Let `alpha` be a fixed constant number such that 0<`alpha`<`(pi)/(2)` ,The function F is defined by ,`F(theta)=int_(0)^( theta)x cos(x+alpha)dx` .If `theta` lies in the interval `[0,(pi)/(2)]` ,then the maximum value of `F(theta)` is

Let alpha be a real number such that 0<=alpha<=pi If f(x)=cos x+cos(x+alpha)+cos(x+2 alpha) takes some constant number c for any x in R , then the value of [c+alpha] is equal to (Note: [y] denotes greatest integer less than or equal to y.)

Let f (x) = log _e (sinx ), ( 0 lt x lt pi ) and g(x) = sin ^(-1) (e ^(-x)), (x ge 0) . If alpha is a positive real number such that a = ( fog)' ( alpha ) and b = (fog ) ( alpha ) , then (A) aalpha ^(2) - b alpha - a = 0 (B) a alpha ^(2) - b alpha - a = 1 (C) a alpha ^(2) + b alpha - a = - 2 alpha ^(2) (D) a alpha ^(2) + b alpha + a = 0

Let alpha,beta be fixed complex numbers and z is a variable complex number such that |z-alpha|^(2)+|z-beta|^(2)=k. Find out the limits for ' k 'such that the locus of z is a circle.Find also the centre and radius of the circle

Let alpha and beta be two distinct complex numbers, such that abs(alpha)=abs(beta) . If real part of alpha is positive and imaginary part of beta is negative, then the complex number (alpha+beta)//(alpha-beta) may be

If f(x) is a polynomial of degree n(gt2) and f(x)=f(alpha-x) , (where alpha is a fixed real number ), then the degree of f'(x) is

If alpha is a complex constant such that alpha^(2)+z+bar(alpha)=0 has a real root then

Let xgt0 be a fixed reacl number. Then the integral underset(0)overset(oo)(f)e^(-1)|x-t|dt is equal to -

The time dependence of a physical quantity P is given by P= P_0 exp (-alpha t^(2)) , where alpha is a constant and t is time. The constant alpha