If the circle \(x^2+y^2-6 x-12 y+1=0\) cuts another circle \(\mathrm{C}\) orthogonally and the centre of the circle \(\mathrm{C}\) is \((-4,2)\), then its radius is

Consider the following statements: Assertion (A): The direction ratios of a line \(\mathrm{L}_1\) are \(2,5,7\) and the direction ratios of another line \(\mathrm{L}_2\) are \(\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}\). Then the lines \(\mathrm{L}_1, \mathrm{~L}_2\) are parallel Reason (R): If the direction ratios of a line \(L_1\) are \(a_1, b_1, c_1\), the direction ratios of a line \(\mathrm{L}_2\) are \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2\) and \(\mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2\) \(+c_1 c_2=0\), then the lines of \(L_1, L_2\) are parallel Which one of the following is True?

The angle of \(\triangle A B C\) are in an arithmetic progression. If the larger sides \(a, b\) satisfy the relation \(\frac{\sqrt{3}}{2} < \frac{b}{a} < 1\), then the possible values of the smallest side are

If ${}^{22}{P}_{r+1}: {}^{20}{P}_{r+2}=11: 52$ then $r=$

$5$ persons entered a lift cabin on the ground floor of a $7$ floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. The probability of all the $5$ persons leaving the cabin at different floors, is

If adj \(\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right]=\left[\begin{array}{ccc}5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n\end{array}\right]\), then \(m+n=\)

Consider the simultaneous linear equations \(\mathrm{AX}=\mathrm{B}\) and \(\mathrm{AY}=\mathrm{Q}\). If \(\mathrm{A}\) is an invertible matrix and \(\mathrm{B}\) is the unique solution of \(\mathrm{AY}=\mathrm{Q}\), then the solution of \(\mathrm{AX}=\mathrm{B}\) is

The incorrect statement about the oxidation states of group 14 elements is

Statement (I) : By increasing the potential difference between cathode and anode continuously in a photoelectric experiment, the photo current always increases continuously. Statement (II) : If two photons A and B of energies 2.5 \(\mathrm{eV}\) and \(3.5 \mathrm{eV}\) respectively fall on a metal surface of work function \(2.0 \mathrm{eV}\), then the ratio of maximum kinetic energies emitted between A and B is 3 . Statement (III) : The maximum energy needed by an electron to come out from metal surface is called the work function of the metal. Which of the following is correct?

The length of the common chord of the two circles \(x^2+y^2-4 x-8 y+4=0\) and \(x^2+y^2-8 x-12 y+16=0\) is

If \(\quad \cos 2 x=(\sqrt{2}+1)\left(\cos x-\frac{1}{\sqrt{2}}\right), \cos x \neq \frac{1}{2}\), then \(x \in\)

A balance is made using a uniform metre scale of mass 100 g and two plates each of mass 200 g fixed at the two ends of the scale and the balance is pivoted at 45 cm mark of the scale. The error when 300 g weight is placed in the plate at 0 cm to weigh vegetables placed in the plate at 100 cm is

If the radius of the earth becomes \(x\) times its present value, the new period of rotation in hours is