The range of the functions. Observe the following lists.
                    List - I                                 List – II

             A)   x - [x]                                1) [0, 5]

             B) $\sqrt{25 - x^{2}}$                           2) $[\frac{1}{3}, 1]$

             C) $\frac{1}{2 - \cos 3 x}$                          3) (0, 5)

             4) sin2x + cos4x                    4) [0, 1)

                                                                 5) $[\frac{3}{4}, 1]$

                The correct match for list I from list II

Very difficult

Solution

(A) range of $x - [x] = [0, 1]$
            (B) Range of $\sqrt{25 - x^{2}} = [0, 5]$

            (C ) $\frac{1}{3} \leq \frac{1}{2 - \cos 3 x} \leq 1 = [\frac{1}{3}, 1]$

            (D) $\sin^{2} x + \cos^{2} x (1 - \sin^{2} x)$

                        $= 1 - \frac{1}{4} \sin^{2} 2 x$ Range $= [\frac{3}{4}, 1]$

                4, 1, 2, 5

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