AP EAMCET · Maths · Trigonometric Ratios & Identities
Match the ranges of the functions given in List - I with those of the items given in List - II
| List - I | List - II |
| (I) \(3 \sin ^2 x+4 \cos ^2 x-2\) | (a) \(\left[\frac{1}{4}, 1\right]\) |
| (II) \(\cos ^2 x+\sin ^4 x\) | (b) \(\left[-\frac{1}{4}, \frac{1}{4}\right]\) |
| (III) \(\sin ^6 x+\cos ^6 x\) | (c) \([1,2]\) |
| (IV) \(\cos x \cos \left(\frac{2 \pi}{3}+x\right) \cos \left(\frac{2 \pi}{3}-x\right)\) | (d) \(\left[\frac{3}{4}, 1\right]\) |
| (e) \([0,1]\) |
- A (1) \(\rightarrow\) (c) (II) \(\rightarrow\) (a) (III) \(\rightarrow\) (d) (IV) \(\rightarrow\) (b)
- B (1) \(\rightarrow\) (c) (II) \(\rightarrow\) (d) (III) \(\rightarrow\) (a) (IV) \(\rightarrow\) (b)
- C (1) \(\rightarrow\) (b) (II) \(\rightarrow\) (d) (III) \(\rightarrow\) (a) (IV) \(\rightarrow\) (e)
- D (1) \(\rightarrow\) (b) (II) \(\rightarrow\) (e) (III) \(\rightarrow\) (d) (IV) \(\rightarrow\) (c)
Answer & Solution
Correct Answer
(B) (1) \(\rightarrow\) (c) (II) \(\rightarrow\) (d) (III) \(\rightarrow\) (a) (IV) \(\rightarrow\) (b)
Step-by-step Solution
Detailed explanation
\(\text {(I)} \begin{aligned} 3 \sin ^2 x+4 \cos ^2 x-2 & =3 \sin ^2 x+3 \cos ^2 x+\cos ^2 x-2 \\ & =3+\cos ^2 x-2=\cos ^2 x+1 \end{aligned}\) Also, \(0 \leq \cos ^2 x \leq 1 \Rightarrow 1 \leq \cos ^2 x+1 \leq 2\) \(\Rightarrow 1 \leq 3 \sin ^2 x+4 \cos ^2 x-2 \leq 2\)…
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