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 (log)_(10)(x^3+y^3)-(log)_(10)(x^2+y^2-x y)lt=2,w h e r ex ,y are positive real number, then find the maximum value of x ydot

If f((x+2y)/3)=(f(x)+2f(y))/3AAx ,y in Ra n df^(prime)(0)=1,f(0)=2, then find f(x)dot

Let R="{"x , y")": x , y in R ,x^2+y^2lt=25} and R^(prime)={(x , y): x , y in R , ygeq4/9x^2}dot Then find the domain and range of RnnR^(prime)dot

The value of |[yz, z x,x y],[ p,2q,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

Let g: RvecR be a differentiable function satisfying g(x)=g(y)g(x-y)AAx , y in R and g^(prime)(0)=aa n dg^(prime)(3)=bdot Then find the value of g^(prime)(-3)dot

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 y=1/(t^2+t-2),w h e r et=1/(x-1), then find the number of points where f(x) is discontinuous.

If from a point P , tangents P Qa n dP R are drawn to the ellipse (x^2)/2+y^2=1 so that the equation of Q R is x+3y=1, then find the coordinates of Pdot

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

The line A x+B y+C=0 cuts the circle x^2+y^2+a x+b y+c=0 at Pa n dQ . The line A^(prime)x+B^(prime)x+C^(prime)=0 cuts the circle x^2+y^2+a^(prime)x+b^(prime)y+c^(prime)=0 at Ra n dSdot If P ,Q ,R , and S are concyclic, then show that |a-a ' b-b ' c-c ' A B C A ' B ' C '|=0