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The period of sin^(2) theta , is...

The period of `sin^(2) theta` , is

A

`pi^(2)`

B

`pi`

C

`2pi`

D

`pi//2`

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The correct Answer is:
B
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To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))/(cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))xx (cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is

OBJECTIVE RD SHARMA ENGLISH-REAL FUNCTIONS -Chapter Test
  1. Let f(x)=x+1 and phi(x)=x-2. Then the value of x satisfying |f(x)+phi(...

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  2. The domain of definition of the function f(x)=tan((pi)/([x+2])), is wh...

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  3. The range of the function f(x)=sin [log (sqrt(4-x^(2))/(1-x)) is :

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  4. The range of the function y=(x+2)/(x^2-8x-4)

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  5. The range of the function f(x)=1+sinx+sin^(3)x+sin^(5)x+…… when x in ...

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  6. The period of the function f(x)=|sin 3x|+| cos 3x| , is

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  7. The function f(x)={{:( 1, x in Q),(0, x notin Q):}, is

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  8. Which of the following functions has period pi ?

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  9. The function f(x)=x[x] , is

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  10. If f(x) and g(x) are periodic functions with the same fundamental per...

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  11. The range of the function f(x)=cosec^(-1)[sinx] " in " [0,2pi], where ...

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  12. If f(sinx)-f(-sinx)=x^(2)-1 is defined for all x in R , then the val...

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  13. Let f:[pi,3pi//2] to R be a function given by f(x)=[sinx]+[1+sinx]+[2...

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  14. Let the function f(x)=3x^(2)-4x+8log(1+|x|) be defined on the interval...

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  15. If f:[-4,0]->R is defined by f(x) = e^x + sin x, its even extension to...

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  16. Which one of the following is not periodic ?

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  17. The domain of the function f(x)=(sin^(-1)(3-x))/(log(e)(|-x|-2)), is

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  18. The domain of f(x)=log5|log(e)x| , is

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  19. The period of sin^(2) theta , is

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  20. f(x)=""^(16-x)C(2x-1)+""^(20-3x)P(4x-5)

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