Question 1

Consider the following in respect of inverse circular functions : I. ^ -1 (-x)=- ^ -1 x II. ^ -1 (-x)= ^ -1 x III. ^ -1 (-x)= - ^ -1 x IV. ^ -1 (-x)= - ^ -1 x How many of the above are correct ?

Accepted Answer

Statement I: ^ -1 (-x) = - ^ -1 x is a standard property of inverse trigonometric functions and is correct. Statement II: The correct property is ^ -1 (-x) = - ^ -1 x. Hence, this statement is incorrect. Statement III: The correct property is ^ -1 (-x) = - ^ -1 x. Hence, this statement is incorrect. Statement IV: ^ -1 (-x) = - ^ -1 x is a standard property and is correct. Exactly two of the given statements are correct.

Question 2

What is [2 ^ -1 1 2 - 4 ] equal to ?

Accepted Answer

Let = ^ -1 1 2 , which implies = 1 2 . We need to evaluate (2 - 4 ). First, calculate (2 ) using the double angle formula: (2 ) = 2 1 - ^2 = 2 ( 1 2 ) 1 - ( 1 2 )^2 = 1 1 - 1 4 = 4 3 Now, apply the tangent subtraction formula: (2 - 4 ) = (2 ) - ( 4 ) 1 + (2 ) ( 4 ) Substitute (2 ) = 4 3 and ( 4 ) = 1: (2 - 4 ) = 4 3 - 1 1 + 4 3 (1) = 1 3 7 3 = 1 7 Answer: 1 7

Question 3

If ^ -1 k + ^ -1 1 2 = 4 , then what is the value of k?

Accepted Answer

Given ^ -1 k + ^ -1 1 2 = 4 Using the formula ^ -1 x + ^ -1 y = ^ -1 ( x+y 1-xy ), we get: ^ -1 ( k + 1 2 1 - k 2 ) = 4 2k + 1 2 - k = ( 4 ) 2k + 1 2 - k = 1 2k + 1 = 2 - k 3k = 1 k = 1 3 Answer: 1 3

Question 4

What is the value of [ 1 2 ^ -1 ( 2 3 ) ]?

Accepted Answer

Let = ^ -1 ( 2 3 ) = 2 3 = 6 The given expression becomes ( 1 2 ) = ( 12 ) ( 12 ) = (15^ ) = (45^ - 30^ ) Using the formula (A - B) = A - B 1 + A B (15^ ) = 45^ - 30^ 1 + 45^ 30^ (15^ ) = 1 - 1 3 1 + 1 3 = 3 - 1 3 + 1 Rationalizing the denominator: (15^ ) = ( 3 - 1)^2 ( 3 + 1)( 3 - 1) = 3 + 1 - 2 3 3 - 1 (15^ ) = 4 - 2 3 2 = 2 - 3 Answer: 2 - 3

Question 5

Consider the following for the two (02) items that follow: Let y = ^ -1 (x - 4x^3 27 ). What is y equal to?

Accepted Answer

Given y = ^ -1 (x - 4x^3 27 ) Substitute x = 3 , which gives = x 3 or = ^ -1 ( x 3 ) The expression becomes: y = ^ -1 (3 - 4(3 )^3 27 ) y = ^ -1 (3 - 4(27 ^3 ) 27 ) y = ^ -1 (3 - 4 ^3 ) Using the trigonometric identity (3 ) = 3 - 4 ^3 : y = ^ -1 ( (3 )) y = 3 Substituting back the value of : y = 3 ^ -1 ( x 3 ) Answer: 3 ^ -1 ( x 3 )

Question 6

If k is a root of x^2 - 4x + 1 = 0, then what is ^ -1 k + ^ -1 1 k equal to?

Accepted Answer

The given equation is x^2 - 4x + 1 = 0. The roots of the equation are given by x = 4 16 - 4 2 = 2 3 . Since 3 1.732, both roots 2 + 3 and 2 - 3 are positive. Therefore, k > 0. Using the standard inverse trigonometric identity for x > 0: ^ -1 x + ^ -1 ( 1 x ) = 2 Substituting x = k, we get: ^ -1 k + ^ -1 ( 1 k ) = 2 Answer: 2

Question 7

What is ^ -1 9 + cosec ^ -1 ( 41 4 ) equal to?

Accepted Answer

Let A = ^ -1 9 A = 9 A = 1 9 Let B = cosec ^ -1 ( 41 4 ) cosec B = 41 4 B = 4 41 From this, B = 4 ( 41 )^2 - 4^2 = 4 41 - 16 = 4 5 Using the formula (A + B) = A + B 1 - A B : (A + B) = 1 9 + 4 5 1 - ( 1 9 ) ( 4 5 ) (A + B) = 5 + 36 45 45 - 4 45 (A + B) = 41 41 = 1 Since A, B (0, 2 ), we have A + B = 4 Answer: 4

Inverse Trigonometric Functions — Previous Year Questions & Solutions