Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The number of elements in the set \ x [- , ] : ^6 x + ^4 x = 1\ (1) is 1 (Q) The number of elements in the set \ x [- 2 , 2 ] : ^2 x + ^6 x = 1 \ (2) is 2 (R) The number of elements in the set \ x [- , ] : ^2 ( x 2 ) - ^2 x = 1 2 \ (3) is 3 (S) The number of elements in the set \ x [-2 , 2 ] : 6 ^2 ( x 2 ) - 3x = 3 \ (4) is 4 (5) is 5
B. (P) (5), (Q) (3), (R) (2), (S) (4)
For (P): ^6 x + ^4 x = 1 ^6 x + (1 - ^2 x)^2 = 1 ^6 x + 1 - 2 ^2 x + ^4 x = 1 ^2 x ( ^4 x + ^2 x - 2) = 0 ^2 x ( ^2 x - 1)( ^2 x + 2) = 0 This gives ^2 x = 0 or ^2 x = 1. In the interval [- , ], the solutions are x \ - , - 2 , 0, 2 , \ . Number of solutions = 5. For (Q): ^2 x + ^6 x = 1 1 - ^2 x + ^6 x = 1 ^2 x ( ^4 x - 1) = 0 This gives ^2 x = 0 or ^2 x = 1. In the interval [- 2 , 2 ], the solutions are x \ - 2 , 0, 2 \ . Number of solutions = 3. For (R): ^2 ( x 2 ) - ^2 x = 1 2 1 + x 2 - (1 - ^2 x) = 1 2 1 + x - 2 + 2 ^2 x = 1 2 ^2 x + x - 2 = 0 Solving the quadratic equation for x, we get: x = -1 17 4 Since x [-1, 1], we reject the negative root. Thus, x = 17 - 1 4 . Since 0 Number of solutions = 2. For (S): 6 ^2 ( x 2 ) - 3x = 3 3(1 - x) - (4 ^3 x - 3 x) = 3 3 - 3 x - 4 ^3 x + 3 x = 3 -4 ^3 x = 0 x = 0 In the interval [-2 , 2 ], the solutions are x \ - 3 2 , - 2 , 2 , 3 2 \ . Number of solutions = 4. Therefore, the correct matching is (P) (5), (Q) (3), (R) (2), (S) (4). Answer: (P) (5), (Q) (3), (R) (2), (S) (4)
Related: Mathematics — Trigonometric Equations · All PYQ Banks