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

If 4+ p=111 -1 ( 2p-11+22p-1 )= , then is equal to .

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

If k = ( 4 + 12 -1 ( 23 ) ) + ( 12 -1 ( 23 ) ), then the number of solutions of the equation -1(kx - 1) = -1x - -1x is .

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

Let the maximum value of ( -1x )2 + ( -1x )2 for x [- 32, 1 2 ] be mn 2, where (m, n) = 1. Then m + n is equal to .

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 )

If y = ( 3 + -1 x2 ), then (x - y)2 + 3y2 is equal to .

Let S = \x : -1x = + -1x + -1(2x + 1) \. Then x S(2x - 1)2 is equal to .

( -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, )