Let = 3 ^ -1 ( 6 11 ) and = 3 ^ -1 ( 4 9 ), where inverse trigonometric functions take only the principal values. Given below are two statements: Statement I: ( + ) > 0. Statement II: ( ) In the light of the above statements, choose the correct answer from the options given below:
A. Both Statement I and Statement II are true
We are given = 3 ^ -1 ( 6 11 ) and = 3 ^ -1 ( 4 9 ). First, we find the range of . Since 1 2 6 Multiplying by 3, we get 2 Since lies in the second quadrant, ( ) Next, we find the range of . Since 0 3 Multiplying by 3, we get Thus, lies in the third quadrant. Now, we determine the sign of ( + ). Adding the inequalities for and : 2 + 3 2 The angle ( + ) lies in the interval ( 3 2 , 9 4 ), which covers the fourth quadrant and a part of the first quadrant. In both of these regions, the cosine function is strictly positive. Therefore, ( + ) > 0. Thus, Statement I is true. Both Statement I and Statement II are true. Answer: Both Statement I and Statement II are true
Related: Mathematics — Inverse Trigonometric Functions · All PYQ Banks