The line \(y=m x+c\) intercepts the circle \(x^2+y^2=r^2\) in two distinct points, if

In \(\triangle A B C\), with usual notation, observe the two statements given below :
(I) \(r r_1 r_2 r_3=\Delta^2\)
(II) \(r_1 r_2+r_2 r_3+r_3 r_1=s^2\)
Which of the following is correct?

Let \(A\) be a \(4 \times 4\) matrix and \(P\) be its adjoint matrix. If \(|P|=\left|\frac{A}{2}\right|\), then \(\left|A^{-1}\right|=\)

The value of \(\sqrt{42+\sqrt{42+\sqrt{42+\ldots \ldots .}}}\) is equal

Match the following
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=p x+q \\
(p \neq 0), \forall x \in R
\end{array} & \begin{array}{l}
\text {I. } f \text { is neither one-one nor onto }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
f: R \rightarrow R^{+} \cup\{0\} \text { is such that } f(x)=x^2, \forall x \in R
\end{array} & \begin{array}{l}
\text {II. } f \text { is both one-one and onto }
\end{array} \\
\hline \text { (C) } & \begin{array}{l}
f: N \rightarrow N \text { is such that } f(n)=n^2+2 n+3, \forall n \in N
\end{array} & \begin{array}{l}
\text {III. } f \text { is one-one but not onto }
\end{array} \\
\hline \text { (D) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=2\left(\cos ^2 5 x+\sin ^2 5 x\right) \\
\forall x \in R
\end{array} & \begin{array}{l}
\text {IV. } f \text { is onto but not one-one }
\end{array} \\
\hline && V. f \text{ is a constant function and also a bijection} \\
\hline
\end{array}\)
The correct answer is

The equation of pair of straight lines parallel to \(X\)-axis and touching the circle \(x^2+y^2-6 x-4 y-12=0\)

If \(f(x)=\left|\begin{array}{ccc}1 & 6+x & 36+x^2 \\ 0 & x-3 & 3 x^2-27 \\ 0 & 2 x-4 & 8 x^2-32\end{array}\right|\), then \(\lim _{x \rightarrow 1} \frac{f(x)}{f(-x)}=\)

The value of the series \(\cos 12^{\circ}+\cos 84^{\circ}\) \(+\cos 132^{\circ}+\cos 156^{\circ}\) is

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

The magnitudes of two vectors are A and \(\mathrm{B}(\mathrm{A}>\mathrm{B})\). If the maximum resultant magnitude of the two vectors is ' \(n\) ' times their minimum resultant magnitude, then \(\frac{\mathrm{A}}{\mathrm{B}}=\)

A body of mass 2 kg is moving towards north with a velocity of \(20 \mathrm{~m} \mathrm{~s}^{-1}\) and another body of mass 3 kg is moving towards east with a velocity of \(10 \mathrm{~m} \mathrm{~s}^{-1}\). The magnitude of the velocity of the centre of mass of the system of the two bodies is

The centre of mass of a homogenous semi-circular plate of radius \(\mathrm{r}\) is located at \(\mathrm{A}\) as shown in the figure. The distance OA is

The space between the plates of a parallel plate capacitor is halved and a dielectric medium of relative permittivity 10 is introduced between the plates. The ratio of the final and initial capacitances of the capacitor is