Question 1

What is the smallest positive x satisfying _ x x+ _ x x=2 ?

Accepted Answer

Let t = _ x x. The given equation becomes t + 1 t = 2. t^2 - 2t + 1 = 0 (t - 1)^2 = 0 t = 1 _ x x = 1 x = x x = 1 The smallest positive value of x is 4 . For x = 4 , x = 1 2 and x = 1 2 , which satisfy the conditions for the base and argument of logarithms (positive and not equal to 1). Answer: 4

Question 2

If 0 x 2 , then what is the number of values of x satisfying the equation x + x = 2 x?

Accepted Answer

The given equation is x + x = 2 x. Rewriting the equation in terms of x and x: x x + 1 x = 2 x For the equation to be defined, x 0, which means x 2 . Thus, x [0, 2 ). Multiplying both sides by x: x + 1 = 2 ^2 x Substituting ^2 x = 1 - ^2 x: x + 1 = 2(1 - ^2 x) 2 ^2 x + x - 1 = 0 Factoring the quadratic equation: (2 x - 1)( x + 1) = 0 This gives x = 1 2 or x = -1. Since x [0, 2 ), x 0. Therefore, x = -1 is rejected. For x = 1 2 , the only solution in the given interval is x = 6 . Thus, there is e

Question 3

For the following two (02) items: Let (2x + 3y) = 1 2 and (3x + 2y) = 3 2 , where - How many values does (y - x) have?

Accepted Answer

Let A = 2x + 3y and B = 3x + 2y. We are given A = 1 2 with - The possible values for A are 3 and - 3 . We are also given B = 3 2 with - The possible values for B are 6 and - 6 . We need to find the number of values of (y - x). Observe that A - B = (2x + 3y) - (3x + 2y) = y - x. The possible values for A - B are: 3 - 6 = 6 3 - (- 6 ) = 2 - 3 - 6 = - 2 - 3 - (- 6 ) = - 6 Since all four values are distinct, (y - x) has four values.

Question 4

For the following two (02) items: Let (2x + 3y) = 1 2 and (3x + 2y) = 3 2 , where - How many values does (x + y) have?

Accepted Answer

Let 2x + 3y = u and 3x + 2y = v. Given u = 1 2 and - Given v = 3 2 and - Adding the two equations, we get u + v = 5(x + y) x + y = u + v 5 . The possible values of u + v are: 3 + 6 = 2 3 - 6 = 6 - 3 + 6 = - 6 - 3 - 6 = - 2 Since all four values of u + v are distinct, there are exactly four distinct values for x + y.

Trigonometric Equations — Previous Year Questions & Solutions