1.

If A = 2 sin2 x – cos 2x, then A lies in the intervalA. [-1, 3]B. [1, 2]C. [-2, 4]D. None of these

Answer»

Correct answer is A.

Given A = 2 sin2 x – cos 2x

[using cos 2x = 1 – 2 sin2 x]

so A = 2 sin2 x – cos 2x = 2 sin2 x –[ 1 – 2 sin2 x]

= 2 sin2 x -1 + 2 sin2 x]

= 4 sin2 x – 1

Now A = 2 sin2 x – cos 2x = 4 sin2 x – 1

As we know sin x lies between -1 and 1

-1 ≤ sin x ≤ 1

0 ≤ sin2x ≤ 1

Multiplying the inequality by 4

0 ≤ 4 sin2 x ≤ 4

Subtracting 1 from the inequality

-1 ≤ (4 sin2 x – 1) ≤ 3

From the above inequation, we can say that

A = (4 sin2 x – 1) belongs to the closed interval [-1, 3]



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