The length of the intercept on the line $4x-3y-10=0$ by the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}-20=0$ is

Let the points \(\mathrm{P}_1\left(\frac{\pi}{4}\right), \mathrm{P}_2\left(\frac{3 \pi}{4}\right), \mathrm{P}_3\left(\frac{5 \pi}{4}\right)\) and \(\mathrm{P}_4\left(\frac{7 \pi}{4}\right)\) given in parametric form, lie on the hyperbola \(\frac{x^2}{9}-\frac{y^2}{16}=1\). Then these four points in that order form

\(\frac{x^4}{x^3-3 x+2}\) is a

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) \(-1\)

The minimum value of \(27 \tan ^2 \theta+3 \cot ^2 \theta\) is

Two balls are drawn from an urn containing 7 white, 6 red and 8 black balls one after the other without replacement. Then the probability that atleast one of them is white, is

If the equation \(2 x^2+k x y-6 y^2+3 x+y+1=0\), \((k>0)\) represents a pair of straight lines, then their point of intersection is

In Young's double slit experiment, the slits separated by \(0.6 \mathrm{~mm}\) are illuminated with light of \(6600 Å\). Interference pattern is obtained on a screen placed at \(4 \mathrm{~m}\) from slits. The minimum distance from the central maximum at which the average intensity is \(50 \%\) of the maximum value is

A circular disc of diameter 0.8 m and mass 4 kg is rolling on a smooth horizontal plane. If 2.56 N m torque is acting on the disc, then its angular acceleration is

\(\int\left[\frac{1}{\log x}-\frac{1}{(\log x)^2}\right] d x=\)

Two concentric loops \(A\) and \(B\) of same radius \(2 \pi \mathrm{~cm}\) are placed at right angles to each other. If the currents flowing through A and B are 3 A and 4 A respectively, then the net magnetic field at their common centre is

From a collection of eight cards numbered 1 to 8 , if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is

If the function \(f(x)\), defined below, is continuous everywhere, then \(k\) equals \(f(x)=\left\{\begin{array}{cc}\frac{2^x-1}{\sqrt{1+x}-1}, & -1 \leq x < \infty \\ k, & x=0\end{array}\right.\)