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)`

Find the value of 3/2x^2+5/3x^3+7/4x^4 +. . . +oo if 0 ltx lt 1

If cos ^-1 x+cos ^(-1) y+cos ^(-1) z=pi and 0ltx, y ,zlt1 ,show that x^2+y^2+z^2+2 x y z=1

f(x) = {{:(xsinx " if " 0 ltx le(pi)/(2)),((pi)/(2)sin(pi + x )" if " (pi)/(2) lt x lt pi):} then f(x) is discontinuous at

If : f(x) {:(=1", ... "0 ltx le (3pi)/4), (= 2 .sin ((2x)/9) ", ..." (3pi)/4lt x lt pi):} then :

Let f(x) = {{:({1+|sin x|}^(a//|sin x|)", " pi/6 lt x lt 0),(" b, " x = 0 ),(e^(tan 2x//tan 3x) ", "0ltx ltpi/6):} Determine a and b such that f(x) is continous at x = 0.

The interval [0,4] isdivided into n equal sub-intervals by the points x_(0),x_(1),x_(2),…,x_(n-1),x_(n)"where" 0=x_(0)ltx_(1)ltx_(2)ltx_(3)lt…ltx_(n)=4 If deltax=x_(i)-x_(i-1)"for"i=1,2,3,...,n, "then"lim_(deltaxrarr0)sum_(i-1)^(n)x_(i)deltax is equal to

The interval [0,4] isdivided into n equal sub-intervals by the points x_(0),x_(1),x_(2),…,x_(n-1),x_(n)"where" 0=x_(0)ltx_(1)ltx_(2)ltx_(3)lt…ltx_(n)=4 If deltax=x_(i)-x_(i-1)"for"i=1,2,3,...,n, "then"lim_(deltaxrarr0)sum_(i-1)^(n)x_(i)deltax is equal to

Let x_(1),x_(2) be the roots of x^(2)-3x+a=0andx_(3),x_(4) be the roots of x^(2)-12x+b=0 . If x_(1)ltx_(2)ltx_(3)ltx_(4)andx_(1),x_(2),x_(3),x_(4) are in GP then ab equals

Which one of the following is correct ? If 4 ltx^(2) le9 , then