Let `n and k` be positive such that `n leq (k(k+1))/2`.The number of solutions `(x_1, x_2,.....x_k), x_1 leq 1, x_2 leq 2, ........,x_k leq k`, all integers, satisfying `x_1 +x_2+.....+x_k = n`, is .......

Let n and k be positive integers such that n gt (k(k+1))/2 . The number of solutions (x_(1),x_(2), . ..x_(k)),x_(1) ge 1, x_(2) ge 2,, . . .x_(k) ge k for all integers satisfying x_(1)+x_(2)+ . . .+x_(k)=n is:

Let n and k be positive integers such that n ge K(K + 1)/2 . Find the number of solutions ( x_(1) , x_(2) , x_(3),………., x_(k) ) x_(1) ge 1, x_(2) ge 2, ……….. X_(k) ge k , all integers satisfying the condition x_(1) + x_(2) + x_(3) + ………. X_(k) = n .

Let n and k be positive integers such that n ge K(K + 1)/2 . Find the number of solutions ( x_(1) , x_(2) , x_(3),………., x_(k) ) x_(1) ge 1, x_(2) ge 2, ……….. X_(k) ge k , all integers satisfying the condition x_(1) + x_(2) + x_(3) + ………. X_(k) = n .

The number of real solutions (x, y) , where |y| = sin x, y = cos^-1 (cosx),-2pi leq x leq 2pi is

The number of integral values of x satisfying |x^2-1|leq3 is

If the arithmetic mean of the observations x_1,x_2,x_3,.............x_n is 1 , then the arithmetic mean of x_1/k,x_2/k,x_3/k,.............x_n/k , (k >0) is

The objective function z = x_1 + x_2 , subject to x_1 + x_2 leq 10,-2x_1 + 3x_2 leq 15,x_1 leq 6,x_1 ,x_2 geq 0 has maximum value of the feasible region.

Let f(x) be a function defined by f (x)=int_1^x x(x^2-3x+2)dx, 1 leq x leq 4 . Then, the range of f(x) is

If M is the mean of n observations x_(1)-k, x_(2)-k, x_(3)-k,…., x_(n)-k , where k is any real number, then what is the mean of x_(1), x_(2), x_(3) …., x_(n) ?