\(A\left(x_1, y_1\right)\) is the internal centre of similitude and \(\mathrm{B}\left(x_2, y_2\right)\) is the external centre of similitude of two circles \(C_1\) and \(C_2\) whose centres are \(\mathrm{P}(\alpha, \beta)\) and \(\mathrm{Q}(\gamma, \delta)\) respectively. If \(\mathrm{PA}=3, \mathrm{AB}=5, \mathrm{QB}=2\), then ratio of the radii of the two circles is

If \(x=\frac{9 t^2}{1+t^4}\) and \(y=\frac{16 t^2}{1-t^4}\) then \(\frac{d y}{d x}=\)

Consider the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-6\hat{k}, \overrightarrow{b}=6\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{c}=3\hat{i}-6\hat{j}-2\hat{k}$ Assertion (A): The three vectors do not form a triangle Reason (R): The three vectors are non-coplanar The correct option among the following is

The values of \(p\) and \(q\) so that the function \[ f(x)=\left\{\begin{array}{cl} (1+\mid \sin x)^{\frac{p}{\sin x}}, \frac{-\pi}{6} < x < 0 \ q & , x=0 \ e^{\frac{\sin 2 x}{\sin 3 x}} & , 0 < x < \frac{\pi}{6} \end{array}\right. \] is continuous at \(x=0\), are

The line \(2 x+y-3=0\) divides the line segment joining the points \(A(1,2)\) and \(B(-2,1)\) in the ratio \(a: b\) at the point \(C\). If the point \(C\) divides the line segment joining the points \(P\left(\frac{b}{3 a},-3\right)\) and \(Q\left(-3,-\frac{b}{3 a}\right)\) in the ratio \(p: q\), then \(\frac{p}{q}+\frac{q}{p}=\)

If $x=2+2^{\frac{2}{3}}+2^{\frac{1}{3}},$ then $x^3-6x^2+6x=$

If $x=\frac{4}{5}+\frac{3}{5}i, y=\frac{\sqrt{3}}{\sqrt{8}}-\frac{\sqrt{5}}{\sqrt{8}}i,$ then $\left(x^2+\frac{1}{x^2}\right)\left(y^2-\frac{1}{y^2}\right)=$

\(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\) and \(B=\left[\begin{array}{ll}x & y \\ 1 & 2\end{array}\right]\) are two matrices such that \((A+B)(A-B)=A^2-B^2\). If \(C=\left[\begin{array}{ll}x & 2 \\ 1 & y\end{array}\right]\) then \(\operatorname{Trace}(C)=\)

If \(z=\cos \alpha+i \sin \alpha ; 0 < \alpha < \frac{\pi}{4}\), then \(\left|\frac{1+z^4}{1-z^3}\right|=\)

The incomplete Lewis representation of \(\mathrm{CO}_3^{2-}\) is given below. The formal charge on atoms marked as a, b, and c, respectively, are

Consider a uniform horizontal solid cylinder of mass \(10 \mathrm{~kg}\) such that its length is 9 times its radius. Let the radius be \(40 \mathrm{~cm}\). Calculate the moment of inertia of the cylinder about a line passing through its edge and perpendicular to its axis.

The standard reduction potentials of \(\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Ag}\) are respectively \(-0.76,0.34\) and \(0.8 \mathrm{~V}\). The following cells were constructed : (1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}\) (2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}\) (3) \(\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}\) What is the correct order of \(E_{\text {cell }}^{\circ}\) of these cells?

Which of the following elements has the highest work function value?