If `pi` < `theta` <`2pi` and z=1+cos`theta`+`i`sin`theta` , then write the value of `|z|`

If x in [0,2pi]" then "y_(1)=(sin x)/(|sin x|), y_(2)=(|cos x|)/(cos x) are identical functions for x in : I. (0,pi/2)" "II. (pi/2, pi)," "III. (pi,(3pi)/(2))," "IV. ((3pi)/(2),2pi)

Let f(x) = {{:(-2 sin x,"for",-pi le x le - (pi)/(2)),(a sin x + b,"for",-(pi)/(2) lt x lt (pi)/(2)),(cos x,"for",(pi)/(2) le x le pi):} . If f is continuous on [-pi, pi) , then find the values of a and b .

The possible values of theta in (0,pi) such that sin (theta) + sin (4theta) + sin(7theta) = 0 are (1) (2pi)/9 , i/4 , (4pi)/9, pi/2, (3pi)/4 , (8pi)/9 (2) pi/4, (5pi)/12, pi/2 , (2pi)/3, (3pi)/4, (8pi)/9 (3) (2pi)/9, pi/4 , pi/2 , (2pi)/3 , (3pi)/4 , (35pi)/36 (4) (2pi)/9, pi/4, pi/2 , (2pi)/3 , (3pi)/4 , (8pi)/9

The possible values of theta in (0,pi) such that sin (theta) + sin (4theta) + sin(7theta) = 0 are (1) (2pi)/9 , i/4 , (4pi)/9, pi/2, (3pi)/4 , (8pi)/9 (2) pi/4, (5pi)/12, pi/2 , (2pi)/3, (3pi)/4, (8pi)/9 (3) (2pi)/9, pi/4 , pi/2 , (2pi)/3 , (3pi)/4 , (35pi)/36 (4) (2pi)/9, pi/4, pi/2 , (2pi)/3 , (3pi)/4 , (8pi)/9

If t a nalpha=5/2 , tanbeta=-7/9 where alpha in [pi/2,pi] , then alpha+beta lies in the interval. [(3pi)/2,(7pi)/3] (b) [(13pi)/6,(5pi)/2] [(13pi)/6,(7pi)/3] (d) [pi/6,pi/3]

For x in (0,(5pi)/2) , define f(x)""=int_0^xsqrt(t)sint"dt" Then f has : local maximum at pi and 2pi . local minimum at pi and 2pi local minimum at pi and local maximum at 2pi . local maximum at pi and local minimum at 2pi .

1.The inverse of cosine function is defined in the intervals (A) [ -pi , 0 ] (B) [ -pi/2,0 ] (C) [ 0,pi/2 ] (D) [ pi/2 , pi ]

int_-pi^(3pi) cot^-1(cotx)dx= (A) pi^2 (B) 2pi^2 (C) 3pi^2 (D) none of these

If int_(pi//6)^(pi//3)sinxdx=kint_(pi//6)^(pi//3)cosxdx , then k is _______.

int_0^(pi//2)xsinx\ dx is equal to a. pi//2 b. pi//4 c. pi d. 1