2sin(\(\frac{\pi}{22}\))sin(\(\frac{3\pi}{22}\))sin(\(\frac{5\pi}{22}\))sin(\(\frac{7\pi}{22}\))sin(\(\frac{9\pi}{22}\)) is equal to
\(\frac{3}{16}\)
\(\frac{1}{16}\)
\(\frac{1}{32}\)
\(\frac{9}{32}\)
To solve the problem, we need to evaluate the expression:
\(2 \cdot \sin\left(\frac{\pi}{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)\).
This expression is a classic problem involving the product of sine terms that can be solved using trigonometric identities and symmetry properties of the sine function.
We can use a trigonometric identity known as the multiple angle identity for sine, which states:
\(\prod_{k=1}^{n} \sin\left(\frac{k\pi}{2n+1}\right) = \frac{1}{2^n}\), for \(n = 5\).
Applying this identity to our problem, the product of sine terms:
\(\sin\left(\frac{\pi}{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)\)
results in:
\(\frac{1}{2^5}\) which simplifies to \(\frac{1}{32}\).
Since the expression is multiplied by 2, we have:
\(2 \times \frac{1}{32} = \frac{1}{16}\).
Hence, the evaluated expression is \(\frac{1}{16}\).
Therefore, the correct answer is \(\frac{1}{16}\), which matches option B.
\[ 2 \sin \left( \frac{\pi}{22} \right) \sin \left( \frac{3\pi}{22} \right) \sin \left( \frac{5\pi}{22} \right) \sin \left( \frac{7\pi}{22} \right) \sin \left( \frac{9\pi}{22} \right) \]
Using the symmetry of angles: \[ = 2 \sin \left( \frac{11\pi - 10\pi}{22} \right) \sin \left( \frac{11\pi - 8\pi}{22} \right) \sin \left( \frac{11\pi - 6\pi}{22} \right) \sin \left( \frac{11\pi - 4\pi}{22} \right) \sin \left( \frac{11\pi - 2\pi}{22} \right) \]
Simplifying: \[ = 2 \cdot \frac{\cos \pi}{11} \cdot \frac{2\cos \pi}{11} \cdot \frac{\cos 3\pi}{11} \cdot \frac{\cos 4\pi}{11} \cdot \frac{\cos 5\pi}{11} \]
\[ = \frac{2 \sin \left( \frac{32\pi}{11} \right)}{2^5 \sin \left( \frac{\pi}{11} \right)} \]
Finally: \[ = \frac{1}{16} \]
If \[ \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx \] is equal to \[ -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C, \] where \( p_i, q_i \) are positive integers with \( \gcd(p_i,q_i)=1 \) for \( i=1,2,3,4 \), then the value of \[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} \] is ___________.
Two circular discs of radius \(10\) cm each are joined at their centres by a rod, as shown in the figure. The length of the rod is \(30\) cm and its mass is \(600\) g. The mass of each disc is also \(600\) g. If the applied torque between the two discs is \(43\times10^{-7}\) dyne·cm, then the angular acceleration of the system about the given axis \(AB\) is ________ rad s\(^{-2}\).

Match the LIST-I with LIST-II for an isothermal process of an ideal gas system. 
Choose the correct answer from the options given below:
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\(F(\frac{dy}{dt},y,t) = 0\)
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.

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\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
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