Show that `2sinx+tanxgeq3x ,w h e r e0lt=x

The vertices of a triangle are (0,0), (x ,cosx), and (sin^3x ,0),w h e r e0ltxltpi/2 the maximum area for such a triangle in sq. units is (a) (3sqrt(3))/(32) (b) (sqrt(3)/32) (c) 4/32 (d) (6sqrt(3)) /(32)

Show that 2^(4n+4)-15n-16 ,w h e r en in N is divisible by 225.

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval

Prove that int_0^oo[cot^(-1)x]dx ,w h e r e[dot] denotes the greatest integer function.

If x in R ,a n da ,b ,c are in ascending or descending order of magnitude, show that (x-a)(x-c)//(x-b)(w h e r ex!=b) can assume any real value.

Solve for xa n dy :sqrt(3)sinx+cosx=8y-y^2-18 ,w h e r e0lt=xlt=4pi, y in R

Discuss monotonocity of f(x)=x/(sinx)a n d g(x)= x / (tanx), w h e r e

Fill in the blanks If 2+isqrt(3) is a root of the equation x^2+p x+q=0,w h e r epa n dq are real, then (p ,q)=(______,_____) .

If the roots of the equation x^3+P x^2+Q x-19=0 are each one more that the roots of the equation x^3-A x^2+B x-C=0,w h e r eA ,B ,C ,P ,a n dQ are constants, then find the value of A+B+Cdot

Evaluate: int_0^(2pi)[sinx]dx ,w h e r e[x] denotes the greatest integer function.