The normal to a circle \(S=0\) at \(P(1,3)\) is \(x+2 y=7\) and it has another normal at \(Q(3,5)\) which is the polar of the point \(A\left(7,-\frac{1}{2}\right)\) with respect to the circle \(x^2+y^2-4 x+6 y-12=0\). Then, the equation of the circle \(S=0\) is

If 2 cards drawn at random from a well shuffled pack of 52 playing cards are from the same suit, then the probability of getting a face card and a card having a prime number is

If \(\frac{x^2+a x+3}{x^2+x+1}\) takes real values for all real values of \(x\) then \(a\) lies in the interval

If the line \(x+y=2\) cuts the circle \(x^2+y^2+2 x-4 y+4=0\) at two points A and B then the radius of the circle passing through \(\mathrm{A}, \mathrm{B}\) and orthogonal to \(x^2+y^2-2 x-4 y-4=0\) is

\(n\) is a positive integer and not a multiple of 3 . If \(\omega\) is non real cube root of unity, then \(\omega^n+\omega^{2 n}\) is equal to

The line \(2 x+y=1\) is a tangent to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>b)\). If this line passes through the point of intersection of a directrix and the positive \(X\)-axis, then the eccentricity of that hyperbola is

If $A=\left(\begin{array}{lll}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right),$ then $adj(adjA)$ is equal to

A person fails 4 times in a game when he plays 9 times. If he plays 15 times, the probability of having success at most one is

\(f(x)=e^x \sin x\), then \(f^{(6)}(x)\) is equal to :

Match the following methods for treating hardness of water with their corresponding reagents used. \begin{array}{|c|c|c|} \hline List-I & & List II \ \hline A. Treatment with washing soda & \mathrm{I}. & \mathrm{Ca}(\mathrm{OH})_2 \ \hline B. Calgon's method & II. & \mathrm{Na}_2 \mathrm{CO}_3 \ \hline C. Clark's method & III & \mathrm{NaAlSiO}_4 \ \hline D. Zeolite/permutit process & IV & \mathrm{Na}_6 \mathrm{P}_6 \mathrm{O}_{18} \ \hline \end{array} The correct match is \(\begin{array}{llllllll}\text { A } & \text { B } & \text { C } & \text { D } & \text { A } & \text { B } & \text { C } & \text { D }\end{array}\)

According to significant figure convention, the result obtained by adding \(12.11,18.0\) and 1.012 is

\[ \int_0^2 x^3(2-x)^4 d x= \]

The number of points at which the function \(f(x)=\frac{\sqrt{11+|x|-6 \sqrt{2+|x|}}}{6-2 \sqrt{2+|x|}}\) is discontinuous in \((-\infty, \infty)\) is