The range of the functions. Observe the following lists. List−I List−IIA) x−x 1) 0, 5B) 25−x2 2) 13, 1C) 12−cos3x 3) 0, 5D) sin2x+cos4x 4) 0, 1 5) 34, 1The correct match for list I from List II

A) x−x=x∴ 0≤x<1⇒ Range=0, 1B) fx=25−x2 Domain=−5, 5∴ −5≤x≤5⇒ 0≤x2≤25⇒ 0≥−x2≥−25⇒ 25≥25−x2≥0⇒ 5≥25−x2≥0∴ Range = 0, 5C) −1≤cos3x≤1⇒ 1≥−cos3x≥−1⇒ 3≥2−cos3x≥1⇒ 13≤12cos3x≤1∴ Range =13, 1D) fx=sin2x+cos4x⇒ cos4c−cos2x+1⇒ cos2x2−212cos2x+14−14+1⇒cos2x−122+340≤cos2x≤1⇒ −12≤cos2x−12≤12⇒ 0≤cos2x−122≤14⇒ 34≤cos2x−122≤1∴ Range=34, 1

The range of the functions. Observe the following lists. List−I List−IIA) x−x 1) 0, 5B) 25−x2 2) 13, 1C) 12−cos3x 3) 0, 5D) sin2x+cos4x 4) 0, 1 5) 34, 1The correct match for list I from List II

A) x−x=x∴ 0≤x<1⇒ Range=0, 1B) fx=25−x2 Domain=−5, 5∴ −5≤x≤5⇒ 0≤x2≤25⇒ 0≥−x2≥−25⇒ 25≥25−x2≥0⇒ 5≥25−x2≥0∴ Range = 0, 5C) −1≤cos3x≤1⇒ 1≥−cos3x≥−1⇒ 3≥2−cos3x≥1⇒ 13≤12cos3x≤1∴ Range =13, 1D) fx=sin2x+cos4x⇒ cos4c−cos2x+1⇒ cos2x2−212cos2x+14−14+1⇒cos2x−122+340≤cos2x≤1⇒ −12≤cos2x−12≤12⇒ 0≤cos2x−122≤14⇒ 34≤cos2x−122≤1∴ Range=34, 1

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$\begin{array}{l} L i s t - I L i s t - I I \\ A) x - [x] 1) [0, 5] \\ B) \sqrt{25 - x^{2}} 2) [\frac{1}{3}, 1] \\ C) \frac{1}{2 - \cos 3 x} 3) (0, 5) \\ D) \sin^{2} x + \cos^{4} x 4) [0, 1) \\ 5) [\frac{3}{4}, 1] \\ T h e c o r r e c t m a t c h f o r l i s t I f r o m L i s t I I \end{array}$

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$4 1 2 5$

$4 3 2 5$

$3 1 4 2$

$4 3 5 2$

answer is A.

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Detailed Solution

$\begin{array}{l} A) x - [x] = \{x\} ∴ 0 \leq \{x\} < 1 \Rightarrow R a n g e = [0, 1) \\ B) f (x) = \sqrt{25 - x^{2}} D o m a i n = [- 5, 5] \\ ∴ - 5 \leq x \leq 5 \\ \Rightarrow 0 \leq x^{2} \leq 25 \\ \Rightarrow 0 \geq - x^{2} \geq - 25 \\ \Rightarrow 25 \geq 25 - x^{2} \geq 0 \\ \Rightarrow 5 \geq \sqrt{25 - x^{2}} \geq 0 \\ ∴ R a n g e = [0, 5] \end{array}$

$\begin{array}{l} C) - 1 \leq \cos 3 x \leq 1 \\ \Rightarrow 1 \geq - \cos 3 x \geq - 1 \\ \Rightarrow 3 \geq 2 - \cos 3 x \geq 1 \\ \Rightarrow \frac{1}{3} \leq \frac{1}{2 \cos 3 x} \leq 1 \\ ∴ R a n g e = [\frac{1}{3}, 1] \\ D) f (x) = \sin^{2} x + \cos^{4} x \Rightarrow \cos^{4} c - \cos^{2} x + 1 \\ \Rightarrow {(\cos^{2} x)}^{2} - 2 (\frac{1}{2}) \cos^{2} x + \frac{1}{4} - \frac{1}{4} + 1 \\ \Rightarrow {(\cos^{2} x - \frac{1}{2})}^{2} + \frac{3}{4} \end{array}$

$\begin{array}{l} 0 \leq \cos^{2} x \leq 1 \\ \Rightarrow - \frac{1}{2} \leq \cos^{2} x - \frac{1}{2} \leq \frac{1}{2} \\ \Rightarrow 0 \leq {(\cos^{2} x - \frac{1}{2})}^{2} \leq \frac{1}{4} \\ \Rightarrow \frac{3}{4} \leq {(\cos^{2} x - \frac{1}{2})}^{2} \leq 1 \\ ∴ R a n g e = [\frac{3}{4}, 1] \end{array}$