NCERT Exemplar - MCQs

Chapter 6 Class 12 Application of Derivatives
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## (C) cos x                            (D) cos 3x

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Question 8 Which of the following functions is decreasing on (0,π/2) (A) sin 2x (B) tan x (C) cos x (D) cos 3x To check decreasing, we check if π^β² (π)<π in (0,π/2) Option A π(π₯)=π ππ 2π₯ Differentiating w.r.t. π π^β² (π)=2 πππ  2π₯ Let 2π = ΞΈ β΄ fβ(π₯) = 2 cos ΞΈ When 0 < x < π/2 , then 0 < ΞΈ < π Now, So, sinβ‘2π₯ is neither increasing nor decreasing in the interval (0,π/2). Option B π(π₯)=π‘ππ π₯ Differentiating w.r.t. π fβ(π) = sec2 π₯ For 0 < ΞΈ < π/π cos ΞΈ > 0 Putting π=2π₯ cosβ‘2π₯>0 2 cosβ‘2π₯>0 β΄ π^β² (π)>π For π/π < ΞΈ < π cos ΞΈ < 0 Putting π=2π₯ cosβ‘2π₯<0 2 cosβ‘2π₯<0 β΄ π^β² (π)<π As square of any number is always positive So, fβ(π₯) > 0 for all values of π₯ β΄ f is strictly increasing on (0 , π/2). Option C π(π₯)=πππ  π₯ Differentiating w.r.t. π π^β² (π)=βπ ππ π₯ Since, sin π > 0 for π₯ β (0 , π/2) So, β sin π < 0 for π₯ β (0 , π/2) β΄ fβ (π₯) < 0 for π₯ β (0 , π/2) So, f is strictly decreasing in (0 , π/2). Option D π(π₯)=πππ  3π₯ Differentiating w.r.t. π fβ (π) = β3 sin 3π₯ Let 3π = ΞΈ β΄ fβ (π₯) = β3 sin ΞΈ When 0 < x < π/2 , then 0 < ΞΈ < ππ/π For 0 < ΞΈ < π sin ΞΈ > 0 Putting π=3π₯ sinβ‘3π₯>0 β3 sinβ‘3π₯<0 β΄ π^β² (π)<π For π < ΞΈ < ππ/π sin ΞΈ < 0 Putting π=3π₯ sinβ‘3π₯<0 β3 sinβ‘3π₯>0 β΄ π^β² (π)>π So, cos 3π₯ is neither increasing nor decreasing in the interval (0,π/2). Hence, only πππ π is decreasing in the interval (0,π/2). So, the correct answer is (C).

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.