Let `f(x)=|(a+x,b+x,c+x),(I+x,m+x,n+x),(p+x,q+x,r+x)|.` Show that `f prime prime(x)=0` and that `f(x)=f(0)+kx` where k denotes the sum of all the co-factors of the elements in `f(0).`

Let f(x)=|{:(p+x,q+x,r+x),(l+x,m+x,n+x),(a+x,b+x,c+x):}| then

If F(x) =[(cos x, -sin x,0),(sin x,cos x,0),(0,0,1)] show that F(x)F(y)=F(x+y)

Let Q(x)=f(x)+f(1-x) and f'(x)<0,0<=x<=

Let f(x+y)=f(x)+f(y) for all real x,y and f'(0) exists.Prove that f'(x)=f'(0) for all x in R and 2f(x)=xf(2)

If f:(0,pi)rarr R and is given by f(x)=sum_(k=1)^(n)[1+sin((kx)/(n))] where [] denotes the integral part of x, then the range of f(x) is

A function f(x) is such that f(x+1)=sqrt(f(x)+2) , where x in I^+ uu {0} and f(0)=0 then f(x) is a/an

Let f(x)=x^(2)+ax+b, where a,b in R. If f(x)=0 has all its roots imaginary then the roots of f(x)+f'(x)+f(x)=0 are

Let f(x+y)=f(x)f(y) for all x and y . Suppose f(5)=2 and f^(prime)(0)=3 , find f^(prime)(5) .

If f(x) = |[x+p,x+a,x+a],[x+b,x+q,x+a],[x+b,x+b,x+r]| and g(x)=(p-x)(q-x)(r-x) , then the value of f(0) is

if f(x)=(sin[x])/([x]+1),x>0 and f(x)=((sin[x])/(2)[x])/([x]),x<0 and f(x)=k,x=0 where [x] denotes the greatest integer less than or equal to x=0, then in order that the continuous at x=0, the value of k is equal to