the set A nn (B uu C) is 274 xx 400 , then l is equal to ______.

Let A = { n in N | n^(2) le n + 10,000 } B = { 3k + 1 | k in N } and C = { 2 k |k in K } , then the sum of all the elements of the set A cap (B - C) is equal to ______

Let A_(1) , A_(2) ,…..,A_(n) ( n lt 2) be the vertices of regular polygon of n sides with its centre at he origin. Let veca_(k) be the position vector of the point A_(k) ,k = 1,2,….,n if |sum_(k=1)^(n-1) (veca_(k) xx veca_(k) +1)|=|sum_(k=1)^(n-1) (vecak.vecak+1)| then the minimum value of n is

Let f(n)=(k=1)?^(n)k^(2)(nC_(k))^(2) then the value of f(5) equals