If y(θ) = 2cosθ+cos2θ / cos3θ+4cos2θ+5cosθ+2, then at θ = π/2, y′′ + y′ + y is equal to

Question

If \(y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}\), then at \(\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+\mathrm{y}\) is equal to:

(1) \(\frac{3}{2}\)

(2) 1

(3) \(\frac{1}{2}\)

(4) 2

Answer 1

Correct option is (4) 2

\(y=\frac{2 \cos \theta+2 \cos ^{2} \theta-1}{4 \cos ^{3} \theta-3 \cos \theta+8 \cos ^{2} \theta-4+5 \cos \theta+2}\)

\(y=\frac{\left(2 \cos ^{2} \theta+2 \cos \theta-1\right)}{\left(2 \cos ^{2} \theta+2 \cos \theta-1\right)(2 \cos \theta+2)}\)

\(y=\frac{1}{2}\left(\frac{1}{1+\cos \theta}\right)\)

\(\Rightarrow \theta=\frac{\pi}{2}, y=\frac{1}{2}\)

\(y^{\prime}=\frac{1}{2}\left(\frac{-1}{(1+\cos \theta)^{2}} \times(-\sin \theta)\right)\)

\(\Rightarrow \theta=\frac{\pi}{2} , y=\frac{1}{2}\)

\(y^{\prime \prime}=\frac{1}{2}\left[\frac{\cos \theta(1+\cos \theta)^{2}-\sin \theta(2)(1+\cos \theta)(-\sin \theta)}{(1+\cos \theta)^{4}}\right]\)

\(\Rightarrow \theta=\frac{\pi}{2}, y = 1\)

\(\Rightarrow y''+y'+y = 1 + \frac12+\frac12 = 4\)