If `0ltx<1,` then prove that= `(1-2x)/(1-x+"x^2)+(2x-4x^3)/(1-x^2+x^4)+(4x^3-8x^7)/(1-x^4+x^8)+....oo=(1+2x)/(1+x+x^2)`

If f(x) = (A)/(pi)(1)/(16+x^(2)),0 ltx lt oo is a probability density function of a continuous random variable X then the value of A is ……… .

For 0ltx ,yltpi, the number of ordered pairs (x, y) satisfying system equations cot^2(x-y)-(1+sqrt(3))cot(x-y)+sqrt(3)="0 and cosy="(sqrt(3))/2 is

If f (x) = {{:(x - 5 ," if ",x le1),(4x^(2)-9," if ",1 ltx lt 2),(3x + 1," if ",x ge 2):} , then the right hand derivative of f(x)at x = 2 is . . . . .

Find all points of discontinuity of f, where f is defined by f(x)={{:(|x|+3," if "x le -3),(-2x," if "-3 ltx lt 3),(6x+2," if "x ge 3):} .

If the line a x+b y+c=0 is a normal to the curve x y=1, then a >0,b >0 a >0,b >0 (d) a<0,b<0 none of these

Let f(x,y) = 0 if xy ne 0 and f (x,y) = 1 if xy = 0 . (i) Calculate : (delf)/(delx)(0,0),(delf)/(dely)(0,0) (ii) Show that f is not continuous at (0,0)

Find the sum of 0.40 + 0.43 + 0.46 + .... + 1.

A Charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii r_1 and r_2 (figure 30-E4). Find the electric field at a point P at a distance x away from the centre for r_1ltxltr_2 . Draw a rough graph showing the electric field as a function of x for 0ltx lt2r_2 .

The number of integral value of x satistying sqrt( x^2+10 x-16)ltx-2 is

Find the area of the region in which the points satisfy the inequaties 4ltx^2+y^2lt16 and 3x^2-y^2ge0 .