\(\lim _{x \rightarrow \pi} \frac{1-\sin x / 2}{\left(\cos \frac{x}{2}\right)\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)}=\)

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

If the angle between the circles \(x^2+y^2+2 x-4 y+1=0\) and \(x^2+y^2-4 x-2 y+c=0\) is \(\frac{\pi}{4}\), then \(c=\)

If \(A B C D E F\) is a regular hexagon, where two adjacent sides \(\mathbf{A B}\) and \(\mathbf{B C}\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Then \(\mathbf{C D}\) is

The differential equation representing the family of circles having their centres on Y-axis is \(\left(y_1=\frac{d y}{d x}\right.\) and \(\left.y_2=\frac{d^2 y}{d x^2}\right)\)

If \(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) are coplanar, then \(\lambda\) is equal to :

If the slope of one of the pair of lines represented by \(2 x^2+3 x y+K y^2=0\) is 2 , then the angle between the pair of lines is

\(\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ}=\)

Which of the following reaction is not an example of hydrolysis reaction?

In hydrogen spectrum, the shortest wavelengths of Lyman and Balmer series are \(\lambda_1\) and \(\lambda_2\) respectively. The Rydberg constant of hydrogen is

If \(a_0, a_1, \ldots, a_{11}\) are in an arithmetic progression with common difference \(d\), then their mean deviation from their arithmetic mean is

The capacitance of a spherical capacitor is 100 pF. If the spacing between the two spheres is 1 cm, then the radius of the inner sphere of the capacitor is

The displacement of a damped harmonic oscillator is given by \(x(\mathrm{t})=\mathrm{e}^{-0.1 \mathrm{t}} \cos (10 \pi \mathrm{t}+\varphi)\). Here \(\mathrm{t}\) is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to: