Inverse Trigonometric Functions

For n N, if -13 + -14 + -15 + -1n = 4, then n is equal to .

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