I n = ∫ π 2 0 cos n x cos ( n x ) d x , n ϵ N then √ I 2001 : I 2002 can be the eccentricity of Options: A . &nbspParabola B . &nbspEllipse C . &nbspCircle D . &nbspHyperbola

Answer: Option D : D I n + 1 = ∫ π 2 0 cos n + 1 x cos ( n + 1 ) x d x = ∫ π 2 0 cos n + 1 x ( cos n x cos x − sin n x sin x ) d x ∴ I n + 1 = I n − I n + 1 ⇒ 2 I n + 1 = I n ∴ I n : I n + 1 = 2

In=∫π20cosnxcos(nx)dx,nϵN then √I2001:I2002 can b

Question

I_{n} = \int_{0}^{\frac{π}{2}} {cos}^{n} x cos (n x) d x, n ϵ N

then

\sqrt{I_{2001} : I_{2002}}

can be the eccentricity of

Options:

A . Parabola

B . Ellipse

C . Circle

D . Hyperbola

Answer: Option D
:
D

I_{n + 1} = \int_{0}^{\frac{π}{2}} {cos}^{n + 1} x cos (n + 1) x d x = \int_{0}^{\frac{π}{2}} {cos}^{n + 1} x (cos n x cos x - sin n x sin x) d x ∴ I_{n + 1} = I_{n} - I_{n + 1} \Rightarrow 2 I_{n + 1} = I_{n} ∴ I_{n} : I_{n + 1} = 2

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