If `x^2+y^2=R^2(w h e r eR >0)a n dk=(y prime prime)/((1+yprime^2)^3)` then find `k` in terms of `R` alone.

If f((x+2y)/3)=(f(x)+2f(y))/3\ AA\ x ,\ y in R and f^(prime)(0)=1 , f(0)=2 , then find f(x) .

The value of |[yz, z x,x y],[ p,2p,3r],[1, 1, 1]|,w h e r ex ,y ,z are respectively, pth, (2q) th, and (3r) th terms of an H.P. is a.-1 b. 0 c. 1 d. none of these

Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a)P=y+x (b)P=y-x (c)P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d)P-Q=x+y-y^(prime)-(y^(prime))^2

If the line x+2y=3 cuts a chord of length r unit with the circle x^2+y^2=r^2 then find r^2 .

If a x^2+2h x y+b y^2+2gx+2f y+c=(l x+m y+n)(l prime x+m^(prime)y+n^(prime)) and Delta_1=|(l,l prime,0),(m,m prime,0),(n,n prime,0)|and Delta_2=|(l prime,l,0),(m prime,m,0),(n prime,n,0)|, then the product Delta_1 Delta_2 is

If R={(x,y): x,y in W, x ^(2)+y^(2)=25} , then find the domain and range or R.

A sum of Rs. 1300 is divided among P, Q, R and S such that (P^(prime)s s h a r e)/(Q^(prime)s s h a r e)=(Q^(prime)s s h a r e)/(R^(prime)s s h a r e)=(R^(prime)s s h a r e)/(S^(prime)s s h a r e)=2/3dot How much is Ps share? R s .140 b. R s .160 c. R s .240 d. R s .320

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

If R_(2)={(x,y)|x and y are integers and x^(2)+y^(2)=64} is a relation.Then find R_(2).