Inverse Trigonometric Functions

Let =3 -1 ( 611 ) and =3 -1 ( 49 ), where inverse trigonometric functions take only the principal values. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are true
2. Both Statement I and Statement II are false
3. Statement I is true but Statement II is false
4. Statement I is false but Statement II is true

Let 0< <1, = 13 and -1(1- )+ -1(1- )= 4. Then 6( + ) is equal to :

1. 6
2. 7
3. 8
4. 9

If ( -1 (x 2 ) )= ( -1 1-x2 ), x (0, 1), then the value of x is :

1. 12
2. 13
3. 23
4. 58

Let [ ] denote the greatest integer function. If the domain of the function f(x)= -1 ( 4x+2[x]3 ) is [ , ], then 12( + ) is equal to :

1. 6
2. 8
3. 9
4. 4

Considering the principal values of inverse trigonometric functions, the value of the expression (2 -1 ( 2 13 ) - 2 -1 ( 3 10 ) ) is equal to :

1. 3356
2. - 3356
3. - 1663
4. 1663

If the domain of the function f(x) = -1 ( 1x2 - 2x - 2 ) is (- , ] [ , ] [ , ), then + + + is equal to

1. 4
2. 2
3. 5
4. 3

The number of solutions of -1 4x + -1 6x = 6, where - 12 6 < x < 12 6, is equal to :

1. 2
2. 0
3. 3
4. 1

If the domain of the function f(x) = -1 ( 2x - 511 - 3x ) + -1(2x2 - 3x + 1) is the interval [ , ], then + 2 is equal to :

1. 5
2. 2
3. 3
4. 1

The value of -1 ( 1+ 2(2)-1 (2) ) - -1 ( 1+ 2 ( 12 )+1 ( 12 ) ) is equal to

1. - 32
2. + 52
3. - 54
4. + 32

Considering the principal values of the inverse trigonometric functions, -1 ( 32x + 12 1 - x2 ), - 12 < x < 1 2, is equal to

1. -5 6 - -1x
2. 5 6 - -1x
3. 6 + -1x
4. 4 + -1x

The sum of the infinite series -1 ( 74 )+ -1 ( 194 )+ -1 ( 394 )+ -1 ( 674 )+ is :

1. 2+ -1 ( 12 )
2. 2- -1 ( 12 )
3. 2- -1 ( 12 )
4. 2+ -1 ( 12 )

( -1 35 + -1 513 + -1 3365 ) is equal to:

1. 3365
2. 1
3. 3265
4. 0

Let [x] denote the greatest integer less than or equal to x. Then the domain of f(x) = -1(2[x] + 1) is:

1. (- , )
2. (- , ) - \0\
3. (- , -1] [0, )
4. (- , -1] [1, )

If > > >0, then the expression -1 \ + (1+ 2)( - ) \+ -1 \ + (1+ 2)( - ) \ + -1 \ + (1+ 2)( - ) \ is equal to :

1. 3
2. 2-( + + )
4. 0

If 2 x 3 4, then -1 ( 1213 x+ 513 x ) is equal to

1. x+ -1 512
2. x- -1 43
3. x+ -1 45
4. x- -1 512

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16 ( ( -1x )2+ ( cosec-1x )2 ) is :

1. 24 2
2. 18 2
3. 22 2
4. 31 2

If the domain of the function f(x) = -1 ( x - 12x + 3 ) is R - ( , ), then 12 is equal to :

1. 40
2. 36
3. 24
4. 32

Given that the inverse trigonometric function assumes principal values only. Let x, y be any two real numbers in [-1, 1] such that -1x - -1y = , - 2 . Then, the minimum value of x2 + y2 + 2xy is

1. 0
2. -1
3. 12
4. -12

For , , 0, if -1 + -1 + -1 = and ( + + )( - + )=3 , then equals

1. 3
2. 32
3. 1 2
4. 3-12 2

If a= -1( (5)) and b= -1( (5)), then a2+b2 is equal to

1. 25
2. 4 2+25
3. 8 2-40 +50
4. 4 2-20 +50

Let x= mn (m, n are co-prime natural numbers) be a solution of the equation (2 -1x )= 19 and let , ( > ) be the roots of the equation mx2-nx-m+n=0. Then the point ( , ) lies on the line

1. 3x-2y=-2
2. 3x+2y=2
3. 5x+8y=9
4. 5x-8y=-9

Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying -1(x)+ -1(2x)= 4 is :

1. more than 2
2. 2
3. 0
4. 1